Fonctionnelles de processus de Lévy et diffusions en milieux aléatoires

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1 Fonctionnelles de processus de Lévy et diffusions en milieux aléatoires Gregoire Vechambre To cite this version: Gregoire Vechambre. Fonctionnelles de processus de Lévy et diffusions en milieux aléatoires. Mathématiques générales [math.gm]. Université d Orléans, 216. Français. <NNT : 216ORLE238>. <tel > HAL Id: tel Submitted on 31 May 217 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 UNIVERSITÉ D ORLÉANS École doctorale Mathématiques, Informatiques, Physique théorique et Ingéniérie des systèmes Laboratoire : MAPMO THÈSE présentée par : Grégoire VÉCHAMBRE soutenue le : 3 Novembre 216 pour obtenir le grade de : Docteur de l université d Orléans Discipline/ Spécialité : Mathématiques FONCTIONNELLES DE PROCESSUS DE LÉVY ET DIFFUSIONS EN MILIEUX ALÉATOIRES THÈSE dirigée par : Pierre ANDREOLETTI MCF, Université d Orléans - Directeur de thèse Rapporteurs : Jean BERTOIN Professeur, Université de Zurich Zhan SHI Professeur, Université Pierre et Marie Curie Jury : Romain Abraham Pierre Andreoletti Jean Bertoin Thomas Duquesne Zhan Shi Arvind Singh Professeur, Université d Orléans MCF, Université d Orléans Professeur, Université de Zurich Professeur, Université Pierre et Marie Curie Professeur, Université Pierre et Marie Curie Chargé de recherches, Université de Paris Sud

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4 THÈSE DE DOCTORAT DE L UNIVERSITÉ D ORLÉANS Spécialité Mathématiques Présentée par Grégoire VÉCHAMBRE Pour obtenir le grade de DOCTEUR de l UNIVERSITÉ D ORLÉANS Sujet de la thèse : Fonctionnelles de processus de Lévy et diffusions en milieux aléatoires i

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6 Remerciements Ce manuscrit est l aboutissement de trois années de travail, et je tiens à exprimer ma gratitude envers tous ceux qui ont permis à ma thèse de se dérouler dans de bonnes conditions jusque à son terme. Je tiens en premier lieu à exprimer ma reconnaissance à mon directeur de thèse, Pierre Andreoletti, qui m a amené, au début de ma thèse, sur le sujet du temps local d une diffusion en milieu aléatoire et qui m a ensuite laissé la liberté de bifurquer sur l étude des fonctionnelles exponentielles. Ce fut très agréable de travailler avec lui et je le remercie pour sa disponibilité, ses conseils, et son implication dans ma thèse. Je remercie vivement Jean Bertoin et Zhan Shi d avoir accepté de rapporter cette thèse. Je suis également très reconnaissant à Romain Abraham, Thomas Duquesne et Arvind Singh qui me font l honneur de faire partie du jury de ma soutenance. Je remercie également Alexis Devulder qui a collaboré à la rédaction du premier article présenté dans ce travail, notamment pour ses relectures très minutieuses du papier. Je le remercie également de m avoir invité à donner un exposé au séminaire de probabilités-statistiques de Versailles, plus tôt cette année. Ayant passé trois ans à travailler au sein du laboratoire MAPMO dans d excellentes conditions, je tiens à exprimer ma gratitude à tous les membres et personnels du laboratoire pour leur convivialité et pour le bon fonctionnement du laboratoire. Je remercie tous les doctorants et anciens doctorants du laboratoire : Sylvain, Alaa, David, Mathilde, Sébastien, Binh, Amina, Rémi, Manon, Tien, Nhat, Han, Julie, Zhang, et plus particulièrement Lan et Hieu j ai une pensée spéciale pour les soirées passées à regarder des films japonais. Je tiens enfin à remercier mes parents, mon frère Benjamin, ma soeur Manon, et aussi mon grand-père qui m a donné très tôt le goût des sciences et l ambition de suivre son exemple. iii

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8 Table des matières 1 Introduction Diffusion en milieu aléatoire Définition et propriétés de la diffusion Temps local et point favori Processus de Lévy et fonctionnelles exponentielles Processus de Lévy Processus de Lévy spectralement négatif Conditionnement à rester positif Fonctionnelles exponentielles Description des résultats obtenus Chapitre 2 : Convergence en loi en milieu brownien drifté Chapitre 3 : Fonctionnelles exponentielles Chapitre 4 : Convergence en loi en milieu Lévy spectralement négatif Chapitre 5 : Comportement presque sûr en milieu Lévy spectralement négatif Quelques perspectives Renewal structure and local time for diffusions in random environment Introduction Presentation of the model Results Notation Path decomposition and Valleys Path decomposition in the neighborhood of the h t -minima m i Definition of h t -valleys and of standard h t -minima m j, j N Contributions for hitting and local times Negligible parts for hitting times Negligible parts for local times Supremum of the local time outside the valleys Local time inside the valley [ L j, L ] j but far from mj Approximation of the main contributions v

9 Table des matières 2.4 Convergence toward the Lévy process Y 1,Y 2 and continuity Preliminaries Proof of Proposition Continuity of some functionals of Y 1,Y 2 in J 1 topology Supremum of the Local time - and other functionals Supremum of the local time proof of Theorem Favorite site proof of Theorem Results and additional arguments from the paper [3] Some estimates on the diffusion X Some estimates on the potential W κ and its functionals Appendix Some estimates for Brownian motion, Bessel processes, Wκ and their functionals Exponential functionals of spectrally one-sided Lévy processes conditioned to stay positive Introduction Results The example of drifted brownian motion conditioned to stay positive Preliminary results on V and finiteness of IV Exponential functionals and excursions theory V and V shifted at a last passage time Finiteness, exponential moments, and self-decomposability Finiteness and exponential moments : Proof of Theorem Decomposition of the law of IV Asymptotic tail at : Proof of Theorems 3.1.2, and Laplace transform of IV Tail at of IV : proof of Theorems and Connection between IV and IV : proof of Proposition Smoothness of the density : Proof of Theorem The spectrally positive case Finiteness, exponential moments : Proof of Theorem Tails at of IZ : Proof of Theorem Path decomposition of a spectrally negative Lévy process, and local time of a diffusion in this environment Introduction Main results Facts and notations Supremum of the local time when κ > The local time at hitting times Proof of Theorems and vi

10 Table des matières 4.3 Path decomposition of a spectrally negative Lévy process h-extrema, h-valleys and some processes conditioned to stay positive Law of the valleys Standard valleys Exponential functionals of the bottom of a standard valley Asymptotic of the h-minima sequence Supremum of the local time when < κ < Proof of Theorem Proof of Proposition and consequences Proof of Proposition Proof of Proposition and consequences Some estimates on V, V, ˆV and the diffusion in V Estimates on V Estimates on V Estimates on ˆV Estimates on the first ascend of h from the minimum Estimates on the valleys Estimates on the diffusion in potential V Proof of some facts Almost sure behavior for the local time of a diffusion in a spectrally negative Lévy environment Introduction Main results Sketch of proofs and organisation of the paper Facts and notations Almost sure behavior when < κ < Traps for the diffusion Decomposition of the diffusion into independent parts The limsup The liminf Almost sure behavior when κ > The liminf The limsup Some lemmas Properties of V, V and ˆV Contribution of the valleys to the traveled distance Proof of some facts and lemmas Almost sure constantness of limsup and liminf Bibliographie 275 vii

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12 Chapitre 1 Introduction Dans cette thèse nous nous sommes intéressés à trois types de problèmes : Étude de la convergence en loi du supremum du temps local d une diffusion transiente en milieu brownien ou Lévy spectralement négatif. Étude du comportement presque sûr du supremum du temps local d une telle diffusion. Étude des fonctionnelles exponentielles de certains processus de Lévy conditionnés à rester positif. Bien que le troisième problème ait été traité séparément et semble peu lié au deux autres, il joue en réalité un rôle crucial pour l étude de ces derniers. Nous montrerons notamment qu il existe un lien très explicite entre le comportement presque sûr du supremum du temps local que nous étudions et les propriétés que nous établissons sur les fonctionnelles exponentielles des processus de Lévy conditionnés à rester positif. Chapitre 2 : Nous étudions la convergence en loi du supremum du temps local et du point favori d une diffusion transiente dans environnement brownien κ-drifté W κ où < κ < 1. Les lois limites obtenues sont exprimées comme des fonctionnelles d un subordinateur κ-stable bidimensionnel. Notre étude consiste en particulier à mettre en évidence une structure de renouvellement pour la diffusion. Chapitre 3 : Nous étudions les propriétés des fonctionnelles exponentielles + e X t dt oùx est un processus de Lévy spectralement négatif ou positif conditionné à rester positif. Nous avons en particulier étudié leur finitude, leur auto-décomposabilité, l existence de moments exponentiels, leur queue en, l existence et la régularité de leur densité. Chapitre 4 : Nous étudions la convergence en loi du supremum du temps local et du point favori d une diffusion transiente en environnement Lévy spectralement négatif. Il s agit en grande partie d une généralisation des résultats du Chapitre 2 à des environnements plus généraux. Ce travail passe en particulier par l étude des h-vallées de l environnement et de la répartition asymptotique des h-minima. 1

13 1.1. DIFFUSION EN MILIEU ALÉATOIRE Chapitre 5 : Nous étudions le comportement presque sûr du supremum du temps local d une diffusion transiente en environnement Lévy spectralement négatif. Nous montrons en particulier que ce comportement est lié à la queue en de la fonctionnelle exponentielle de l environnement conditionné à rester positif. La suite de cette introduction présente les différents objets étudiés ainsi que les principaux résultats obtenus. 1.1 Diffusion en milieu aléatoire La notion de marche ou de diffusion dans un environnement aléatoire généralise la notion de processus Markovien dans le sens où une telle marche ou diffusion est un mélange de processus Markoviens. Ces marches ou diffusions modélisent le problème d un déplacement aléatoire dans un milieu qui est lui-même aléatoire et qui présente, selon le modèle choisi, une certaine hétérogénéité qui influe grandement sur le comportement global du processus. Dans le cas discret cela revient à considérer une marche aléatoire au plus proche voisin dont les probabilités de transitions sont elles-même choisies aléatoirement. De telles marches s appellent marches aléatoires en milieu aléatoire abrégé en RWRE pour le terme anglais Random Walk in Random environnement et ont été introduites en 1967 par Chernov [23] pour modéliser la réplication de l A.D.N. Plus récemment, Lubensky et Nelson [49] font aussi usage de ces marches pour modéliser d autres phénomènes en génétique. Notons également Temkin [71] pour des applications de ces marches à la métallurgie Définition et propriétés de la diffusion Dans le cas continu auquel nous nous sommes intéressés, on considère le processus de diffusion X t t qui se déplace dans un potentiel aléatoire Vx, x R, c està-dire la solution de l équation différentielle stochastique suivante : { dxt = 1 2 V X t dt+db t X = où B est un mouvement brownien indépendant de V. Dans ce qui suit nous prenons comme potentiel V un mouvement brownien drifté ou non, ou plus généralement un processus de Lévy indexé par R et nul en par convention. Pour de tels potentiels qui ne sont pas dérivables l équation est purement formelle et la diffusion X doit être définie conditionnellement à V par son générateur : A V = 1 2 evx d dx e Vx d dx. Sous des hypothèses de régularité du potentiel, le calcul stochastique permet d exprimer une diffusion ayant ce générateur comme un mouvement brownien changé en 2

14 1.1. DIFFUSION EN MILIEU ALÉATOIRE temps et en espace, ce qui autorise des calculs explicites sur la diffusion. Plus précisément, soit B un mouvement brownien indépendant de V, définissons S V x := x evu du et pour s τb, + e Vu du, où τb,a est le temps d atteinte de a par B, posons T V s := s e 2VS 1 V Bu du. On montre qu en posant Xt := S 1 V BT 1 V t, on définit une diffusion qui, conditionnellement à V, a pour générateur A V. Pour l étude d une telle diffusion comme pour celle des RWRE il faut tenir compte d une part de l aléa du au milieu et d autre part de celui du au déplacement aléatoire, cela implique de travailler avec plusieurs mesures de probabilité. On note en général P V pour la loi de X conditionnellement à l environnement V. Sous cette loi, appelée communément loi quenched, la diffusion X est un processus Markovien. On note P pour la loi de probabilité dite annealed qui est la moyennisation de la loi quenched sur tous les environnements : P := P V.PdV où PdV désigne ici la loi de l environnement. Le caractère Markovien de X est perdu sous la loi annealed mais cette dernière bénéficie des éventuelles propriétés d invariance que présente la loi de l environnement. L étude d un processus en environnement aléatoire nécessite de bien choisir la mesure de probabilité utilisée, on parle selon le cas de méthodes quenched ou de méthodes annealed. Si V = W est mouvement brownien sans drift dans ce cas la diffusion est appelé processus de Brox la diffusion X est récurrente et converge en loi : Theorem Brox [17], Pour le processus de Brox, logt 2 Xt L m, où m est une variable aléatoire non dégénérée dépendant de l environnement. Notons que le problème analogue pour le cas discret a aussi été étudié par Sinai [63] qui, en 1982, avait obtenu la convergence en loi des RWRE récurrentes avec la même renormalisation. La distribution limite des RWRE récurrentes a été explicitée indépendamment par Kesten [47] et Golosov [42] en 1986, il s agit de la même distribution limite que pour le processus de Brox. La lenteur du processus de Brox et des RWRE récurrentes par rapport respectivement au mouvement brownien et à la marche aléatoire simple classique qui ont une vitesse en t vient de l hétérogénéité des environnements aléatoires. Ceux-ci présentent en effet des puits de potentiel au fond desquels les diffusions ou les RWRE resteront piégées un certain temps avant d en sortir, ce qui va considérablement les ralentir. Les puits de potentiels jouent un rôle extrêmement important pour étudier la localisation des processus en milieu aléatoire et déterminer les points qu ils ont le plus visités. La preuve du Théorème se base d ailleurs sur la localisation de la diffusion au fond d un certain puits de potentiel et la loi limite m correspond à la distribution asymptotique du fond de ce puits. 3

15 1.1. DIFFUSION EN MILIEU ALÉATOIRE Si Vx = W κ x := Wx κ x est mouvement brownien drifté, la diffusion X 2 est transiente et son comportement, qui dépend de la valeur de κ, a été étudié par Kawazu et Tanaka [45] : Theorem Kawazu, Tanaka [45], On suppose que V = W κ : si < κ < 1 alors t κ Xt converge en loi vers une distribution de Mittag- Leffler d indice κ, si κ = 1 alors t/logt 1 Xt converge en probabilité vers 1/4, si κ > 1 alors t 1 Xt converge presque sûrement vers κ 1/4. Ces résultats ont par la suite été précisés par Kawazu et Tanaka [46], Tanaka [7] et Hu, Shi et Yor [44] qui, lorsque κ > 1, exhibent plusieurs comportements possibles et prouvent des théorèmes de type central limite. Notons aussi que, comme pour le Théorème 1.1.1, le Théorème a également été précédé d un résultat analogue pour les RWRE prouvé en 1975 par Kesten, Kozlov et Spitzer [48]. Ils observent, pour les RWRE transientes, les mêmes trois types de régimes possibles en fonction d un paramètre κ, dépendant de l environnement et qui joue un rôle analogue à celui du cas continu. Dans le cas des RWRE transientes à vitesse nulle, leur résultat a été récemment retrouvé et explicité par Enriquez, Sabot et Zindy [35] avec d autres méthodes. Le comportement de la diffusion a également été étudié dans le cas où le potentiel est un processus de Lévy par Carmona [18], Cheliotis [21] et Singh [66], [64], [65]. Dans [66], Singh généralise notamment les résultats de Kawazu et Tanaka au cas où l environnement V est un processus de Lévy spectralement négatif. Notons également que la diffusion en milieu aléatoire a été étudiée en dimension supérieure à 1 par Tanaka [68] et Mathieu [5], [51] Temps local et point favori Pour la diffusion X que nous venons de définir, il existe un phénomène de localisation dans les puits ou vallées du potentiel. Le taux d occupation d un point x de l espace par la diffusion avant le temps t est "mesuré" par L X t,x où L X est le processus temps local de X, c est-à-dire la version continue en temps et càd-làg en espace de la densité de la mesure d occupation de la diffusion : t t, f L, fx s ds = fyl X t,ydy. Cette relation implique en particulier l approximation 1 L X t,x = lim ǫ ǫ t R 1 Xs [x,x+ǫ] ds qui donne du sens à l affirmation selon laquelle L X t,x "mesure" le taux d occupation de x jusqu au temps t, et qui peut être utilisée comme définition alternative du temps local. 4

16 1.1. DIFFUSION EN MILIEU ALÉATOIRE Nous nous sommes intéressés plus particulièrement à L X, le supremum du temps local au temps t : L Xt = supl X t,x. x R Dans le cas où le potentiel V de la diffusion est mouvement brownien sans drift c est-à-dire dans le cas du processus de Brox Shi [61] a étudié le comportement presque sûr de L X et montré que P-p.s. lim sup t + L X t tlogloglogt Plus tard Andreoletti et Diel [5] ont établi la convergence en loi de L X t/t : Theorem Andreoletti, Diel [5], 21. Pour le processus de Brox, L X t t L t e Rx dx, où L désigne la convergence en loi sous la probabilité annealed P et où Rx := R 1 x1 x +R 2 x1 x, R 1 et R 2 étant deux processus de Bessel de dimension 3, issus de et indépendants. Diel [29] a ensuite poursuivi l étude de Shi [61] en obtenant les renormalisations exactes pour les limites supérieures et inférieures ainsi qu un encadrement de ces limites : Theorem Diel [29], 211. Pour le processus de Brox, P-p.s. lim sup t + L X t tlogloglogt e2 2 et j2 64 liminf t + L X t t/logloglogt e2 π 2 4, où j est la plus petite racine strictement positive de la fonction de Bessel J. Dans le cas où le potentiel V de la diffusion est W κ, le mouvement brownien κ-drifté, Devulder [28] a obtenu par des méthodes annealed la convergence en loi et le comportement presque sûr de L X dans les cas κ = 1 et κ > 1 : Theorem Devulder [28], 216. Si V = W κ alors, pour les convergences en loi : si κ = 1, L Xt/t 1/κ si κ > 1, L Xt/t 1/κ L F1,1/2, t + L t + Fκ,4κ2 κ 1/8 1/κ, où, pour α,s >, Fα,s est la distribution de Fréchet de paramètres α et s, c està-dire la loi de fonction de repartition Fα,s[,t] = e s/tα

17 1.1. DIFFUSION EN MILIEU ALÉATOIRE Pour les convergences presque sûres : pour a une fonction positive croissante, si κ > 1 on a + n=1 { 1 < + nan = + limsup t + et P-p.s lim inf t + Si κ = 1, P-p.s liminf t + { L X t tat = P-p.s /κ + L X t t/loglogt = 1/κ 4κ2 κ 1/8 1/κ L X t 1/2. t/logtloglogt Dans le cas où < κ < 1 les méthodes annealed sont beaucoup moins efficaces et Devulder [28] obtient des résultats partiels pour le comportement presque sûr du temps local. Il prouve notamment que la renormalisation de la limsup doit être plus grande que t : P-a.s. lim sup t + L X t t = +, et que la renormalisation de la liminf est au plus t/loglogt et plus grande que t/logt 1/κ loglogt 2/κ+ǫ pour tout ǫ > : P-a.s. lim inf t + ǫ >, P-a.s. liminf t + où Cκ est une constant positive non explicite. L X t Cκ, t/loglogt L X t = +, t/logt 1/κ loglogt2/κ+ǫ Peu de choses sont connus pour des environnements plus généraux. Notons tout de même Diel et Voisin [3] qui généralisent les résultats de [5] et donc en particulier la convergence en loi du Théorème au cas où l environnement V est un processus de Lévy α-stable avec α [1,2]. Dans le cas des RWRE le temps local au point x et à l instant n, noté ξn,x, correspond tout simplement au nombre de visite du point x jusqu à l instant n. Le supremum du temps local à l instant n est noté ξ. Pour les RWRE récurrentes, la convergence en loi de ξ n/n a été établie par Gantert, Peres et Shi [4] qui obtiennent également la limsup : P-p.s. lim supξ n/n = C, n + où C est une constante explicite dépendant de la loi de l environnement. La liminf a ensuite été étudiée par Dembo, Gantert, Peres et Shi [26] qui obtiennent un résultat analogue à celui du Théorème établi par Diel [29] pour le cas continu récurrent : P-p.s. < liminf n + ξ n n/logloglogn < +. 6

18 1.1. DIFFUSION EN MILIEU ALÉATOIRE Remark On constate une différence entre le résultat et celui du cas continu récurrent donné par et par le Théorème Cela provient du fait que le puits de potentiel contenant la diffusion au temps t peut être beaucoup plus abrupte dans le cas continu que dans le cas discret, ce qui autorise de plus grandes valeurs pour le supremum du temps local et induit cette différence de renormalisation. Pour les RWRE transientes, le comportement presque sûr du supremum du temps local, plus précisément la limsup, a été étudié par Gantert et Shi [41]. Ils distinguent deux cas : < κ 1 et κ > 1, où κ dépend de la loi de l environnement et est défini comme dans [48] où est prouvé l analogue du Théorème pour les RWRE transientes. Pour κ > 1 ils obtiennent un résultat analogue à établi par Devulder [28] : soit a une fonction positive croissante, on a + n=1 { { 1 < + nan = + limsup ξ n t + nan = P-p.s /κ + Lorsque < κ 1 leur résultat est similaire à celui du cas récurrent : P-p.s. < limsupξ n/n < n + Pour en revenir au cas de la diffusion continue, un problème très lié à la fois au comportement asymptotique de L X t et à celui de la localisation de la diffusion est de déterminer le comportement du point favori au temps t, c est-à-dire le point F t en lequel est atteint le supremum du temps local au temps t : F t := inf{x R, L X t,x L X t,x = L Xt} La localisation des diffusions se fait généralement autour de minima locaux du potentiel. On appelle h-minimum la position du fond d un puits de potentiel de hauteur au moins h voir le Chapitre 2, un peu avant le Théorème pour une définition rigoureuse des h-minima, et le puits de potentiel associé est appelé h- vallée. La notion de h-minimum du mouvement brownien a été introduite et étudiée par Neveu et Pitman [53]. Pour le processus de Brox, Cheliotis [22] a montré que le point favorif t est proche en probabilité du minimum de lalogt-vallée contenant, qui est également le point près duquel est localisée la diffusion au temps t. Ce résultat est généralisé par Diel et Voisin dans [3] pour la diffusion en milieu Lévy α-stable avec α [1,2]. Notons que le problème du point favori a aussi été étudié pour les RWRE récurrentes par Hu et Shi [43]. Notre travail se situe dans la continuité des résultats présentés dans cette section. Nous étudions des problèmes de convergence en loi du supremum du temps local et du point favori aux Chapitres II et IV, et nous étudions des problèmes de comportement presque sûr au Chapitre 5 où, notamment, nous améliorons les résultats partiels 1.1.9, et dans un contexte plus général. 7

19 1.2. PROCESSUS DE LÉVY ET FONCTIONNELLES EXPONENTIELLES 1.2 Processus de Lévy et fonctionnelles exponentielles Processus de Lévy Un processus de Lévy réel est un processus stochastique valant à l origine, à accroissements indépendants et stationnaires, dont les trajectoires sont presque sûrement càd-làg. Nous nous référons notamment aux livres de Bertoin [8] et Sato [59] pour une vision d ensemble sur ces processus et pour les preuves des propriétés mentionnées dans cette section. Il est connu que la loi d un processus de Lévy réel Yt, t est caractérisée par les lois 1-dimensionnelles de celui-ci. Pour t, la fonction caractéristique de Yt peut être mise sous la forme E [ e iξyt] = e tψ Y ξ, où ψ Y est appelé exposant caractéristique de Y. ψ Y est donné par la formule de Lévy-Khintchine : ψ Y ξ = Q 2 ξ2 +iγξ + e iξx 1 iξx1 x <1 νdx, R oùq,γ R etν est une mesure de Lévy surr, c est-à-dire une mesure supportée par R\{} telle que 1 x 2 νdx < +. Un tel triplet Q,γ,ν est unique et est appelé triplet générateur de Y. Réciproquement, à chaque triplet Q,γ,ν vérifiant les propriétés ci-dessus correspond un unique processus de Lévy réel. Pour x R, Y x désigne en général le processus Y issu de x, qui est égal en loi à x+y. Les exemples les plus simples de processus de Lévy sont le mouvement brownien éventuellement drifté et les processus de Poisson composés. Dans la formule 1.2.1, le premier terme correspond à une composante Gaussienne, le deuxième à une composante de drift et le troisième à une composante de sauts compensée par un drift. Mouvement brownien et processus de Poisson composés sont donc en quelque sorte les ingrédients de base à partir desquels peuvent être construits tous les processus de Lévy. Une telle représentation des processus de Lévy est en effet donnée par la décomposition de Lévy-Ito qui illustre en décomposant Y comme une somme de deux processus indépendants : un mouvement brownien éventuellement drifté et une limite de processus de Poisson composés dont les petits sauts sont éventuellement compensés par un drift. Du fait de ses accroissements indépendants et stationnaires, un processus de Lévy est généralement considéré comme l analogue continu d une marche aléatoire. Si en particulier E[ Y1 ] < + alors Y satisfait à la loi des grand nombres : Yt/t p.s. t + E[Y1]. 8

20 1.2. PROCESSUS DE LÉVY ET FONCTIONNELLES EXPONENTIELLES Plus généralement un processus de Lévy réel non nul possède, comme pour les marches aléatoires réelles, trois types de comportements asymptotiques possibles : lim Yt =, liminf t + t + Yt = et limsup t + Yt = +, lim Yt = +, t + où toutes les limites sont ici des limites presque sûres. Dans le deuxième cas on dit généralement que Y oscille. Un critère général pour déterminer le comportement de Y est donné par la valeur des intégrales I := + 1 t 1 PYt < dt et I + := + 1 t 1 PYt > dt. Si I < + alors Y converge vers, si I + < + alors Y converge vers +, et si I = I + = + alors Y oscille Processus de Lévy spectralement négatif Nous considérons le cas d un processus de Lévy réel V ne possédant presque sûrement aucun saut positif. Si Q, γ, ν désigne son triplet générateur, l absence de sauts positifs se traduit par le fait que ν],+ [ =. Un tel processus V est dit spectralement négatif. L exemple le plus simple de processus de Lévy spectralement négatif est celui du mouvement brownien, drifté ou non. Il est possible de montrer que Vt admet une transformée de Laplace sur le demi-plan complexe positif {λ C, Rλ } et dont la forme est la suivante : t,λ, E [ e λvt] = e tψ V λ. Ψ V est appelé exposant de Laplace de V et est un prolongement de ψ V sur le demiplan complexe positif : ψ V ξ = Ψ V iξ. Par prolongement analytique, l expression de Ψ V se déduit de : Ψ V λ = Q 2 λ2 +γλ+ e λx 1 λx1 x <1 νdx Comme cela est habituellement le cas dans l étude des processus de Lévy spectralement négatifs, nous excluons le cas où V est monotone. La restriction de Ψ V à la demi-droite des nombre réels positifs a alors la propriété de tendre vers l infini en +, de plus elle est convexe et nulle en. Il est donc possible de définir pour tout q positif : Φ V q := sup{λ, Ψ V λ = q}. On montre voir [8] que V tend vers, oscille ou tend vers + selon que Ψ V + <, Ψ V + = ou Ψ V + >. Dans le cas où V tend vers, le supremum du processus V sur [,+ [ suit une loi exponentielle de paramètre Φ V. Pour A un borélien, définissons le premier temps de passage de V dans A : τv,a := inf{t, Vt A}, 9

21 1.2. PROCESSUS DE LÉVY ET FONCTIONNELLES EXPONENTIELLES et nous notons plus simplement τv,y pour τv,{y}. Comme nous venons de le voir, le comportement asymptotique des processus de Lévy spectralement négatifs se détermine assez simplement. L intérêt de ces processus vient souvent du fait qu ils atteignent chaque niveau positif continûment : τv,[y,+ [ = τv,y, ce qui permet d appliquer facilement la propriété de Markov en τv,[y,+ [ et confère de bonnes propriétés au processus des temps d atteinte τv,[y,+ [, y qui est un subordinateur i.e. un processus de Lévy croissant dont la transformée de Laplace s exprime en fonction de Φ V : y,λ, E [ e λτv,[y,+ [] = e yφ V λ Le fait de pouvoir appliquer facilement la propriété de Markov en τv,[y,+ [ permet également de montrer que les processus de Lévy spectralement négatifs possèdent une fonction d échelle, W, qui vérifie y > x >, PτV x,[y,+ [ < τv x,],] = Wx/Wy. Cette fonction joue un role important pour conditionner le processus V à rester positif. Comme cela a été mentionné à la fin de la Section 1.1.1, Singh [66] a généralisé les résultats de Kawazu et Tanaka [45] et Hu, Shi et Yor [44] pour la diffusion dans un environnement Lévy spectralement négatif. De même qu au Théorème 1.1.2, plusieurs régimes distincts peuvent se présenter. Le κ dans ce théorème, qui représente le drift du mouvement brownien et détermine dans quel régime on se trouve, doit, dans le cas d un environnement Lévy spectralement négatif, être remplacé par κ := Φ V qui joue un rôle similaire. Comme nous le verrons aux Chapitres IV et V, κ := Φ V joue également un rôle similaire au drift de l environnement brownien pour ce qui est du comportement du temps local d une diffusion en environnement Lévy spectralement négatif Conditionnement à rester positif Nous introduisons ici la notion de conditionnement à rester positif pour les processus décrits à la section précédente. Nous renvoyons le lecteur à [8] pour les preuves ou pour plus de détails. Soit V un processus de Lévy spectralement négatif et non monotone. On définit un semi-groupe Markovien p t par la relation t,x,y >, p tx,dy := WyP V x t dy, inf V x > /Wx. [,t] Pour x >, on note Vx pour le processus Markovien issu de x, à valeur dans ],+ [ et dont le semi-groupe de transition est p t. Vx est communément appelé processus V issu de x conditionné à rester positif. Cette appellation est justifiée par le fait que pour tous y > x > le processus Vxt, t τvx,y est égal en loi au 1

22 1.2. PROCESSUS DE LÉVY ET FONCTIONNELLES EXPONENTIELLES processusv x t, t τv x,y conditionnellement à{τv x,y < τv x,],]}. En d autres termes, pour les processus tués en leur temps d atteinte de y, Vx a même loi que V x conditionné dans le sens usuel à rester positif. Dans le cas où V converge vers + la même relation reste vraie en remplaçant le temps d atteinte de y par +. Dans les cas où V oscille ou tend vers, il n est pas possible de conditionner V x à rester positif au sens usuel pendant un temps infini, puisque l événement par rapport auquel on veut conditionner est de probabilité nulle. Il est toutefois notable que même dans ces cas le processus Vx soit défini et ait un temps de vie infini. Il est d ailleurs possible de montrer que dans tous les cas, le processus Vx converge presque sûrement vers +. On peut montrer qu il existe un processus V, noté plus simplement V, issu de qui est la limite en loi des processusvx lorsquextend vers.v est appelé processus V conditionné à rester positif, c est un processus de Feller dont la restriction à ],+ [ des lois de transition est encore donnée par le semi-groupe p t. Le processus V apparait de façon fondamental dans la loi des excursions du processus de Lévy V réfléchi en son infimum, c est cette propriété qui le fait apparaitre dans l étude des diffusions en milieu aléatoire. En effet, ces diffusions ont tendance à rester piégées un certain temps autour des minima locaux du potentiel et il est donc indispensable de connaitre la loi du potentiel à ces endroits. Pour ce qui est du cas brownien drifté, il est connu que les processus W κ et W κ ont même loi. Ceci peut se voir par exemple à partir de l EDS satisfaite par W κ pour la preuve voir par exemple [37], Lemme 6 : dx t = db t +κcothκx t dt, X =. Dans l étude de W κ il est donc suffisant de se restreindre au cas où κ > Fonctionnelles exponentielles La fonctionnelle exponentielle d un processus de Lévy réel Y est définie par : IY := + e Yt dt et a été intensément étudiée, en particulier par Bertoin et Yor [1], [11] ou encore Carmona et al. [19], [2]. Une monographie assez complète sur le sujet a été écrite par Bertoin et Yor [12]. Cette fonctionnelle intervient dans de nombreux domaines des probabilités : l étude des processus Markoviens auto-similaires, les mathématiques financières, et également l étude des diffusions en milieu aléatoire. En effet, lorsque V est un processus de Lévy réel indexé par R, les valeurs en respectivement + et de la fonction S V dite fonction d échelle de la diffusion en milieu V, et définie en Section ont même loi que respectivement I V et IV. La connaissance des conditions de finitude pour IY et de ses queues de distribution 11

23 1.2. PROCESSUS DE LÉVY ET FONCTIONNELLES EXPONENTIELLES est donc essentielle pour l étude des diffusions en milieu aléatoire. La finitude de IY est reliée au comportement asymptotique de Y décrit à la fin de la Section Theorem voir par exemple [12]. Soit Y un processus de Lévy réel. Si Y tend vers + alors IY est finie presque sûrement. Si Y tend vers ou oscille alors IY est infinie presque sûrement. Pour les queues de distributions de IY, un résultat connu et particulièrement utile pour l étude des diffusions en milieu aléatoire est le suivant : Theorem Rivero [58], 25, voir aussi [12]. Soit Y un processus de Lévy réel non arithmétique c est-à-dire qu il n existe aucun r > tel que presque sûrement Y1 rz. On suppose qu il existe θ > tel que E[e θy1 ] = 1 et E[ Y1 e θy1 ] < +. Il existe alors une constante positive C telle que PIY > x t + Cx θ. Une étude plus générale de la queue à droite de la fonctionnelle IY a été effectuée par Maulik et Zwart [52], notamment dans le cas où la condition de Cramér E[e θy1 ] = 1 n est pas vérifiée. D autres choses ont été étudiée sur les fonctionnelles exponentielles des processus de Lévy, notamment le calcul des moments [19], [11] voir aussi [12], l absolue continuité [2], [9], et les propriétés de la densité [2], [54]. Dans le cas d une diffusion dans un potentiel Lévy V qui tend vers, le Théorème permet de montrer que la diffusion est presque sûrement transiente vers +. Le Théorème permet d étudier le temps de sortie d une vallée et de montrer par exemple que ce dernier est dans le domaine d attraction d une loi stable, de sorte que, dans certains cas, les temps d atteintes de la diffusion convergent vers une loi stable ceci peut fournir une approche pour prouver ou généraliser le premier point du Théorème Comme nous l avons mentionné dans la section précédente,v apparait lorsqu on considère la loi du potentiel V autour de ses minima locaux. De plus, l expression du temps passé par la diffusion à l intérieur d un puits de potentiel fait intervenir l intégrale prise dans le puits de exp Ṽ, où Ṽ est le potentiel re-centré au fond du puits. Ceci fait donc apparaitre la fonctionnelle exponentielle de V dans l étude de la diffusion en milieu V. Pour l étude précise du temps local d une diffusion en milieu V nous avons donc besoin de connaître certaines propriétés de la fonctionnelle exponentielle de V. Soit donc V un processus de Lévy spectralement négatif et non monotone. Au Chapitre 3 nous nous sommes intéressés à la fonctionnelle IV := + e V t dt. Un cas particulièrement simple d une telle fonctionnelle est celui où V est un mouvement brownien drifté. On rappelle que W κ désigne le mouvement brownien 12

24 1.3. DESCRIPTION DES RÉSULTATS OBTENUS κ-drifté et que W κ et W κ ont même loi de sorte qu on peut supposer κ positif. Il est connu que IW κ est fini presque sûrement et sa transformée de Laplace est connue explicitement : E[e λiw κ ] = 2 2λ κ 2 κ Γ1+κI κ 2 2λ, où I κ est une fonction de Bessel modifiée. Cette expression de la transformée de Laplace de IW κ a été démontrée et utilisée par Andreoletti et Devulder [3] pour l étude de la diffusion en milieu brownien drifté voir le Lemme 4.2 dans [3], voir aussi le Lemme au Chapitre 2. L étude de cette transformée de Laplace voir par exemple Section au Chapitre 3 permet de déduire l existence de moments exponentiels pour IW κ, l existence d une densité de classe C, ainsi que de déterminer sa queue à gauche : log P IWκ x 2 x x Dans la suite, nous comparons nos résultats généraux obtenus pour IV avec ceux mentionnés ici pour IW κ. Nous verrons en particulier qu il existe, dans le cas général, une grande variété de comportements pour la queue à gauche de IV, et que cette queue détermine très précisément le comportement presque sûr du temps local d une diffusion en environnement V. 1.3 Description des résultats obtenus Chapitre 2 : Convergence en loi en milieu brownien drifté Ce travail est issu d une collaboration avec Pierre Andreoletti et Alexis Devulder et a fait l objet d un article [4] accepté pour publication dans le revue ALEA. L objectif de ce chapitre est d établir la convergence en loi du supremum du temps local L X dans le cas d une diffusion en milieu aléatoire dont le potentiel est W κ, le mouvement brownien κ-drifté avec < κ < 1. Avant d énoncer les résultats obtenus il nous faut au préalable introduire quelques objets. Soit R κ une variable aléatoire ayant même loi que la somme de deux copies indépendantes de la variableiwκ définie en Section SoitY 1,Y 2 le subordinateur bidimensionnel κ-stable dont la mesure de Lévy ν est donnée par : x >,y >, ν[x,+ [ [y,+ [ = C ] 2 [R y κe κ κ1 Rκ y + C 2 R x x κp κ > y,1.3.1 x où C 2 est une constant positive. 13

25 1.3. DESCRIPTION DES RÉSULTATS OBTENUS Pour Z un processus croissant et càd-làg, et s, on pose respectivement Zs, Z s et Z 1 s pour respectivement la limite à gauche de Z en s, le plus grand saut de Z avant s, et l inverse généralisée de Z en s : Zs = lim Zr, r < s Z s := sup Zr Zr, Z 1 s := inf{u, Zu > s}, r s où inf = + par convention. On définit le couple de variables aléatoires I 1,I 2 par : I 1 := Y1Y 2 1 1, I 2 := 1 Y 2 Y2 1 1 Y 1Y2 1 1 Y 1 Y2 1 1 Y 2 Y2 1 1 Y 2 Y2 1 1, Le résultat que nous obtenons sur la convergence en loi de L X peut alors s écrire : Theorem Pour la diffusion dans le potentiel W κ avec < κ < 1 on a, L Xt/t L t + I := maxi 1,I 2. La preuve de ce résultat repose sur la décomposition de l environnement en h t - vallées définies à la fin de la Section 1.1.2, c est-à-dire en puits de potentiel d une hauteur donnée h t dépendante du temps t. Ces h t -vallées sont indépendantes et visitées successivement par la diffusion sans retour en arrière, elles possèdent chacune un fond le h t -minimum qui leur est associé près duquel la diffusion va rester piégée un certain temps avant de s échapper de la vallée. Chacune des h t -vallées visitées contient donc un pic de temps local situé en son minimum, et elle correspond à une dépense de temps de la part de la diffusion. Les contributions des vallées successives au temps local et au temps dépensé par la diffusion forment une suite iid dont on montre que la somme renormalisée par t des termes converge vers un le subordinateur κ-stable Y 1,Y 2. Il s agit alors d exprimer la quantité d intérêt le supremum du temps local au temps t comme une fonctionnelle de la suite iid, de montrer que cette fonctionnelle est continue puis d utiliser le continus mapping theorem pour en déduire la convergence de la quantité d intérêt vers cette fonctionnelle appliquée à Y 1,Y 2. La forme de la loi limite peut alors s expliquer de la façon suivante : I 1 est définie comme le plus grand saut dey 1 avant le saut qui fait quey 2 dépasse1, elle représente donc le plus grand pic de temps local avant l entrée de la diffusion dans la vallée qui la contient à l instant t. Pour obtenir la loi limite il faut également tenir compte de la contribution de la dernière vallée, celle qui contient la diffusion à l instant t. Le pic de temps local dans cette dernière vallée est représenté par Y 1 Y2 1 1 Y 1 Y2 1 1 la première composante du saut de Y 1,Y 2 qui fait que Y 2 dépasse 1, or il ne faut garder qu une certaine proportion de ce pic de temps local : la partie qui date d avant l instant t. La proportion de temps passé avant l instant t par la diffusion dans la dernière vallée est représentée par 1 Y 2 Y /Y 2 Y Y 2 Y 1 En multipliant cette proportion par Y 1 Y Y 1 Y qui représente le pic

26 1.3. DESCRIPTION DES RÉSULTATS OBTENUS de temps local on obtient I 2 qui représente alors la contribution de la dernière vallée au temps local avant l instant t. Il est alors naturel que la loi limite soit le maximum des variables I 1 et I 2. Il se trouve que la suite iid des contributions des vallées successives au temps local et au temps dépensé par la diffusion contient des informations sur d autres quantités intéressantes. Il y a notamment F t, la position au temps t du point favori de la diffusion défini en Il est donc possible d appliquer la méthode que nous avons développée pour étudier ces autres quantités. On a en particulier le théorème suivant sur la convergence en loi du point favori : Theorem Pour la diffusion dans le potentiel W κ avec < κ < 1 on a, F t/xt L t + B U [,1] +1 B, où B est une variable de Bernoulli de paramètre PI 1 < I 2, et U [,1] est une variable aléatoire de loi uniforme sur [,1], indépendante de B. La loi limite du Théorème peut aussi s expliquer intuitivement : avec une probabilité d environ PI 1 < I 2 le supremum du temps local est atteint avant la dernière vallée et il correspond à un maximum de variables iid, la position de ce maximum est alors uniforme. Sur l événement complémentaire, le supremum du temps local est atteint dans la dernière vallée, celle qui par définition contient la diffusion au temps t, F t/xt est alors proche de 1. La forme de la loi limite obtenue illustre le rôle prépondérant des vallées en ce qui concerne le comportement du temps local de la diffusion, ceci explique que les méthodes annealed utilisées dans [28] soient peu adaptées au cas < κ < 1, malgré leur grande efficacité pour le cas κ > 1. Au Chapitre 2, nous présentons également les convergences d autres quantités qui nous semblent intéressantes : le supremum du temps local juste avant l entrée dans la dernière vallée et le supremum du temps local juste après la sortie de celle-ci Chapitre 3 : Fonctionnelles exponentielles Ce travail a fait l objet d un article [72] actuellement en révision. L objectif principal de ce chapitre est l étude de la fonctionnelle exponentielle IV définie en Section 1.2.4, ceci afin d établir les propriétés nécessaires pour la généralisation des résultats du Chapitre 2 au cas d une diffusion dans un potentiel Lévy spectralement négatif. De plus, comme nous le verrons au Chapitre 5, le comportement presque sûr du supremum du temps local de la diffusion est étroitement lié à la queue à gauche de la variableiv qu il convient donc d étudier précisément. L extension au cas de certains processus de Lévy conditionnés à rester positif des résultats connus pour les fonctionnelles exponentielles des processus de Lévy est également une motivation à part entière de ce travail. Nous nous intéressons donc, 15

27 1.3. DESCRIPTION DES RÉSULTATS OBTENUS dans ce chapitre, à d autres propriétés telles que l existence et la régularité de la densité pour la fonctionnelle IV. Dans ce chapitre nous supposons que les hypothèses habituelles sur V faites en Section V est spectralement négatif et non monotone sont satisfaites. La première question qui se pose dans l étude de la fonctionnelle IV est sa finitude. De plus, pour l étude du temps local de la diffusion en milieu V il est utile d établir l existence de moments exponentiels pour IV. Nous prouvons le résultat suivant : Theorem La variable aléatoire IV est finie presque sûrement, elle admet une espérance finie E[IV ] et [ ] λ < 1/E[IV ], E e λiv < Un point clé dans l étude deiv est de la voir comme la solution d une équation affine aléatoire : on montre que pour tout y > il existe une variable aléatoire A y telle que IV L = A y +e y IV, où les deux termes du membre de droite sont indépendants. Cela montre en particulier que IV est une variable aléatoire positive auto-décomposable et que par conséquent elle est absolument continue et unimodale. À y > fixé, des itérations infinies de la relation permettent d écrire IV comme la somme d une série entière aléatoire IV L = k e ky A y k, où les coefficients A y k sont iid de même loi que Ay. Cette décomposition est particulièrement utile pour l étude des propriétés IV, puisqu elle permet de se ramener à l étude de la variable A y. Les résultats suivants sont fondamentaux pour l application aux diffusions en milieu aléatoire : ils décrivent la queue à gauche de la fonctionnelle IV. L idée est de relier cette queue au comportement asymptotique de Ψ V. Theorem Supposons qu il existe α > 1 et une constante positive C tels que pour tout λ suffisamment grand on ait Ψ V λ Cλ α. Alors pour tout δ ],1[, on a pour x suffisamment petit P IV x exp δα 1/Cx 1/α Supposons qu il existe α > 1 et une constante positive c tels que pour tout λ suffisamment grand on ait Ψ V λ cλ α. Alors pour tout δ > 1, on a pour x suffisamment petit P IV x exp δα α/α 1 /cx 1/α

28 1.3. DESCRIPTION DES RÉSULTATS OBTENUS Avant d énoncer d autres résultats il nous faut définir une façon de quantifier le comportement asymptotique de Ψ V. En nous inspirant de ce qui se fait pour l étude, par exemple, de la dimension de Hausdorff des trajectoires des processus de Lévy, nous définissons : { σ := sup β := inf } α, lim λ + λ α Ψ V λ =, { } α, lim λ + λ α Ψ V λ =. Si Ψ V est à variation α-régulière pour un α [1,2] on a σ = β = α. Rappelons que Q désigne la composante gaussienne de V dans son triplet générateur. Il est possible de voir sur l expression que, lorsque Q >, Ψ V λ Qλ 2 /2 et donc que Ψ V est à variation 2-régulière et, lorsque Q =, 1 σ β 2. Nos résultats sur les queues à gauche de IV peuvent être énoncés comme suit : Theorem β > β, lim x x 1/β 1 log P IV x =, si σ > 1, σ ]1,σ[, lim x x 1/σ 1 log P IV x = Le théorème précédent donne pour PIV x une borne inférieure impliquant σ et une borne supérieure impliquantβ. Dans le cas oùψ V est à variationα-régulière pour un α ]1,2] on sait que σ = β = α, ce qui renforce la précision du résultat. En renforçant les hypothèse sur la régularité de Ψ V on peut encore préciser ce résultat : Theorem On suppose qu il existe une constante positive C et α ]1,2] tels que Ψ V λ λ + Cλ α, alors log P IV x Cx 1 x α 1 Ce théorème est vrai en particulier lorsque, pour un α ]1,2], V est un processus de Lévy α-stable sans sauts positifs avec adjonction ou non d un drift. En particulier, la queue donnée dans l expression est un cas particulier du Théorème En effet, lorsque V = W κ, sa composante gaussienne vaut 1 et donc Ψ V λ λ 2 /2. L équivalent donné par le Théorème est alors le même que l équivalent Remark Puisque Ψ V λ/λ 2 a toujours une limite finie en +, le Théorème implique l existence d une constante positive K dépendante de V telle que pour tout x suffisamment petit P IV x e K/x. α 1. 17

29 1.3. DESCRIPTION DES RÉSULTATS OBTENUS Remark Le Théorème s applique lorsque β = 1, et β peut alors être choisi de façon à ce que 1/β 1 soit aussi grand qu on veut. Lorsque V est à variation bornée on peut prouver un résultat plus précis : PIV x est nulle pour x suffisamment petit. Remark On rappelle que IV est unimodale. Si était un mode, alors on aurait PIV x cx pour une certaine constante positive c et tout x suffisamment petit, ce qui contredirait Ainsi la densité de IV est croissante sur un voisinage de. Ceci implique que les Théorèmes 1.3.4, 1.3.5, et les Remarques 1.3.7, sont vrais si on remplace la fonction de répartition PIV. par la densité de IV. Lorsque V tend vers + nous établissons que la queue à gauche de IV est la même que la queue à gauche de IV. Ceci implique que tous nos résultats sur la queue à gauche de IV s appliquent aussi à la queue à gauche de IV : Proposition Lorsque V tend vers +, les Théorèmes 1.3.4, 1.3.5, et les Remarques 1.3.7, sont vraies si on remplace IV par IV. La dernière proposition illustre le fait que l étude de la fonctionnelle exponentielle d un processus de Lévy conditionné à rester positif permet de déduire des résultats pour la fonctionnelle exponentielle du processus de Lévy correspondant. Nous savons déjà, grâce à l auto-décomposabilité, que IV est absolument continue. Une question intéressante, bien qu elle ne se pose pas pour l étude de la diffusion en milieu aléatoire, est de connaitre la régularité de cette densité. Le résultat suivant est une condition sur Ψ V pour la régularité de la densité : Theorem Si σ > 1 et β sont tels que 2β 2 3σβ +σ +β 1 <, alors la densité de IV appartient à l espace de Schwartz. Ses dérivées à tout ordre convergent vers en + et en. Lorsque IV est finie, le théorème précédent permet également d étudier la régularité de sa densité. On a en effet le corollaire suivant : Corollary Lorsque V tend vers + et est tel que σ > 1 et que est satisfaite, alors la densité de IV est de classe C et ses dérivées à tout ordre convergent vers en + et en. Remark Si Ψ V est à variation α-régulière pour un α ]1,2] alors σ = β = α et la condition devient α 1 2 <, ce qui est toujours vrai pour α > 1. On peut donc appliquer le Théorème et le Corollaire En d autres termes, le fait d être à variation α-régulière pour Ψ V implique la régularité des densités de IV et de IV si cette fonctionnelle est finie lorsque α > 1. 18

30 1.3. DESCRIPTION DES RÉSULTATS OBTENUS Nous venons de présenter des résultats concernant IV. Lorsque V converge vers certains de ces résultats sont fondamentaux pour l étude, aux Chapitres IV et V, du temps local de la diffusion dans le potentiel V. Bien qu elle joue un rôle moins déterminant, la fonctionnelle exponentielle de V conditionné à rester positif intervient également dans cette étude et il nous faut donc aussi étudier les fonctionnelles exponentielles des processus de Lévy spectralement positifs conditionnés à rester positif. Dans la suite de cette section, Z est un processus de Lévy spectralement positif qui converge vers +, la définition de Z conditionné à rester positif, noté Z, est donnée au début du Chapitre 3. Nous étudions la fonctionnelle exponentielle IZ. Le premier résultat concerne sa finitude et l existence de moments exponentiels : Theorem La variable aléatoire IZ est finie presque sûrement et admet des moments exponentiels, c est-à-dire, [ ] λ >, E e λiz < +. La présence de sauts positifs pour Z et Z alourdit les queues à gauche de IZ et IZ par rapport à celle obtenue pour IV. On a en effet : Theorem Si Z est à variation non bornée et que sa mesure de Lévy est non nulle, alors il existe une constante positive c telle que e clogx2 PIZ x P IZ x. La minoration pour PIZ x ne nécessite pas l hypothèse de la variation non bornée. Remark Si la mesure de Lévy de Z est nulle, on voit sur la formule de Lévy-Khintchine que Z est un mouvement brownien drifté. La queue en de IZ est alors donnée par le Théorème qui s applique avec α = 2 et elle est plus fine que celle donnée par le Théorème dans le cas où Z a des sauts. La présence ou l absence de sauts joue donc un rôle déterminant pour la queue à gauche de la fonctionnelle exponentielle. Il y a deux raisons pour lesquelles l étude du cas spectralement positif est moins poussée que celle du cas spectralement négatif. La première est que nous n avons pas, dans le cas spectralement positif, de décomposition du type pour la fonctionnelleiz, or presque toute notre étude du cas spectralement négatif repose sur une telle décomposition. La deuxième raison est que nous n avons pas besoin, pour les applications aux diffusions en milieu aléatoire, de résultats aussi précis dans le cas spectralement positif que dans le cas spectralement négatif. En effet, lorsqu on étudie la diffusion dans un environnement Lévy spectralement négatif V qui tend vers, il apparait une variable aléatoire R dont la loi est la convolution de celles deiv et dei V. Les résultats précédents montrent que pour certaines choses le comportement de IV est dominant par rapport à celui de I V, lorsque V a des sauts. En particulier, la queue à gauche de R est la même que celle de IV. 19

31 1.3. DESCRIPTION DES RÉSULTATS OBTENUS Chapitre 4 : Convergence en loi en milieu Lévy spectralement négatif Ce travail a fait l objet d un article [74] actuellement soumis. L objectif de ce chapitre est d établir des résultats de convergence en loi du supremum du temps local L X t et du point favori F t dans le cas général d une diffusion en milieu aléatoire dont le potentiel V est un processus de Lévy spectralement négatif et non monotone qui converge vers en +. Comme cela a été mentionné à la Section 1.2.2, il est connu que pour de tels potentiels κ = Φ V = sup{λ, Ψ V λ = } joue un rôle similaire, pour le comportement de la diffusion, à celui du drift du mouvement brownien W κ voir notamment [66]. Lorsque κ > 1, nous utilisons une égalité en loi entre le temps local et un processus d Ornstein-Uhlenbeck généralisé, selon une méthode inspirée de [66]. Ceci permet de généraliser la convergence obtenue par Devulder [28] dans le cas d un potentiel brownien drifté. Soient K et m définis comme dans [66] : K := E [ + κ 1 ] e Vt dt et m := 2 Ψ V 1 > Lorsque κ > 1, nous obtenons la convergence en loi suivante pour le supremum du temps local : Theorem Si κ > 1, L Xt/t 1/κ L t + Fκ,2Γκκ2 K/m 1/κ, où F.,. désigne la loi de Fréchet définie en Remark Si V = W κ pour un κ > 1, alors K = 2 κ 1 /Γκ voir Exemple 1.1 dans [66] et m = 4/κ 1. La loi limite donné par le Théorème est alors exactement la même qu en L intérêt de la méthode utilisée est qu elle permet aussi d étudier la convergence en loi du point favori de la diffusion : Theorem Si κ > 1, mf t/t où U [,1] désigne une loi uniforme sur [,1]. L t + U [,1], Lorsque < κ < 1, la loi limite que nous exhibons pour L X t/t est proche de celle obtenue au Chapitre 2. Rappelons que la finitude des fonctionnelles IV et I V est assurée par les Théorèmes et de la Section Soit R une variable aléatoire dont la loi est la convolution des lois de IV et de I V. 2

32 1.3. DESCRIPTION DES RÉSULTATS OBTENUS R est l analogue de R κ définie à la Section en effet, si V = W κ les lois de IV et de I V coincident. Rappelons que C la constante apparaissant dans le Théorème qui donne la queue à droite de la fonctionnelle I V, et posons C := C/2 + u κ e u/2 du. Soit Y 1 le subordinateur κ-stable dont l exposant de Laplace est C Γ1 κλ κ : t,λ, E [ e λy 1t ] = e tc Γ1 κλ κ. On considère le processus de Lévy Y 1,Y 2 où la composante Y 2 est définie en multipliant chaque saut de Y 1 par une copie indépendante de R. Y 1,Y 2 peut également être définie par sa mesure de Lévy comme à la Section : x >,y >, ν[x,+ [ [y,+ [ = C y κe [ R κ 1 R y x ]+ R C x κp > y. x Il est possible de voir que ces deux définitions du processus Y 1,Y 2 sont équivalentes et que ce dernier a pour transformée de Laplace t,α,β, E [ e αy 1t βy 2 t ] = e tc Γ1 κe[α+βr κ]. On a alors la convergence en loi de L X t/t et la loi limite s exprime grâce à la même fonctionnelle qu à la Section On définit le couple aléatoire I 1,I 2 par I 1 := Y1Y 2 1 1, I 2 := 1 Y 2 Y2 1 1 Y 1Y2 1 1 Y 1 Y2 1 1 Y 2 Y2 1 1 Y 2 Y2 1 1, et la convergence en loi dont est l objet le supremum du temps local s écrit Theorem Si < κ < 1, V est à variation non bornée et V1 L p pour un p > 1 alors L L Xt/t I := maxi 1,I 2. t + Ce théorème inclut le cas où V = W κ et est donc une généralisation du Théorème La différence entre le cas d un environnement brownien drifté et le cas plus général d un environnement Lévy spectralement négatif vient de la variable R dont la queue à gauche peut grandement varier selon les cas. Ceci induit une multitude de possibilités pour le comportement presque sûr du temps local. Nous verrons en particulier au Chapitre 5 comment la renormalisation de L X t dépend de la loi de la variable R en ce qui concerne le comportement presque sûr. La preuve du Théorème nécessite une étude précise de l environnement V afin de généraliser les arguments de la preuve du Théorème qui repose sur des estimés bien connus et explicites pour le mouvement brownien drifté W κ. Il nous faut en particulier généraliser la décomposition du potentiel en h-vallées et étudier les propriétés de ces vallées, la distance entre deux h-minima consécutifs est par exemple un élément important à connaitre. La hauteur h des vallées considérées 21

33 1.3. DESCRIPTION DES RÉSULTATS OBTENUS doit être adaptée à l échelle de temps et notamment tendre vers l infini lorsque le temps t tend vers l infini. Nous avons donc besoin d étudier la distribution asymptotique des h-minima. Notons m 1,m 2,... la suite des h-minima, nous déterminons le comportement asymptotique de cette suite dans le théorème suivant : Theorem Lorsque h tend vers l infini, la suite renormaliséee κh m 1,m 2,... converge en loi vers la suite des temps de sauts d un processus de Poisson standard de paramètre q qui dépend explicitement de la loi de V. Si V = W κ alors q = κ 2 /2. L intérêt de ce théorème est de nous informer sur la distance typique entre deuxhminima consécutifs. Ceci permet en particulier de donner une explication heuristique au fait que les méthodes et résultats obtenus sont différents dans les cas < κ < 1 et κ > 1. Comme le montre le Théorème , la distance entre deux h-minima consécutifs est de l ordre de e κh. Lorsque < κ < 1, cette distance est telle qu on peut négliger ce qui se passe en dehors des fonds des h-vallées, si bien que les principales contributions au temps local et au temps dépensé par la diffusion sont localisées au fond des h-vallées et fortement corrélées. Ceci se traduit par l apparition du subordinateur Y 1,Y 2 qui ne s accroit que par sauts chacun représentant une vallée et dont les deux composantes sont corrélées l une représente les pics de temps local dans les vallées et l autre les dépenses de temps associées à ces vallées. Lorsque κ > 1, la distance entre deux h-minima consécutifs est si grande que le temps passé par la diffusion entre deux h-minima n est pas négligeable comparée au temps passé à proximité de ces h-minima. En particulier il existe également des pics de temps local dans la zone entre deux h-minima. Il est alors impossible d utiliser les puits de potentiel pour localiser les pics du temps local, contrairement au cas < κ < 1. Ceci explique la distribution asymptotiquement uniforme du point favori donnée au Théorème De plus, le cas κ > 1 correspond au cas où X est à vitesse positive. Le temps local au temps t est alors proche du temps local en un temps d atteinte qui lui-même est égal en loi au processus d Ornstein-Uhlenbeck généralisé, introduit dans [66]. Le supremum du temps local peut alors être représenté comme la hauteur maximale dans un ensemble iid d excursions d un processus Markovien. Ceci explique la loi de Fréchet au Théorème , puisqu une telle loi est classiquement l une des trois limites possibles pour les maxima de suites iid Chapitre 5 : Comportement presque sûr en milieu Lévy spectralement négatif Ce travail a fait l objet d un article [73] qui sera prochainement soumis. L objectif de ce chapitre est d étudier le comportement presque sûr du supremum du temps local L X t pour une diffusion en milieu aléatoire dont le potentiel est V, un processus de Lévy spectralement négatif et non monotone qui converge vers en +. Ici encore nous posons κ := Φ V = sup{λ, Ψ V λ = }. Un cas 22

34 1.3. DESCRIPTION DES RÉSULTATS OBTENUS particulièrement intéressant est celui des valeurs extrêmement grandes de L X t dans le cas < κ < 1. Ces dernières sont reliées aux propriétés de la variable R qui intervient dans la loi limite du Théorème et qui peut être étudiée grâce aux résultats du Chapitre 3. Nous caractérisons le comportement presque sûr de L X t lorsque < κ < 1 et κ > 1. En particulier, la restriction de nos résultats au cas où V = W κ avec < κ < 1 améliore les résultats partiels 1.1.9, et de Devulder [28] en donnant la renormalisation exacte ainsi que la valeur exacte de la constante pour la limsup. Pour la liminf on obtient aussi la renormalisation exacte et un majorant explicite pour la constante. Commençons par énoncer les résultats obtenus lorsque κ > 1. L étude de ce cas est assez classique et utilise la même méthode qu au Chapitre 4, c est-à-dire l exploitation des propriétés du processus d Ornstein-Uhlenbeck généralisé introduit par Singh dans [66]. Pour la limsup on a : Theorem Soit f une fonction positive et décroissante. Lorsque κ > 1, on a { { + ft κ < + dt t = + limsup ftl X t = t + t 1/κ + P-p.s. 1 Ce résultat est l analogue, pour la diffusion en milieu Lévy, de qui a été montré par Gantert et Shi [41] pour les RWRE transientes. Dans le cas où V = W κ pour κ > 1, notre résultat est une reformulation de établi par Devulder [28] dont il est donc une généralisation. Pour la liminf, il y a une valeur positive explicite. Soient K et m définis en On a : Theorem Lorsque κ > 1, on a P-presque sûrement lim inf t + L X t t/loglogt 1/κ = 2Γκκ2 K/m 1/κ. Exemple : Si V = W κ pour κ > 1 alors on rappelle que K = 2 κ 1 /Γκ et m = 4/κ 1 voir Remarque La limite du Théorème est alors 4κ 2 κ 1/8 1/κ, ce qui coïncide précisément avec Le cas < κ < 1 est beaucoup plus subtil et nécessite de pousser encore plus loin les méthodes utilisées aux Chapitres II et IV. La loi limite de L X t/t est donnée par le Théorème et elle dépend implicitement de la loi de la variable R, définie comme la convolution des lois de IV et de I V. Dans l étude de la limsup de L X t, on relie le comportement asymptotique presque sûr de L X t avec la queue à gauche de IV ou de R dans le cas où V = W κ. Cette queue est reliée au comportement asymptotique de Ψ V par les résultats du Chapitre 3. La proposition suivante permet de relier précisément les queues des fonctionnelles exponentielles trouvées au Chapitre 3 avec la limsup de L X t. 23

35 1.3. DESCRIPTION DES RÉSULTATS OBTENUS Theorem On suppose que < κ < 1, V est à variation non bornée, V1 L p pour un p > 1 et V possède des sauts négatifs. S il existe γ > 1 et C > tels que pour x suffisamment petit P IV x exp C, x 1 γ 1 alors on a P-presque sûrement lim sup t + L X t tloglogt γ 1 C1 γ S il existe γ > 1 et C > tels que pour x suffisamment petit P IV x exp C, x 1 γ 1 alors on a P-presque sûrement lim sup t + L X t tloglogt γ 1 C1 γ On suppose maintenant que V = W κ avec < κ < 1, alors les implications précédentes et sont vraies, mais avec R à la place de IV. La proposition précédente fait la distinction entre le cas où V possède des sauts négatifs et celui où V est un mouvement brownien drifté W κ. Notons que ces deux cas sont complémentaires : la formule permet de voir que si V n admet pas de sauts négatifs alors V est un mouvement brownien drifté qui s écrit QW κ où Q, comme dans la la formule 1.2.2, est la composante gaussienne de V. Par un changement d échelle, ce cas se ramène à celui où V = W κ, d où l alternative considérée dans la proposition. La différence du résultat entre les deux cas vient de la présence ou non de symétrie pour l environnement. En effet, la loi de la variable R est la convolution des lois de IV et de I V. Si V possède des sauts négatifs seule la queue deiv est à prendre en compte dans la queue à gauche dercomme le montrent les Théorèmes et Si V = W κ, notons que W κ a même loi que W κ qui a lui-même même loi que W κ voir Section 1.2.3, ainsi V et V ont même loi, et la loi de R est alors la convolution de deux lois identiques dont aucune ne peut être négligée. Remark Les limsup dans la proposition précédente sont presque sûrement égales à des constantes appartenant à [, + ] et les inégalités et sont des inégalités pour ces constantes. Il en est de même pour tous les résultats à suivre : toutes les limsup et liminf considérées sont presque sûrement égales à des constantes. Ce fait sera justifié à la fin du Chapitre 5. 24

36 1.3. DESCRIPTION DES RÉSULTATS OBTENUS En combinant le Théorème avec ce que le Chapitre 3 nous apprend sur la queue à gauche de IV, plus précisément le Théorème 1.3.5, on obtient : Theorem Si < κ < 1, V est à variation non bornée et V1 L p pour un p > 1 alors on a P-presque sûrement et β > β, limsup t + L X t tloglogt β 1 si de plus σ > 1 alors σ ]1,σ[, limsup t + =, L X t tloglogt σ 1 = Le Théorème est très précis dans le sens où la connaissance exacte de la queue à gauche de IV ou de R si V = W κ entraine la connaissance de la renormalisation exacte de L X t et de la valeur exacte de la limsup. En se plaçant sous les hypothèses des Théorèmes et on obtient des résultats plus précis pour la limsup : Theorem On suppose que < κ < 1, V est à variation non bornée, V1 L p pour un p > 1 et V possède des sauts négatifs. S il existe deux constantes positives c < C et α ]1,2] tels que cλ α Ψ V λ Cλ α pour λ suffisamment grand. On a alors P-presque sûrement c α α limsup t + L X t tloglogt α 1 C α 1 α 1. Si, plus précisément, il existe une constante positive C et α ]1,2] tels que Ψ V λ Cλ α pour λ grand, on a alors P-presque sûrement lim sup t + L X t tloglogt α 1 = C α 1 α 1. On suppose maintenant que V = W κ avec < κ < 1, on a alors P-presque sûrement L X lim sup t t + tloglogt = 1 8. Remark En combinant le Théorème et la Remarque on a que si < κ < 1, V est à variation non bornée et V1 L p pour un p > 1, alors on a toujours lim sup t + L X t tloglogt < +. En d autres termes, tloglogt est la renormalisation maximale parmi toutes les autres possibles pour la limsup. 25

37 1.3. DESCRIPTION DES RÉSULTATS OBTENUS Remark De même que cela avait été constaté par Shi [61] et Diel [29] dans le cas récurrent voir Remarque 1.1.6, on constate ici aussi une différence entre la renormalisation des RWRE transientes à vitesse nulle donnée par et celles des diffusions transientes à vitesse nulle données par les résultats précédents. Cette différence peut s expliquer de la même façon que dans le cas récurrent, c est-à-dire par le fait que les vallées peuvent être beaucoup plus abrupte dans le cas continu avec un potentiel à variation non bornée que dans le cas discret, et donc les pics de temps local peuvent potentiellement être beaucoup plus grands dans le premier cas. Les résultats précédents montrent que la renormalisation del X t pour lalimsup dépend directement du comportement asymptotique de Ψ V. En particulier, le Théorème montre que pour un environnement α-stable drifté sans sauts positifs et avec α > 1, la renormalisation de L X t est tloglogtα 1. On voit qu il existe, pour des environnements Lévy spectralement négatifs, une plus grande variété de comportements presque sûrs qu avec des environnements browniens driftés. Même si, pour des raisons techniques, les résultats précédents ne s appliquent pas dans le cas où l environnement V est à variation bornée, on peut légitimement conjecturer que le comportement de L X t reste lié à la queue à gauche de IV qui est donnée par la Remarque Ceci implique la conjecture suivante : Conjecture Si V est à variation bornée, on a P-presque sûrement < limsup t + L X t t < +. Si cette conjecture est vraie on aurait, pour des environnements à variation bornée, la même renormalisation que dans le cas discret transient donné par Ceci ne serait pas surprenant compte tenu du fait que le cas discret ne génère que des potentiels à variation bornée. De plus, si V est un processus de Lévy spectralement négatif, non monotone, à variation bornée, alors V est la différence d un drift positif et d un subordinateur. Les vallées ne peuvent alors pas être plus abrupte que le drift ce qui, conformément à l explication heuristique de la Remarque , impose que la renormalisation de L X t soit la même que dans le cas discret. Pour la liminf, il n y a qu une seule renormalisation possible. Nous prouvons le résultat suivant : Theorem Si < κ < 1, V est à variation non bornée et V1 L p pour un p > 1 alors on a P-presque sûrement < liminf t + L X t t/loglogt 1 κ κe[iv ]+E[I V ] Remark Les espérances E[IV ] et E[I V ] sont finies puisque IV et I V admettent des moments exponentiels finis en vertu des Théorèmes et

38 1.3. DESCRIPTION DES RÉSULTATS OBTENUS Exemple : L expression permet de calculer le premier moment deiw κ : E[IW κ] = 2/1+κ. On rappelle de plus que W κ a même loi quew κ, si bien que E[IW κ]+e[i W κ ] = 4/1+κ. Ainsi, pour la diffusion dans l environnement V = W κ avec < κ < 1, la borne supérieure du Théorème pour la liminf devient 1 κ 2 /4κ. En mettant ceci en relation avec les résultats obtenus par Devulder [28], on voit que l application du Théorème dans le cas spécial d un environnement brownien drifté améliore et complète en donnant une majoration explicite de la liminf. Il est intéressant de se demander pourquoi il existe, selon l environnement choisi, une multitude de renormalisations possibles pour la lim sup, mais seulement une pour la liminf. Nous proposons l explication heuristique suivante : dans chaque vallée, le temps passé est environ égal au pic de temps local au fond de la vallée multiplié par une fonctionnelle exponentielle du fond de la vallée qui est proche de R. La limsup est liée aux grandes valeurs prises par le temps local à un temps fixé t, elle est approchée lorsque le pic de temps local au fond d une vallée est très grand tandis que le temps passé dans cette vallée est de l ordre au plus de t, une telle chose arrive quand la fonctionnelle exponentielle du fond d une certaine vallée est très petite. Un lien très précis entre la lim sup et les petites valeurs d une fonctionnelle exponentielle est d ailleurs fait rigoureusement par le Théorème La lim inf est quant à elle liée aux petites valeurs prises par le temps local à un temps fixé t, elle est approchée lorsque les pics de temps local au fond des premières vallées sont très petits tandis que la somme des temps passés dans ces vallées est de l ordre au moins de t, une telle chose arrive quand les fonctionnelles exponentielles des fonds de quelques-unes de ces vallées sont grandes. On voit alors que la différence de comportement entre la limsup et la liminf vient de la différence entre la queue à gauche et la queue à droite der. La queue à gauche est principalement celle deiv qui dépend du comportement asymptotique deψ V d après les Théorèmes , et il y a plusieurs possibilités pour le comportement de Ψ V, selon le choix de l environnement. D un autre côté, la queue à droite est toujours exponentielle : elle est au plus exponentielle d après les Théorèmes et il sera justifié au Chapitre 3 Remarque qu elle est au moins exponentielle. Ceci explique la différence des comportements entre les limsup et liminf. En comparant les Théorèmes , et on constate que la renormalisation de la limsup est plus grande dans le cas transient à vitesse nulle < κ < 1 que dans le cas transient à vitesse positive κ > 1, ce qui est en accord avec l intuition. Il est cependant surprenant de constater que la renormalisation du cas transient à vitesse nulle est également plus grande que la renormalisation du cas récurrent donnée par et par le Théorème Nous interprétons ce phénomène de la façon suivante : dans le cas récurrent la diffusion est piégée au fond d une vallée assez large, tandis que dans le cas transient à vitesse nulle la diffusion se retrouve successivement piégée au fond de vallées qui sont beaucoup plus étroites. Cette différence de largeur des vallées explique que les valeurs extrêmement grandes du temps local ont tendance à être plus grandes dans le second cas, malgré la transience. 27

39 1.4. QUELQUES PERSPECTIVES Pour la liminf les comportements mis en valeurs dans les différents cas sont en accord avec l intuition : en comparant les Théorèmes 1.1.4, et on voit que la renormalisation de la liminf dans le cas transient à vitesse positive est plus petite que la renormalisation du cas transient à vitesse nulle qui est elle-même plus petite que la renormalisation du cas recurrent. 1.4 Quelques perspectives Nous présentons ici quelques travaux en cours et perspectives de recherche qui s inscrivent dans la continuité des travaux présentés dans cette thèse. Il serait intéressant de poursuivre l étude effectuée au Chapitre 3, sur la fonctionnelle exponentielle IV, notamment en tentant d étendre à ce problème les méthodes d analyse complexe transformée de Mellin utilisées pour l étude des fonctionnelles exponentielles des processus de Lévy non conditionnés. Pour la diffusion en milieu aléatoire, il serait intéressant d utiliser les techniques que nous avons développées dans cette thèse pour étudier le problème du temps semi-local, c est-à-dire déterminer la taille du plus petit intervalle où une proportion β ],1[ du temps a été passée par la diffusion. Cela permettrait de mettre en évidence des comportements spéciaux tels que la concentration dans un intervalle arbitrairement petit ou au contraire un étalement dans l espace des points les plus occupés. Pour β ],1[, définissons St,β, la taille du plus petit intervalle où la diffusion a passé un temps plus grand que βt : { St,β := inf α >, sup x R x+α x α } L X t,ydy βt. Nous nous proposons d étudier la convergence en loi et le comportement presque sûr de St,β. Lorsque < κ < 1, notre étude du supremum du temps local exclu le cas où l environnement est à variation bornée. Il serait intéressant de faire l étude du temps local de la diffusion dans ce cas. Dans ce sens nous pourrions par exemple prouver la Conjecture ou étudier l existence, pour ces diffusions, d un principe d invariance, c est-à-dire déterminer s il existe ou non une proximité entre le temps local de la diffusion et celui d une RWRE associée. 28

40 Chapitre 2 Renewal structure and local time for diffusions in random environment This work comes from a collaboration with Pierre Andreoletti and Alexis Devulder and has been the object of an article [4] published in the journal ALEA. 2.1 Introduction Presentation of the model Let Xt, t be a diffusion in a random càdlàg potential Vx, x R, defined informally by X = and dxt = dβt 1 2 V Xtdt, whereβs, s is a Brownian motion independent ofv. Rigorously,X is defined by its conditional generator given V, 1 d 2 evx e Vx d. dx dx We put ourselves in the case where V is a negatively drifted Brownian motion : Vx = W κ x := Wx κ x, x R, with < κ < 1 and Wx, x R is a two 2 sided Brownian motion. We explain at the end of Section what should be done to extend our results to a more general Lévy potential. We denote by P the probability measure associated to W κ.. The probability conditionally on the potential W κ is denoted by P Wκ and is called the quenched probability. We also define the annealed probability as P. := P Wκ.PW κ dω. 29

41 2.1. INTRODUCTION We denote respectively by E Wκ, E, and E the expectations with regard to P Wκ, P and P. In particular, X is a Markov process under P Wκ but not under P. This diffusion X has been introduced by [6]. It is generally considered as a continuous time analogue of random walks in random environment RWRE. We refer e.g. to [76] for general properties of RWRE. In our case, since κ >, the diffusion X is a.s. transient and its asymptotic behavior was first studied by Kawazu and Tanaka : if Hr is the hitting time of r R by X, Hr := inf{s >, Xs = r}, [45] proved that, for < κ < 1 under the annealed probabilityp,hr/r 1/κ converges in law as r + to a κ-stable distribution see also [44], and [69]. Here we are interested in the local time of X, which is the jointly continuous process Lt,x, t >, x R satisfying, for any positive measurable function f, t fxsds = + fxlt,xdx, t >. One quantity of particular interest is the supremum of the local time of X at time t, defined as L t := suplt,x, t >. x R For Brox s diffusion, that is, for the diffusion X in the recurrent case κ =, it is proved in [5] that the local time process until time t re-centered at the localization coordinate b t see [17] and renormalized by t converges in law under the annealed probability P. This allows the authors of [5] to derive the limit law of the supremum of the local time at time t as t +. We recall their result below in order to compare it with the results of the present paper. To this aim, we introduce for every κ, R κ := + e W κx dx+ + e W κx dx, where W κx, x and W κ, x are two independent copies of the process W κ x, x Doob-conditioned to remain positive. Theorem [5] If κ =, then L t t L 1 R κ, where L denotes convergence in law under the annealed probability P as t +. 3

42 2.1. INTRODUCTION Extending their approach, and following the results of [61], [29] obtains the nontrivial normalizations for the almost sure behavior of the limsup and the liminf of L t as t + when κ =. Notice that corresponding results have been previously established in [26] and [4] for the discrete analogue of X in the recurrent case κ =, the recurrent RWRE generally called Sinai s random walk. One of our aims in this paper is to extend the study of the local time of X in the case < κ < 1, and deduce from that the weak asymptotic behavior of L t suitably renormalized as t +. Before going any further, let us recall to the reader what is known for the slow transient cases. For transient RWRE in the case < κ 1 see [48] for the seminal paper, a result of [41] states the almost sure behavior for the limsup of the supremum of the local time L S n of these random walks denoted by S at time n : there exists a constant c > such that limsup n + L S n/n = c > P almost surely. Contrarily to the recurrent case [4] their method, based on a relationship between the RWRE S and a branching process in random environment, cannot be exploited to determine the limit law of L S n/n. For the transient diffusion X considered here, the only paper dealing with L t is [27], in which it is proved, among other results, that when < κ < 1, limsup t + L t/t = + almost surely. But once again his method cannot be used to characterize the limit law of L t/t in the case < κ < 1. Our motivation here is twofold, first we prove that our approach enables to characterize the limit law of L t/t and open a way to determine the correct almost sure behavior of L t as was done for Brox s diffusion by [61] and [29]. Second we make a first step on a specific way to study the local time which could be used in estimation problems in random environment, see [1], [2], [6], [7], [24], [25], [38]. The method we develop here is an improvement of the one used in [3] about the localization of Xt for large t. Before recalling the main result of this paper [3], we need to introduce some new objects. We start with the notion of h-extrema, with h >, introduced by [53] and studied more specifically in our case of drifted Brownian motions by [37]. For h >, we say that x R is an h-minimum for a given continuous function f, R R, if there exist u < x < v such that fy fx for all y [u,v], fu fx+h and fv fx + h. Moreover, x is an h-maximum for f iff x is an h-minimum for f. Finally, x is an h-extremum for f iff it is an h-maximum or an h-minimum for f. As we are interested in the diffusion X until time t for large t, we only focus on the h t -extrema of W κ, where h t := logt φt, with < φt = ologt, loglogt = oφt, and t φt is an increasing function, as in [3]. It is known see [37] that almost surely, the h t -extrema of W κ form a sequence indexed by Z, unbounded from below and above, and that the h t -minima and h t -maxima alternate. We denote respectively by m j, j Z and M j, j Z the increasing sequences of h t -minima and of h t - maxima of W κ, such that m < m 1 and m j < M j < m j+1 for every j Z. 31

43 2.1. INTRODUCTION Define { N t := max k N, sup Xs m k }, s t the number of positive h t -minima on R + visited by X until time t. We have the following result. Theorem [3] Assume < κ < 1. There exists a constant C 1 >, such that lim P Xt mnt C1 φt = 1. t + This result proves that before time t, the diffusion X visits the N t leftmost positive h t -minima, and then gets stuck in a very small neighborhood of an ultimate one, which is m Nt. An analogous result was proved for transient RWRE in the zero speed regime < κ < 1 by [34]. This phenomenon is due to two facts : the first one is the appearance of a renewal structure which is composed of the times it takes the process to move from one h t -minimum to the following one. The second is the fact that like in Brox s case κ =, the process is trapped a significant amount of time in the neighborhood of the local minimum m Nt. It is the extension of this renewal structure to the sequence of local times at the h t -minima that we study here. We now detail our results Results Let us introduce some notation involved in the statement of our results. Assume that < κ < 1. Denote by D [,+,R 2,J 1 the space of càdlàg functions [,+ R 2 with J 1 -Skorokhod topology and denote by L S the convergence in law for this topology. On this space, define a 2-dimensional Lévy process Y 1,Y 2 taking values in R + R +, which is a pure positive jump process with κ-stable Lévy measure ν given by x >, y >, ν [x,+ [ [y,+ [ = C ] 2 [R y κe κ κ 1 Rκ y + C 2 R x x κp κ > y, x where R κ is defined in and C 2 is a positive constant see Lemma The Laplace transform of R κ is given by E e γrκ 2γ κ/2 2 = κγκi κ 2 γ >, γ as proved in Lemma below, where I κ is the modified Bessel function of the first kind of index κ. Moreover, R κ admits moments of any positive order see also Lemma In particular E[R κ κ ] is finite and ν is well defined. For a given càdlàg function f in D[,+,R, define for any s >,a > : f s := sup fr fr, r s f 1 a := inf{x, fx > a}, 32

44 2.1. INTRODUCTION where fr denotes the left limit of f at r. In words, f s is the largest jump of f before time s, whereas f 1 a is the first time f is strictly larger than a. We also introduce the couple of random variables I 1,I 2 as follows, Y 1 I 1 := Y 1 Y 1 2 1, I 2 := 1 Y 2 Y Y Y 1 Y Y 2 Y Y 2 Y We recall that L denotes convergence in law under the annealed probability P as t +. We are now ready to state our first result. Theorem Assume < κ < 1. We have, L t t L I =: maxi 1,I 2. Contrary to the recurrent case κ =, we have no scaling property for the potential, and the diffusion X cannot be localized in a single valley as we can see in Theorem However in the transient case we can make appear and use a renewal structure. We now give an intuitive interpretation of this theorem, explaining the appearance of the Lévy process Y 1,Y 2. First for any s >, Y 1 s is the limit of the sum of the first se κφt normalized by t local times taken specifically at the se κφt first h t -minima see Proposition below. Similarly, Y 2 s is the limit of the sum of the exit times of the se κφt first h t -valleys, normalized by t, where an h t -valley is a large neighborhood of an h t -minimum. For a rigorous definition of these h t -valleys, see Section and Figure 2.1. So, by definition,i 1 is the largest jump of the processy 1 before the first timey 2 is larger than 1. It can be interpreted as the largest re-normalized local time among the local times at the h t -minima visited by X until time t and from which X has already escaped. That is to say, I 1 is the limit of the random variable sup k Nt 1Lm k,t/t. I 2 is a product of two factors : the first one, 1 Y 2 Y2 1 1, corresponds to the re-normalized amount of time left to the diffusion X before time t after it has reached the ultimate visited h t -minimum m Nt, that is, to t Hm Nt /t. The second factor corresponds to the local time ofx at this ultimateh t -minimumm Nt, that is to say I 2 is the limit of Lt,m Nt /t. Intuitively Y 2 is built from Y 1 by multiplying each of its jumps by an independent copy of the variable R κ. Therefore this second factor can be seen as an independent copy of 1/R κ taken at the instant of the overshoot of Y 2 which makes it larger than 1. Notice that this variable R κ plays a similar role as R of Theorem Indeed as in the case κ =, the diffusion X is prisoner in the neighborhood of the last h t -minimum visited before time t. We prove Theorem by showing first that portions of the trajectory of X re-centered at the local h t -minima, until time t, are made in probability with independent parts. This has been partially proved in [3] but we have to improve their results and add simultaneously the study of the local time. 33

45 2.1. INTRODUCTION Second, we prove that the supremum of the local time is, mainly, a function of the sum of theses independent parts, which converges to a Lévy process. We now provide some details about this. Recall that W κs, s is defined as a continuous process, taking values in R +, with infinitesimal generator given for every x > by 1 2 dx + κ κ d 2 2 coth 2 x dx. d 2 This process W κ can be thought of as a κ/2-drifted Brownian motion W κ Doobconditioned to stay positive, with the terminology of [8], which is called Doob conditioned to reach + before in [37] for more details, see Section 2.1 in [3], where W κ is denoted by R. We call BES3,κ/2 the law of W κs, s. That is, W κs, s is a 3-dimensional κ/2-drifted Bessel process starting from. For any process Ut, t R +, we denote by τ U a := inf{t >, Ut = a}, the first time this process hits a, with the convention inf = +. For a < b, W b κ s, s τ Wb κ a is defined as a κ/2-drifted Brownian motion starting from b and killed when it first hits a. We now introduce some functionals of W κ and W κ, which already appeared in [3], Section 4.1 : Let < δ < 1, define F ± x := G ± a,b := τ W κx τ Wκa b exp±w κsds, x >, exp ±W b κs ds, a < b n t := e κφt1+δ, t >, which is, with large probability, an upper bound for N t as stated in Lemma Let S j,r j,e j,j n t be a sequence of i.i.d. random variables depending on t, with S j, R j and e j independent, S 1 L = F + h t + G + h t /2,h t, R 1 L = F h t /2 + F h t /2 and e 1 L = E1/2 an exponential random variable with parameter 1/2, where F is an independent copy off andf + is independent ofg +, and L = denotes equality in law. Definel j := e j S j andh j := l j R j. Note that to simplify the notation, we do not make appear the dependence in t in the sequel. Intuitively, l j plays the role of the local time at the j-th positive h t -minimum m j if X escapes from the j-th h t -valley before time t, that is, if j < N t. Similarly, H j plays the role of the time X spends in the j-th h t -valley before escaping from it. Define the family of processes Y 1,Y 2 t indexed by t, by s, Y 1,Y 2 t s = Y1s,Y t 2s t := 1 se κφt l j,h j. t j=1 34

46 2.1. INTRODUCTION Recall that L S denotes convergence in law under J 1 -Skorokhod topology. Here is our next result. Proposition Assume < κ < 1. We have under P, as t +, Y 1,Y 2 t L S Y1,Y 2. Once this is proved, we check that we can approximate, in law, the renormalized local time L t/t by a function of Y 1,Y 2 t. We obtain such an expression in Proposition Then to obtain the limit claimed in Theorem 2.1.3, we prove the continuity inj 1 -topology of the involved mapping and apply a continuous mapping Theorem see Section It appears that with this method we can also obtain some other asymptotics. Indeed, we obtain in the following theorem the convergence in law of the supremum of the local time of X before X hits the last h t -minimum m Nt visited before time t, of the supremum of the local time of X before X leaves the last h t -valley visited before time t the one around m Nt approximately at time Hm Nt+1, and of the position of the favorite site. Theorem Assume < κ < 1. We have the following convergences in law under P as t +, L Hm Nt+1 t L Hm Nt t L Y 1 Y 1 2 1, L Y 1 Y = I Let us call F t the position of the first favorite site, that is, F t := inf{x R, Lt,x = L t}. Then, F t Xt L BU [,1] +1 B, where B is a Bernoulli random variable with parameter PI 1 < I 2, and U [,1] is a uniform random variable on [,1], independent of B. We remark that with probability one there is at most one point x such that Lt,x = L t so Ft is actually the favorite site. Note that similar questions about favorite points for X have been studied in the recurrent case κ = by [22]. One question we may ask here is : what happens in the discrete case that is, for RWRE, or with a more general Lévy potential? For RWRE, we expect a very similar behavior because the renewal structures which appear in both cases RWRE and our diffusion X are very similar see [34]. The main difference comes essentially from the functionalr κ, which should be replaced by a sum of exponentials of simple random walks conditioned to remain positive see [36], [34]. 35

47 2.1. INTRODUCTION For a more general Lévy potential, we have in mind for example a spectrally negative Lévy process diffusions in such potentials have been studied by [66]. More work needs to be done, especially for the potential. First, to obtain a specific decomposition of the Lévy s path similar to what is done for the drifted Brownian motion in [37], and also to study the more complicated functional R κ which is less known than in the Brownian case. This is a work in preparation by [74]. The rest of the paper is organized as follows. In Section 2.2, we recall the results of Faggionato on the path decomposition of the trajectories of W κ. Also we recall from [3] the construction of specific h t -minima which plays an important role in the appearance of independence, under P, on the path of X before time t. In Section 2.3, we study the joint process of the hitting times of the h t -minima m j,1 j n t and of local times at thesem j. We show that parts of the trajectory of X are not important for our study, that is, we prove that the time spent outside the h t -valleys, and the supremum of the local time outside the h t -valleys are negligible compared to t. We then prove the main result of this section : Proposition It shows that the joint process exit times, local times can be approximated in probability by i.i.d random variables which are the H j and l j. This part makes use of some technical results inspired from [3], they are summarized in Section 6. In Section 2.4, we prove Proposition 2.1.4, and study the continuity of certain functionals of Y 1,Y 2 which appear in the expression of the limit law I. This section is independent of the other ones, we essentially prove a basic functional limit theorem and prepare to the application of continuous mapping theorem. Section 2.5 is where we make appear the renewal structure in the problem we want to solve. In particular we show how the distribution of the supremum of the local time can be approximated by the distribution of some function of the couple Y 1,Y 2 t, the main step being Proposition Section 2.6 is a reminder of some key results and their extensions extracted from [3]. For some of these results, sketch of proofs or complementary proofs are added in order for this paper to be more self-contained. Finally, Section 2.7 is a reminder of some estimates on Brownian motion, Bessel processes, and functionals of both of these processes Notation In this section we introduce typical notation and tools for the study of diffusions in a random potential. For any process Ut, t R + we denote by L U a bicontinuous version of the local time of U when it exists. Notice that for our main process X we simply write L for its local time. The inverse of the local time for every x R is denoted by σ U t,x := inf{s >, L U s,x t} and in the same way σt,x := σ X t,x. We also denote by U a the process U starting from a, and by P a the law of U a, with the 36

48 2.2. PATH DECOMPOSITION AND VALLEYS notation U = U. Now, let us introduce the following functional of W κ, Ar := r e Wκx dx, r R. We recall that since κ >, A := lim r + Ar < a.s. As in [17], there exists a Brownian motion Bs, s, independent of W κ, such that Xt = A 1 [BT 1 t] for every t, where Tr := r exp{ 2W κ [A 1 Bs]}ds, r < τ B A The local time of the diffusion X at location x and time t, simply denoted by Lt,x, can be written as see [61], eq. 2.5 Lt,x = e Wκx L B T 1 t,ax, t >,x R With this notation, we recall the following expression of the hitting times of X, Hr = T [ τ B Ar ] = r e Wκu L B [τ B Ar,Au]du, r Path decomposition and Valleys Path decomposition in the neighborhood of theh t -minima m i We first recall some results for h t -extrema of W κ. Let V i x := W κ x W κ m i, x R, i N, which is the potential W κ translated so that it is at the local minimum m i. We also define τ i h := sup{s < m i, V i s = h}, h >, τ i h := inf{s > m i, V i s = h}, h > The following result has been proved by [37] [for i and ii], and the last fact comes from the strong Markov property see also [3], Fact 2.1 and its proof. Fact path decomposition of W κ around the h t -minima m i i The truncated trajectories V i m i s, s m i τ i h t, V i m i +s, s τ i h t m i, i 1 are independent. ii Let W κs, s be a process with law BES3,κ/2. All the truncated trajectories V i m i s, s m i τ i h t for i 2 and V j m j +s, s τ j h t m j for j 1 are equal in law to W κ s, s τ W κ h t. iii For i 1, the truncated trajectory V i s+τ i h t, s is independent of Wκ s, s τ i h t and is equal in law to Wκ ht s, s, that is, to a κ/2- drifted Brownian motion starting from h t. 37

49 2.2. PATH DECOMPOSITION AND VALLEYS Definition of h t -valleys and of standard h t -minima m j, j N We are interested in the potential around the h t -minima m i, i N, in fact intervals containing at least [τ i 1 + κh t,m i ]. However, these valleys could intersect. In order to define valleys which are well separated and i.i.d., we introduce the following notation. This notation is used to define valleys of the potential around some m i, which are thanks to Lemma equal to the m i for 1 i n t with large probability. Let h + t := 1+κ+2δh t. As in [3], we define L + :=, m :=, and recursively for i 1 see Figure 2.1, L i := inf{x > L + i 1, W κx W κ L + i 1 h+ t }, τ i h t := inf { x L i, W } κx inf,x]w [ L κ h t, i m i := inf { x L i, W κx = inf [ L i, τ ih W } t] κ, L + i := inf{x > τ i h t, W κ x W κ τ i h t h t h + t }. We also introduce the following random variables for i N : M i := inf{s > m i, W κ s = max mi u L + W κ u}, i L i := inf{x > τ i h t, W κ x W κ m i = 3h t /4}, L i := inf{x > τ i h t, W κ x W κ m i = h t /2}, τ i h := inf{s > m i, W κ x W κ m i = h}, h >, τ i h := sup{s < m i, W κ x W κ m i = h}, h >, L i := τ i h+ t. We stress that these random variables depend on t, which we do not write as a subscript to simplify the notation. Notice also that τ i h t is the same in definitions and with h = h t. Moreover by continuity of W κ, W κ τ i h t = W κ m i +h t. Thus, the m i, i N, are h t -minima, since W κ m i = inf [ L+ i 1, τ ih W t] κ, W κ τ i h t = W κ m i +h t and W κ L + i 1 W κ m i +h t. In addition, L + i 1 < L i m i < τ i h t < L i < L i < L + i, i N, L + i 1 L i < m i < τ i h t < M i < L + i, i N Also by induction, the random variables L i, τ ih t and L + i, i N are stopping times for the natural filtration of W κ x, x, and so L i, L i, i N, are also stopping times. Moreover by induction, W κ L i = inf [, L i ] W κ, W κ m i = inf [, τ i h t] W κ, W κ L + i = inf [, L + i ] W κ = W κ m i h + t,

50 2.2. PATH DECOMPOSITION AND VALLEYS L + i 1 L i L i m i τ i h t M i L i L i L + i h + t 1 2 h t h + t h t 3 4 h t h + t Figure 2.1 Schema of the potential between L + i 1 and L + i, in the case L i < L i. for i N. We also introduce the analogue of V i for m i as follows : Ṽ i x := W κ x W κ m i, x R, i N. We call i th h t -valley the translated truncated potential Ṽ i x, L i x L i, for i 1. The following lemma states that, with a very large probability, the first n t + 1 positive h t -minima m i, 1 i n t + 1, coincide with the random variables m i, 1 i n t +1. We introduce the corresponding event V t := nt+1 i=1 {m i = m i }. Lemma Assume < δ < 1. There exists a constant C 1 > such that for t large enough, P V t C1 n t e κht/2 = e [ κ/2+o1]ht. Ṽ Moreover, the sequence i x+ L + i 1, x L + i L + i 1, i 1, is i.i.d. Proof : This lemma is proved in [3] : Lemma 2.3. The following remark is used several times in the rest of the paper. Remark On V t, we have for every 1 i n t, m i = m i, and as a consequence, Ṽ i x = V i x, x R, τ i h = τ i h and τ ih = τ i h for h >. Moreover, Mi = M i. Indeed, Mi is an h t -maximum for W κ, which belongs to [ m i, m i+1 ] = [m i,m i+1 ] on V t, and there is exactly one h t -maximum in this interval since the h t -maxima and minima alternate, which we defined as M i, so M i = M i. So in the following, on V t, we can write these random variables with or without tilde. 39

51 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES 2.3 Contributions for hitting and local times Negligible parts for hitting times In the following lemma we recall results of [3] which say, roughly speaking, that the time spent by the diffusion X outside the h t -valleys is negligible compared to the amount of time spent by X inside the h t -valleys. This lemma also gives an upper bound for the number of h t -valleys visited before time t. Finally, it tells us that with large probability, up to time t, after first hitting the bottom m j of each h t -valley [ L j, L j ], X leaves this h t -valley on its right, that is on L j, and that X never backtracks in a previously visited h t -valley. We define H x y := inf{s > Hx, Xs = y} Hx for any x and y, which is equal to Hy Hx if x < y. Let U :=, U i := H L i H m i = H mi L i, i 1, { } m k 1 B 1 m := H m k U i < ṽ t, m 1, k=1 where ṽ t := 2t/logh t and i=1 U i = by convention. Finally, we introduce m { } B 2 m := H mj L j < H mj L, j H Lj m j+1 < H Lj L, m 1. j j=1 Lemma For any δ > small enough, we have for all large t, i=1 P [ H m 1 ṽ t ] P [ B1 n t ] 1 C 2 v t, with v t := n t logh t e φt = o1 and C 2 >. Moreover, there exists C 3 > such that for large t, PB 2 n t 1 C 3 n t e δκht, PN t < n t 1 e φt Proof : The first statement is Lemma 3.7 in [3]. The second one follows directly from Lemmata 3.2 and 3.3 in [3]. For the proof of see Lemma Negligible parts for local times We now provide estimates for the local time of X at time t. We first prove that the local time of X outside the first n t h t -valleys is negligible compared to t. Second, we prove that for every 1 j n t the local time of X inside the h t -valley [ L j, L j ] but outside a small neighborhood of m j is also negligible compared to t. 4

52 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES Supremum of the local time outside the valleys The aim of this subsection is to prove that at time t, the maximum of the local time outside the h t -valleys is negligible compared to t. More precisely, let ft := te [κ1+3δ 1]φt and, for m 1, { } B3m 1 := sup LH m 1,x ft x [, m 1 ] B 2 3m := m 1 j=1 m 1 j=1 { B 3 m := B 1 3m B 2 3m. sup LH m j+1,x ft x [ L j, m j+1 ] }, {sup x Lj LH m j+1,x L H Lj,x ft This section is devoted to the proof of the following lemma. Lemma Assume that δ is small enough such that κ1+3δ < 1. There exists C 5 > such that for any large t with w t := e κδφt. PB 3 n t 1 C 5 w t, Its proof is based on Lemma below, for which we introduce the following notation, depending only on the potential W κ : τ 1h := inf{u, W κ u inf [,u] W κ h}, h >, m 1h := inf{y, W κ y = inf [,τ 1 h]w κ }, h >. Throughout the paper, C + resp. c denotes a positive constant that may grow resp. decrease from line to line. Lemma Assume that κ1+3δ < 1. For large t, P sup 1ht]L[Hτ 1h x [,m t,x] > te [κ1+3δ 1]φt }, C + n t eκδφt Proof of Lemma : Thanks to and there exists a Brownian motion Bs, s, independent of W κ, such that L[Hτ 1h t,x] = e Wκx L B [τ B Aτ 1h t,ax], x R

53 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES By the first Ray Knight theorem see e.g. [57], chap. XI, for every α >, there exists a Bessel processes Q 2 of dimension 2 starting from, such that L B τ B α,x is equal to Q 2 2α x for every x [,α]. Consequently, using and the independence of B and W κ, there exists a 2-dimensional Bessel process Q 2 such that L[Hτ 1h t,x] = e Wκx Q 2 2[ Aτ 1 h t Ax ] x τ 1h t In order to evaluate this quantity, the idea is to say that loosely speaking, Q 2 2 grows almost linearly. More formally, we consider the functions kt := e 2κ 1φt, at := 4φt and bt := 6κ 1 φte κht, and define the following events { + } A := A := e Wκu du kt, A 1 := { u,kt], Q 2 2u 2eu [ at+4loglog[ekt/u] ]}, A 2 := { inf [,τ 1 h t]w κ bt }. We know that PA y C + y κ for y > since 2/A is a gamma variable of parameter κ,1 see [32], or [15] IV.48 p. 78, having a density equal to e x x κ 1 1 R+ x/γκ, so P A C+ kt κ = C + e 2φt. Moreover, P A 1 C + exp[ at/2] = C + e 2φt by Lemma Also we know that inf [,τ 1 h]w κ, denoted by β in [37], eq. 2.2 is exponentially distributed with mean 2κ 1 sinhκh/2e κh/2 [37], eq So for large t, P A 2 = P[ inf[,τ 1 h t]w κ > bt] = exp [ btκ/2sinhκh t /2e κht/2 ] e 2φt. Now, assume we are ona A 1 A 2. Due to 2.3.6, we have for every x < τ 1h t, since < Aτ 1h t Ax A kt, L[Hτ 1h t,x] e Wκx 2e[Aτ 1h t Ax] { at+4loglog [ ekt/[aτ 1h t Ax] ]} We now introduce f i := inf{u, W κ u i} = τ Wκ i, i N, and let x < τ 1h t. There exists i N such that f i x < f i+1. Moreover, we are on A 2, so i bt. Furthermore, x < f i+1, so W κ x i + 1 and then e Wκx e i+1 = e Wκf i+1. All this leads to e Wκx [Aτ1h t Ax ] τ = e Wκx 1 h t e Wκu du 42 e x τ 1 h t f i e Wκu Wκf i du

54 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES To bound this, we introduce the event A 3 := bt i= { τ 1 h t f i } e Wκu Wκfi du e 1 κht btn t e κδφt. We now consider τ 1u,h t := inf{y u, W κ y inf [u,y] W κ h t } τ 1h t for u. We have τ 1 h t τ E e Wκu Wκfi 1 f i,h t du E e Wκu Wκfi du = β h t, f i f i by the strong Markov property applied at stopping time f i, where we define β h := τ E 1 h e Wκu du. By , β h C + e 1 κh for large h. Hence for large t by Markov inequality, P bt A 3 i= τ 1 h t P e Wκu Wκfi du > e 1 κht btn t e κδφt f i [bt+1]β h t e 1 κht btn t e κδφt C + n t e κδφt. Now, on 3 j=a j, and lead to L[Hτ 1h t,x] 2e 2+1 κht btn t e κδφt{ at+4loglog [ ekt/[aτ 1h t Ax] ]} We now consider only x m 1h t. By definition of A 2, inf [,τ 1 h t]w κ bt, such that Aτ 1h t Ax = τ 1 h t x τ 1 h t e Wκu du m 1 ht e Wκu du e bt [τ 1h t m 1h t ] e bt on the event 4 i=a i with A 4 := {τ 1h t m 1h t 1}. Since m 1 = m 1h t and τ 1 h t = τ 1h t on {M } by definition of h t -extrema, we have P A 4 P < M < m 1 +P[τ 1 h t m 1 < 1] C + h t e κht +P [ τ W κ h t τ W κ h t /2 < 1 ] C + h t e κht +C + exp[ c h 2 t] 43

55 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES due to [3], eq. 2.8, coming from [37], Fact ii and Now, we have ekt/[aτ 1h t Ax] ekte bt on 4 i=a i, and then, on this event, leads to L[Hτ 1h t,x] 2e 2+1 κht btn t e κδφt{ at+4loglog [ ekte bt]}. C + tφte [κ1+δ 1]φt e κδφt h t, since φt = ologt, h t = logt φt and n t = e κ1+δφt. We notice that for large t, C + φth t e κδφt since loglogt = oφt. Hence, for large t, L[Hτ 1h t,x] te [κ1+3δ 1]φt, on 4 i=a i for every x m 1h t. This gives for large t, P sup 1ht]L[Hτ 1h x [,m t,x] te [κ1+3δ 1]φt P 4 C + i=a i 1 n t e κδφt, due to the previous bounds for P A i, i 4. This proves the lemma. With the help of the previous lemma, we can now prove Lemma Proof of Lemma : The method is to do a coupling, similarly as in the proof of Lemma 3.7 of [3]. Recall the definition of L i < L i < L i+1 just above Also, let τ i+1h t := inf { u L i, W κ u inf [ L i,u] W κ h t } τi+1 h t, i 1, m i+1h t := inf { u L i, W κ u = inf [ L i, τ κ} i+1 ht]w, i 1, A 5 := nt 1 i=1 { τ i+1 h t = τ i+1 h t }, X i u := X u+h Li, X i u := X u+h L i, u, i Let i 1. By the strong Markov property, X i and X i are diffusions in the potential W κ, starting respectively from L i and L i. We denote respectively by L Xi, L X i, H Xi and H X i the local times and hitting times of X i and X i. We have for every x L i, LH m i+1,x LH L i,x LH m i+1,x LH L i,x = L X HX i i m i+1,x. { } Consequently, on A 5 A 6 with A 6 := nt 1 j=1 HXj m j+1 < H Xj L j, for 1 i n t 1, sup LH m i+1,x L H Li,x x R = sup LH m i+1,x L H Li,x L i x m i+1 sup L X HX i i m i+1,x L i x m i+1 sup L X HX i i τ i+1h t,x, L i x m i+1 44

56 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES since m i+1 = m i+1 τ i+1 h t = τ i+1h t on A 5. Now, notice that the right hand side of is the supremum of the local times of X i L i, up to its first hitting time of τ i+1h t L i, over all locations in [, m i+1 L i]. Since X i L i is a diffusion in the potential W κ L i +x W κ L i, x R, which has on [,+ the same law as W κ x, x because L i is a stopping time for W κ, the right hand side of has the same law, under the annealed probability P, as sup x [,m 1 h t]l[hτ 1h t,x]. Consequently, P nt 1 i=1 { sup x R } LH m i+1,x LH L i,x > te [κ1+3δ 1]φt n t [P sup 1ht]L [ x [,m Hτ 1h t,x ] > te +P ] [κ1+3δ 1]φt A 5 +P A6 C + e κδφt by Lemma 2.3.3, since P A 5 C+ n t h t e κht by 2.6.9, P A 6 P B2 n t C 3 n t e δκht by and since φt = ologt. Notice that, as before, m 1 = m 1 = m 1h t on V t {M }. Finally, P sup LH m 1,x > te [κ1+3δ 1]φt x [, m 1 ] C + e κδφt +P V t +P < M < m 1 C + e κδφt also by Lemma 2.3.3, Lemma 2.2.1, and since P < M < m 1 C + h t e κht due to This and prove the lemma Local time inside the valley [ L j, L j ] but far from mj We introduce for t > and j 1, r t := C φt, D j := [ m j r t, m j +r t ], where C > is a constant that can be chosen as large as needed. We also define { m B 4 m := sup L H L j,x L H m j,x } < te 2φt x D j [ τ j h+ t, L j ] j=1 for m 1, where D j is the complementary of D j. Moreover, we recall that L j = τ j h+ t. Lemma There exists C 6 > such that if C is large enough, for large t, P [ B 4 n t ] 1 C 6 n t e 2φt. 45

57 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES Proof : Let j [1,n t ]. Throughout the rest of the paper, for y R, we denote by P Wκ y the law of X starting from y instead of, conditionally on W κ. As we are interested in the local time at x after X reaches m j we work under P Wκ m j. So first, thanks to and , under P Wκ m j, there exists a Brownian motion Bs, s, independent of Ṽ j, such that L [ H L j,x ] = e Ṽ j x L B [τ B A j L j,a j x], x R, where A j x := x m j eṽ j s ds. Let B j. := B j A j L j 2./A j L j. By scaling, and because B is independent from W κ, we notice that conditionally to W κ, Bj is a standard Brownian motion. Therefore, even if W κ appears in the expression of B j, B j is probabilistically independent of W κ. We still denote it by B in the sequel to simplify the notation. With this notation, we have L [ H L j,x ] = e Ṽ j x A j L j L B [τ B 1,A j x/a j L j ], x R In order to bound the factors L B [ τ B 1,. ] and A j L j in , we first introduce A 1 := { sup u R L B [τ B 1,u] e 2φt}, A 2 := { A j L j 2e ht+2φt/κ} We have P A 1 5e 2φt for large t by Lemma eq and Moreover on V t, we have by Remark and Fact ii and iii, A j Lj [ τj h t m j ] e h t + = [ τ j h t m j ] e h t + Lj eṽ js ds τ j h t Lj τ j h t L = e ht τ W κ h t +G + h t /2,h t, e V j s ds where W κ has law BES3,κ/2 and is independent of G + h t /2,h t, which is defined in 2.1.8, and with L j = inf{s > τ j h t,ṽ j s = h t /2} as defined in 2.2.4, and L j := inf{s > τ j h t,v j s = h t /2}. Consequently, P A 2 P τ W κ h t > e 2φt/κ +P G + h t /2,h t > e ht+2φt/κ +P V t C + e 2φt for large t by Lemma eq , Lemma eq and Lemma 2.2.1, and since φt = ologt and loglogt = oφt. Now, we would like to bound the factor e Ṽ j x that appears in To this aim, let A 3 := { τ j [κc φt/8] m j +C φt }, { } A 4 := inf V j κc φt/16, [τ j [κc φt/8],τ j h t] 46

58 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES with τ j and τ j defined in and 2.2.6, and τ j and τ j in and First, using , P A 3 C+ e [κ2 C φt]/16 2 e 2φt if we choose C large enough. Moreover Fact together with applied with h = C φt, α = κ/8, γ = κ/16 and ω = h t /C φt, see also the remark at the end of Lemma give P A 4 2e κ 2 C φt/16 e 2φt for large t. We notice that inf [ mj +C φt, τ j h t]ṽj κc φt/16 on A 3 A 4 V t, since τ j = τ j and V j = Ṽ j on V t thanks to Remark We prove similarly that P A 5 C+ e κ2 C φt/16 2 +P V t 2e 2φt, where A 5 := A 6 := { inf [ τ j ht, m j C φt] } Ṽ j κc φt/16, { } inf Ṽ j h t /2. [ τ j h+ t, τ j ht] Also by 2.6.1, P A 6 e κh t/8. We also know that Ṽ j x h t /2 κc φt/16 for all τ j h t x L j by definition of L j, uniformly for large t. Consequently on 6 i=3a i V t, for all x D j [ τ j h+ t, L j ], we have e Ṽ j x e κc φt/16. Hence on 6 i=1a i V t, we have under P Wκ m j, by and , sup L [ H L j,x ] 2te 1+2/κφt e κcφt/16 < te 2φt, x D j [ τ j h+ t, L j ] if we choosec large enough. So, conditioning byw κ and applying the strong Markov property at time H m j, we get [ [ P sup L H Lj,x ] L [ H m j,x ] ] < te 2φt x D j [ τ j h+ t, L j ] E P Wκ m j 6 i=1 A i V t 1 C+ e 2φt uniformly for large t due to the previous estimates and thanks to Lemma This proves the lemma Approximation of the main contributions In this section we give an approximation of the exit time of eachh t -valley[ L j, L j ] and of the local time at the bottom m j of this h t -valley for every 1 j n t. More precisely, we make a link between the family U j := H L j H m j,lh L j, m j,1 j n t, and the i.i.d. sequence Hj,l j,1 j n t described in the introduction. In the following, F + 1 h t, G + h t /2,h t, F 2 h t /2 and F 3 h t /2 denote independent r.v. with law respectively F + h t, G + h t /2,h t, F h t /2 and F h t /2, defined in and

59 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES Proposition For δ > small enough recall that δ appears in the definitions of n t and h + t, there exist d 1 = d 1 δ,κ > and D 1 d 1 > such that for large t, possibly on an enlarged probability space, there exists a sequence S j,r j,e j, 1 j n t of i.i.d. random variables depending on t, with S j, R j and e j independent for every j and S j L = F + 1 h t +G + h t /2,h t, R j L = F 2 h t /2+F 3 h t /2 and e j L = E1/2 exponential variable with mean 2 such that { Uj P nt j=1 H j εt H j, } LH Lj, m j l j εt l j 1 e D 1h t, where l j := S j e j, H j := R j l j and ε t := e d 1h t. The proof of the above proposition, which is in the spirit of the proofs of Propositions 3.4 and 4.4 in [3], makes use of the following lemma : Lemma For δ > small enough, there exist constants d > and D >, possibly depending on κ and δ, such that the two following statements are true for t > large enough. i There exists a sequence e j, 1 j n t of i.i.d. random variables with exponential law of mean 2 and independent of W κ, such that nt P j=1 { U j H j e d h t Hj, LH L j, m j = L j } 1 e D h t, where L j := e j Lj m j eṽ j x dx, Rj := τ j h t/2 τ j ht/2e Ṽ j x dx and H j := L j Rj for all 1 j n t. Moreover the random variables L j, H j, 1 j n t, are i.i.d. ii Possibly on an enlarged probability space, there exist random variables R j and S j, 1 j n t, such that all the random variables R j, S j, e j, 1 j n t are independent, with S j L = F + 1 h t +G + h t /2,h t, and R j L = F 2 h t /2+F 3 h t /2 for every 1 j n t, such that P nt j=1 { } eṽ jx dx S j m j e d h t S j, Rj = R j 1 e D h t, Lj Proof of Lemma : We start with i. Recall that m j < L j < m j+1 for every j 1, e.g. by By the strong Markov property applied under P Wκ at stopping times H m j, the random variables U j,l[h L j, m j ], 1 j n t, are independent under P Wκ. By the same Markov property and formulas and , the sequence U j,l[h L j, m j ], 1 j n t is equal to the sequence 48

60 2.3. CONTRIBUTIONS FOR HITTING AND LOCAL TIMES H j L j, L j [H j L j, m j ], 1 j n t, where H j L j := Lj e Ṽ j u L B j[ τ B j A j L j,a j u ] du, L j [H j L j, m j ] = L B j[ τ B j A j L j, ], A j u := u m j eṽ j x dx, u R, with B j,1 j n t a sequence of independent standard Brownian motions independent of W κ, such that B j starts at A j m j = and is killed when it first hits A j L j. Recall that L B j { denotes the local time of B j. Define A j := maxu< L L [ B j j τ B j A j L j,a j u ] = },1 j n t. By 2.6.6, there existsc > possibly depending on κ and δ such that P nt j=1 A j 1 e c h t for large t. So for large t, where h j := Lj L j nt P j=1 { H j L j = h j } 1 e c h t, e Ṽ j u L B j[ τ B j A j L j,a j u ] du, 1 j n t. We also notice that for every 1 j n t, hj,l j [H j L j, m j ] is measurable with respect to the σ-field generated by Ṽ j x + L + j 1, x < L + j L + j 1 and Bj, where by and 2.2.8, L+ j 1 < L j < m j < L j < L + j. Hence, the random variables hj,l j [H j L j, m j ], 1 j n t are i.i.d under P by the second fact of Lemma For the same reason, Rj,A j L j, 1 j n t are also i.i.d. For 1 j n t, let B j. := B j A j L j 2. /A j L j. Notice that L B j[ τ B j A j L j,a j u ] [ = A j L j L Bj τ Bj1,A j u/a j L j ], L j u L j Moreover by scaling and because B j is independent from W κ, Bj is, conditionally to W κ, a standard Brownian motion starting from and killed when it first hits 1. Furthermore, even if W κ appears in the expression of B j, Bj is independent of W κ. Then, let e j := L Bj [ τ Bj 1, ] = L B j[ τ B j A j Lj ], /A j Lj Notice that by the first Ray-Knight theorem, e j is exponentially distributed with mean 2. Since B j is independent of W κ, e j is also independent of W κ. Also, the sequence e j, 1 j n t is i.i.d. because the B j are independent and the Rj,A j L j are i.i.d., so L j, H j, 1 j n t, are also i.i.d. Moreover, leads to L j [ Hj Lj, mj ] = A j Lj L Bj [ τ Bj 1, ] = A j Lj ej = L j

61 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY Now, for small ε >, thanks to Lemma 2.6.3, we have for large t, P nt j=1 { hj A j L j e j Rj 2e 1 3εh t/6 A j L j e j Rj } 1 C +n t e c εh t 1 C + e c /2εh t, since n t = e o1ht. Finally, this, together with 2.3.2, and the equality of sequences at the start of this proof show for some D > and d >. So i is proved. We now prove ii. The r.v. Ãj L j = Lj eṽ jx m j dx and R j are not independent, so we want to replace them by r.v. having better independence properties. Applying Lemma with subscript 2 replaced by j for 1 j n t gives the existence of R j and S j, independent and independent of e j, having the law claimed in ii and satisfying with 2 replaced by j. This gives since n t = e o1ht. The fact that we can build these R j and S j with the claimed independence properties follows from the fact that e j, R j,ãj L j,1 j n t are i.i.d. Proof of Proposition : The existence and the law of the e j come from Lemma i. The existence and the law of the R j and S j, and the independence of R j, S j, e j, 1 j n t come from Lemma ii. Moreover, by Lemma i and ii, there exist d 1 > and D 1 > such that for large t, { Uj P nt j=1 e j S j R j εt e j S j R j, } L H Lj, mj ej S j εt e j S j 1 e D 1h t, which proves So Proposition is proved. 2.4 Convergence toward the Lévy process Y 1,Y 2 and continuity Preliminaries We begin this section by the convergence of certain repartition functions. These key results are in the same spirit as the second part of Lemma 5.1 in [3]. Lemma Recall from Proposition that l 1 := e 1 S 1 and H 1 := e 1 S 1 R 1. Then for any ε,1/3, lim sup x κ e κφt P l 1 /t > x C 2 =, t + x [e 1 2εφt,+ [ lim sup t + y [e 1 3εφt,+ [ y κ e κφt P H 1 /t > y C 2 E [ R κ κ] =,

62 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY with C 2 a positive constant see below Moreover, for any α >, e κφt Pl 1 /t x,h 1 /t y converges uniformly when t goes to infinity on [α,+ [ [α,+ [ to ν[x,+ [ [y,+ [, where ν is defined in Proof : Let ε,1/3. Proof of : We first prove that, as t +, x κ e κφt PS 1 /t > x converges uniformly in x [ e 1 εφt,+ [ to a constant c, that is, we prove that lim t + lim t + sup x [e 1 εφt,+ [ sup y [1,+ [ x κ e κφt PS 1 /t > x c = For that, with the change of variables y = e 1 εφt x, we just have to prove that y κ e κεφt P S 1 /e ht+εφt > y c =, but this is equivalent to prove that for any function f : ],+ [ [1,+ [, lim t + ftκ e κεφt P S 1 /e ht+εφt > ft = c First by definition see Proposition 2.3.5, S 1 can be written as the sum of two independent random variables, that we denote by F + 1 h t and G + h t /2,h t for simplicity. That is, S 1 /t = F + 1 h t +G + h t /2,h t /t = e φt e ht F + 1 h t +e ht G + h t /2,h t Since we know the asymptotic behavior of the Laplace transforms of F + h t /e ht and G + h t /2,h t /e ht, the proof of is similar to the proof of a Tauberian theorem. First by and we have, using the independence of F 1 + h t and G + h t /2,h t, 1 [ ] γ >, ω f,t γ := 1 E e γs 1/fte h t +εφt γ t + c γ κ 1 ft κ e κεφt, where c = Γ1 κ2 κ /Γ1+κ. Note that by Fubini, ω f,t is the Laplace transform of the measure du f,t z := 1 R+ zp S 1 /fte ht+εφt > z dz, that is, ω f,t γ = e γz du f,t z. From 2.4.7, we have γ >, ω f,t γ ω f,t 1 t + γκ 1. We can now follow the same line as in the proof of a classical Tauberian theorem, making the link between a Laplace transform and the repartition function, see for example [39] volume 2, section XIII.5, Theorem 1, page 442, we can deduce that z >, U f,t [,z] ω f,t 1 51 t + z 1 κ Γ2 κ.

63 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY Then, e.g. as in the proof of Theorem 4 of the same reference page 446, or using inequalities similar to those at the end of the proof of Lemma 5.1 in [3], we deduce from the monotony of the densities of measures U f,t that P S 1 /fte ht+εφt > z 1 κ z >, ω f,t 1 t + z κ Γ2 κ. Considering this convergence withz = 1 we get exactly2.4.5 forc = c 1 κ/γ2 κ = 2 κ /Γ1+κ, so follows. Now, let a t := e εφt. For any x >, at x κ e κφt Pe 1 S 1 /t > x, e 1 < a t = 2 1 x/u κ e κφt PS 1 /t > x/uu κ e u/2 du, because e 1 has law E1/2 and is independent of S 1. Taking x arbitrary in [e 1 2εφt,+ [, we have x/u [e 1 εφt,+ [ for every u ],a t ], so thanks to we get lim t + sup x [e 1 2εφt,+ [ xκ e κφt Pe 1 S 1 /t > x, e 1 < a t c 2 + u κ e u/2du = Now for t large enough such that y 1, y κ e κφt PS 1 /t > y < 2c see 2.4.3, we have for any x >, x κ e κφt P e 1 S 1 /t > x, e 1 < a t x κ e κφt P e 1 S 1 /t > x = x κ e κφt P e 1 S 1 /t > x, e 1 a t + = 2 1 x κ e κφt PS 1 /t > x/ue u/2 du a t + = 2 1 u κ x/u κ e κφt PS 1 /t > x/u 1 x u e u/2 du a t u κ x/u κ e κφt PS 1 /t > x/u 1 x>u e u/2 du a t e κφt u κ e u/2 du+c u κ e u/2 du a t a t For the second term in the inequality we used the fact that x/u κ e κφt PS 1 /t > x/u < 2c when x u. Since a t = e εφt, the right hand side of converges to when t goes to infinity. Combining this with 2.4.8, we get + xκ e κφt Pe 1 S 1 /t > x 2 1 c u κ e u/2 du =, lim t + sup x [e 1 2εφt,+ [ and this is exactly with C 2 := 2 1 c + u κ e u/2 du = 2 κ Γκ+1c = 4 κ

64 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY Proof of : Let µ R1 be the distribution of R 1. For any y >, a > and t >, we have by independence of e 1 S 1 and R 1, y κ e κφt Pe 1 S 1 R 1 /t > y, R 1 < a = a y/u κ e κφt Pe 1 S 1 /t > y/uu κ µ R1 du. Taking a = a t = e εφt and y arbitrary in [e 1 3εφt,+ [, we have y/u [e 1 2εφt,+ [ for all u ],a t ], so thanks to 2.4.1, we get lim t + =, sup y [e 1 3εφt,+ [ yκ e κφt P at e 1 S 1 R 1 /t > y, R 1 < a t C2 u κ µ R1 du κh t/2 κh t/2 where we used u κ µ R1 du = E[R 1 κ ] E[R κ κ ] <, as explained in the following lines. By definition see before Proposition and and with W κ an independent copy ofwκ,r 1 is equal in law to τ W e W κx dx+ τ W e W κx dx, which itself converges almost surely to R κ defined in when t goes to infinity. This also shows that for each t, R 1 is stochastically inferior to R κ, which admits finite moments of any positive order by Lemma In particular the family R 1 t> is bounded in all L p spaces, and more precisely, E[R 1 p ] E[R κ p ] < for every p R +. So by the dominated convergence theorem, + u κ µ R1 du converges to E[R κ κ ] when t goes to infinity. Hence, lim t + sup y [e 1 3εφt,+ [ y κ e κφt P e 1 S 1 R 1 /t > y, R 1 < a t C2 E [ R κ κ] =. Finally, as the familyr 1 t> is bounded in alll p spaces,e κφt a t u κ µ R1 du converges to as t +. So we can proceed as before as in 2.4.9, integrating with respect to R 1 instead of e 1 and using instead of to remove the event R 1 < a t and we thus get lim t + sup y [e 1 3εφt,+ [ y κ e κφt P e 1 S 1 R 1 /t > y C 2 E [ R κ κ] =, which is

65 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY We now prove the last assertion. For any x >, y >, a > and t >, we have = = e κφt Pe 1 S 1 /t > x, e 1 S 1 R 1 /t > y, R 1 < a a e κφt Pe 1 S 1 /t > x, e 1 S 1 /t > y/uµ R1 du a y/x e κφt Pe 1 S 1 /t > y/uµ R1 du a + e κφt Pe 1 S 1 /t > xµ R1 du, a y/x = 1 a y/x e κφt y/u κ Pe y κ 1 S 1 /t > y/uu κ µ R1 du + 1 x κ a a y/x e κφt x κ Pe 1 S 1 /t > xµ R1 du. Taking a = a t = e εφt and x, y arbitrary in [α,+ [ for some α >, we have y/u,x [e 1 2εφt,+ [ 2, u ],a t ] whenever t is large enough, so, thanks to we get thate κφt Pe 1 S 1 /t > x,e 1 S 1 R 1 /t > y, R 1 < a t converges uniformly in x,y [α,+ [ [α,+ [ toward C 2 x κ P R κ > y/x +C 2 y κ E R κ κ 1 Rκ y/x = ν [x,+ [ [y,+ [. Then as before we can remove the event {R 1 < a t } since e κφt PR 1 a t as t + because the family R 1 t> is bounded in all L p spaces, which gives the last assertion of Lemma Proof of Proposition We start with the finite dimensional convergence. We recall that Y 1,Y 2 t s is defined just before Proposition 2.1.4, and Y 1,Y 2 before We sometimes use the notation Y 1,Y 2 s = Y 1 s,y 2 s and Y 1,Y 2 t s = Y1s,Y t 2s. t Lemma For any k N and s i >,i k, Y 1,Y 2 t s i,i k converges in law as t goes to infinity to Y 1,Y 2 si,i k. Proof : The proof is basic here, however we give some details as we deal with a two dimensional walk which increments depend on t itself. As Y1s t and Y2s t are sums of i.i.d sequences we only have to prove the convergence in law for the couple Y 1,Y 2 t s for any s >. For b, we define Y 1 >b,y 2 >b, obtained from Y1,Y 2 t by keeping only the increments larger than b, that is, Y >b and Y >b 2 s := 1 t Y t i s Y >b se κφt j=1 l j 1 lj /t>b 1 s := 1 t se κφt j=1 H j 1 Hj /t>b for every s and t >. Also let Y b i s := i s for i {1,2}. We first prove that for any s >, lim limsup P Y ε 1,Y ε 2 t s > ε 1 κ2 κ =, ε t + 54

66 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY where for any a = a 1,a 2 R 2, a := max a 1, a 2, with Y ε 1,Y ε 2 t s = Y ε 1 s,y ε 2 s and 1 κ2 κ > since κ < 1. Let ε > and s >. We now give an upper bound for the first moments of Y ε 1 s and Y ε 2 s. Let η > be such that κ 1 3η <. Applying Fubini, we have for large t, e κφt l1 E t 1 l 1 /t ε [ ] = e κφt e1 S 1 E 1 e1 S t 1 /t ε = ε e κφt Pe 1 S 1 /t > x dx e 1 2ηφt e κ 1 2ηφt + e κφt Pe 1 S 1 /t > x dx+ ε ε e 1 2ηφt e κφt Pe 1 S 1 /t > x dx e 1 2ηφt x κ x κ e κφt Pe 1 S 1 /t > x dx The first term in converges towhentgoes to infinity becauseκ 1 2η < η <. Moreover, according to 2.4.1, for t large enough, we have x e 1 2ηφt, x κ e κφt Pe 1 S 1 /t > x 2C 2. For such t, the second term in is less than So, we get for large t, e κφt E 2C 2 ε x κ dx = 2C 2 ε 1 κ 1 κ. l1 t 1 l 1 /t ε e κ 1 2ηφt +C + ε 1 κ Using the same method and applying this time 2.4.2, we get for large t, e κφt H1 E t 1 H 1 /t ε e κ 1 3ηφt +C + ε 1 κ We thus obtain E Y ε 1 s se κ 1 2ηφt +C + sε 1 κ, E Y ε 2 s se κ 1 3ηφt +C + sε 1 κ, then a Markov inequality leads to since κ 1 3η <. The next step is to prove that Y 1 >ε,y 2 >ε t s can be written as the integral of a point process which converges to the desired limit. We have Y >ε 1,Y 2 >ε t = Y >ε s 1 s,y 2 >ε s s s = xptdx, 1 dv, xptdx, 2 dv x>ε 55 x>ε

67 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY where the measurespt 1 andpt 2 are defined bypt 1 := + i=1 δ t 1 l i,e κφt i and similarly Pt 2 := + i=1 δ t 1 H i,e i. Recall that P 1 κφt t and Pt 2 are not independent. We now prove that Pt,P 1 t 2 converges to a Poisson point measure. For that just use Lemma together with Proposition 3.1 in [56] after discretization, it implies thatpt,p 1 t 2 converges weakly to the Poisson random measure denoted by P 1,P 2 with intensity measure given by ds ν. Then using that for any ε >, and T < +, on [,T ε,+ ε,+ ds ν is finite, we have that Y 1 >ε,y 2 >ε t s converges weakly to s s Y 1 >ε,y 2 >ε s := xp 1 dx, dv, xp 2 dx, dv. x>ε We are left to prove that Y 1 >ε,y 2 >ε converges to Y 1,Y 2 when ε. This is a straightforward computation, that we detail for completeness. Let ν 1 [x,+ [ := ν[x,+ [ R + = C 2 /x κ, we have E x ε s xp 1 dx, dv x>ε = s xν 1 dx = Cε 1 κ, x ε s xp1 dx, dv Then a Markov inequality proves that for any s >, the process x ε converges to zero when ε goes to zero in probability. The same is true for s x ε xp2 dx, dv, so we obtain thaty 1 >ε,y 2 >ε s converges in probability toy 1,Y 2 s when ε. We now prove the tightness of DY 1,Y 2 t t, the family of measures induced by processes Y 1,Y 2 t. Lemma The family of laws DY 1,Y 2 t t is tight on D[,+,R 2,J 1. Proof : We only have to prove that the family law of the restriction of the process to the interval [,T], Y 1,Y 2 t [,T] t is tight. To prove this we use the following restatement of Theorem 1.8 in [13] using Aldous s tightness criterion see Condition 1, and equation page 176 in [13] also used in [16] page 1. We have to check the two following statements : 1 for any ε >, there exists a such that for any t large enough, P sup s [,T] Y 1,Y 2 t s a ε. 2 for any ε >, and η > there exists δ, < δ < T and t > such that for t > t, P [ ωy 1,Y 2 t,δ,t η ] ε, with ωy 1,Y 2 t,δ,t := sup r T ωy 1,Y 2 t,δ,t,r, and ωy 1,Y 2 t,δ,t,r := sup r δ u 1 <u<u 2 r+δ T { min Y1,Y 2 t u 2 Y 1,Y 2 t u, Y1,Y 2 t u Y 1,Y 2 t u 1 }. 56

68 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY Also PvY 1,Y 2 t,,δ,t η ε, and PvY 1,Y 2 t,t,δ,t η ε, where vy 1,Y 2 t,u,δ,t := sup u δ u1 u 2 u+δ T{ Y 1,Y 2 t u 1 Y 1,Y 2 t u 2 }. We first check 1 since the process is monotone increasing, P sup Y 1,Y 2 t s a = P Y 1,Y 2 t T a PY 1 T a+py 2 T a. s [,T] Recall that Y 1 >b is obtained from Y 1 where we remove the increments l j /t smaller than b and Y b 1 = Y 1 Y 1 >b. Define N u >b := ue κφt i=1 1 lj /t>b. Let < δ 1 < 1. A Markov inequality yields P Y t 1T a P Y 1 1 T a 2 +P Y >1 2 a E[ Y 1 1 T ] + 1 a E N >1 δ 1 T 1 T a 2 +P Y >1 1 T a 2,N>1 T a δ 1 On {N T >1 a δ 1 } there is at most a δ 1 terms in the sum Y 1 >1 T so P Y 1 >1 T > a/2,n T >1 a δ 1 P l i /t a 1 δ 1 /2 l i /t 1 1 i a δ 1 a δ 1 P l 1 /t a 1 δ 1 /2 l 1 /t a δ 1 2 C 2e κφt a κ1 δ 1 2 κ C 2 e κφt = 2 1+κ a δ 1 κ1 δ 1, for all t large enough thanks to and δ 1 such that δ 1 κ1 δ 1 <. Also, notice that for any b >, N T >b follows a binomial law with parameters Te κφt,pl 1 /t > b. So, using again and , we obtain for t large enough, EN >b T 2C 2 Tb κ, [ ] E Y b 1 T 2C 2 Tb 1 κ Collecting 2.4.2, and , we get the existence of t 1 > such that lim sup PY 1 T a = a + t t 1 The same arguments holds for Y 2 using instead of and instead of so also holds for Y 2 instead of Y 1. We conclude the proof 57

69 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY of 1 by putting and its analogous for Y 2 in We now check 2 We first write, as usual, { ω Y1,Y 2 t,δ,t η } { ω Y b 1,Y b 2 t,δ,t η/2 } { ω t,δ,t } Y 1 >b,y 2 >b η/2. For Y. b, we have P [ ω Y b 1,Y b 2 t,δ,t η/2 ] P [ ω Y b 1,δ,T η/2 ] +P [ ω Y b 2,δ,T η/2 ]. Moreover, by positivity of the increments, P ω Y b 1,δ,T η/2 { P k T/2δ Y b 1 P k T/δ Y b 1 k +12δ Y b 1 k +12δ Y b 1 k2δ η/4 } k2δ η/ For any k, Y b 1 k + 12δ Y b 1 k2δ is the sum of at most 2δe κφt + 1 i.i.d. random variables having the same law as l 1 /t. We get that for any integer k P Y b 1 k +12δ Y b 1 k2δ η/4 P Y b 1 3δ η/4 8C 2 δb 1 κ /η, where the first inequality holds for t large enough so that 2δe κφt 1 and the second from the second expression in replacing T by 2δ. Combining with we get for large t P ωy b 1,δ,T η/2 24C 2 T1+2δb 1 κ /η, [note that δ will be chosen later and will be less than 1]. T and η are fixed so we choose b small enough so that the right hand side of is less than ε/4. A similar estimate can be proved for PωY b 2,δ,T η/2. For Y. >b, we have again PωY >b 1,Y >b 2 t,δ,t η/2 PωY >b 1,δ,T η/2+pωy >b 2,δ,T η/2. Since Y 1 >b is piecewise constant with jumps larger than b, {ωy 1 >b,δ,t > η/2} implies that two jumps larger than b for Y1 t occur in an interval smaller than 2δ. That is {ωy 1 >b,δ,t > η/2} Teκφt j=1 Teκφt {l i>j,i j/e κφt 2δ j l i /t > b}. Applying for t large enough, P Teκφt j=1 Teκφt i>j,i j/e κφt 2δ 58 { } l j l i /t > b 8C2δTb 2 2κ,

70 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY which can be small choosing this time δ = δb properly. Again the same argument can be used for ωy 2 >b,δ,t. To finish the proof, we have to deal with v, as again our processes are increasing, PvY 1,Y 2 t,,δ,t η P Y 1,Y 2 t δ η we can then proceed as for 1 decreasing the value of δ if needed, this also applies to PvY 1,Y 2 t,t,δ,t η. Putting together the two preceding lemmata we obtain Proposition Continuity of some functionals of Y 1,Y 2 in J 1 topology In this section, we study the continuity of some functionals which will be applied later to Y 1,Y 2 t and to the Lévy processes Y 1,Y 2. For our purpose, we are interested in the following mappings. We have already mentioned the first two in the introduction : J : DR +,R DR +,R I : DR +,R,J 1 DR +,R,U f f f f 1 where U denotes uniform convergence on every compact subset of R +. Then we also need the compositions of these two : for any positive a, let J I,a : DR +,R 2 R f = f 1,f 2 f 1 f 1 2 a, J I,a : DR +,R 2 R f = f 1,f 2 f 1 f 1 2 a, J I,a respectively J I,a produces the largest jump of f 1, between and the time just after respectively before f 2 first reaches a,+. We also define K I,a, K I,a, KI,a and K I,a as follows. K I,a : DR +,R 2 R f = f 1,f 2 f 1 f 1 2 a, K I,a : DR +,R 2 R f = f 1,f 2 f 1 f 1 2 a, K I,a : DR +,R 2 R f = f 1,f 2 f 2 f 1 2 a, K I,a : DR +,R 2 R f = f 1,f 2 f 2 f 1 2 a. Finally, with f 1 s := f 1 s f 1 s, define F by F : DR +,R 2 R f = f 1,f 2 inf { s,f2 1 1, f 1 s = f 1 f }. We need this functional F for the characterization of the favorite site

71 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY Lemma J is continuous in the J 1 topology. Proof : This fact is basic. However, we have not found a proof in the literature, so we give some details. To prove the continuity on DR +,R, we only have to prove it for every compact subset of R +, see [75] Theorem So let f DR +,R and T > at which f is continuous, let us prove that J T defined by J T : D[,T],R D[,T],R g g is continuous at the restriction f [,T]. Let ε > and g D[,T],R such that d T f [,T],g ε 2. d T is the usual metric d of the J 1 -topology restricted to the interval [,T]. By definition of d T there exists a strictly increasing continuous mapping of [,T] onto itself, e : [,T] [,T] such that sup es s ε s [,T] 2 and sup ges f [,T] s ε s [,T] 2. So for every s [,T] we have ges f [,T] s = ges ges f [,T] s f [,T] s ges f [,T] s + ges f [,T] s 2 ε 2 = ε, where hs = hs hs. This implies d T JT f [,T],JT g ε. Lemma Fix a >. The mappings J I,a, J I,a, K I,a, K I,a, K I,a and K I,a are continuous for J 1 -topology at every couple f 1,f 2 DR +,R 2 such that i. For any ε >, f 1 and f 2 have a finite number of jumps greater than ε on every compact subset of R +, ii. f 2 is strictly increasing, with a limit equal to +, iii. f 2 =, iv. f 2 has a jump at If 2 a and f 2 If 2 a < a < f 2 If 2 a. Proof : This fact may also be known as we are looking at randomly stopped process, but once again we did not find what we need in the literature [62], [75]. Let f 1 n,f 2 n n be a sequence of elements of DR +,R which converges to f 1,f 2 for the J 1 topology. To prove continuity, we prove that the sequence J I,a f1 n,f 2 n n converges to J I,a f1,f 2, and the equivalent for J I,a. 6

72 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY The first hypothesis guaranties that there exist neighborhoods of If 2 a for which f 1 makes no jump greater than 1/4 times its higher previous jump, that is to say there exists δ ],If 2 a[ notice that If 2 a exists tanks to 2 and is positive thanks to 3 such that f 1 makes no jump greater than Jf 1 If 2 a δ/4 on [If 2 a δ,if 2 a[ and on]if 2 a,if 2 a+δ]. Note also thatjf 1 is constant on [If 2 a δ,if 2 a[ and on ]If 2 a,if 2 a+δ]. Also δ can be made smaller if needed in such a way that If 2 a+δ is a point of continuity of f 1,f 2 and f 1 n,f 2 n n for every n N. By hypothesis df 1 n,f 2 n,f 1,f 2 n + so d n := d [,If 2 a+δ] f 1 n,f 2 n [,If 2 a+δ],f 1,f 2 [,If 2 a+δ] n +, where [,If 2 a+δ] in index means restriction to [,If 2 a+δ]. Also by continuity of J see Lemma we also have djfn,jf 1 1 n + and therefore Jf d 1 n := d [,If 2 a+δ] n, Jf 1 [,If 2 a+δ] [,If 2 a+δ] n +. Let h respectively h + be the largest jump of f 1 just before resp. just after If 2 a. By definition of δ we have h = Jf 1 If 2 a δ,h + = Jf 1 If 2 a+δ. We have two cases, either Jf 1 is continuous at If 2 a or it makes a jump. Case Jf 1 makes a jump, in this case the size of the jump is h + h >. Let α = 8 1 minh,δ,1 f 2 If 2 a,f 2 If 2 a 1, and n N be such that for any n n, d n < α and d n < α. T= If 2 a+δ, there exist two homeomorphisms e n,e n : [,T] [,T] such that : sup s [,T] e n s s d n, sup s [,T] f 1 n e n s,fne 2 n s [,If 2 a+δ] f 1 s,f 2 s d [,If 2 a+δ] n. sup s [,T] e ns s d n, Jf 1 sup s [,T] n [,If 2 a+δ] e ns Jf 1 [,If 2 a+δ] s d n. The second inequality implies that for any n n, fn 2 en If 2 a < a < fn 2 en If 2 a, so as we also have f 2 nif 2 na a f 2 nif 2 na we get If 2 na = e n If 2 a The fourth point implies that for any n n, Jfn 1 e n If 2 a 12 δ Jf 1 If 2 a 12 δ α = h α > 1 2 h

73 2.4. CONVERGENCE TOWARD THE LÉVY PROCESS Y 1,Y 2 AND CONTINUITY The second point and the argument of the previous proof imply that for any n n, each jump of f 1 n on [e n If 2 a δ,e n If 2 a[ is 2α-close to a jump of f 1 on [If 2 a δ,if 2 a[, but such jumps are less than h /4 because of the definition of δ. Thus, f 1 n makes no jump larger than h /2 on the interval [e n If 2 a δ,e n If 2 a. Moreover, the increases of e n and the first and third points imply that e n If 2 a δ e n If 2 a δ/2 e n If 2 a. So, combining this with , we get that Jfn 1 is constant on the interval [e nif 2 a δ/2,e n If 2 a. Now by definition of J I,a, with and then collecting what have just done above yields n n, JI,a f 1 n,fn 2 = Jfn 1 Ifna 2 = Jfn e 1 n If 2 a = Jf 1 n e n If 2 a δ/ From definition of J I,a and the constantness of Jf1 on [If 2 a δ,if 2 a[ we also have J I,a f1,f 2 := Jf 1 If 2 a = Jf 1 If 2 a δ/ Combining , and the fourth point gives that, as n goes to infinity, J I,a f1 n,f 2 n converges to J I,a f1,f 2. For J I,a, we prove in a similar way as above that Jf 1 n is constant on [e n If 2 a, e nif 2 a+δ/2] so, as in we have for n large enough J I,a f 1 n,f 2 n = Jf 1 n e n If 2 a+δ/2, which, combined with the analogous of J I,a f 1,f 2 = Jf 1 If 2 a+δ/2 allows us to conclude, using the fourth point, that J I,a f 1 n,f 2 n converges to J I,a f 1,f 2 as n goes to infinity. Therefore, both J I,a and J I,a are continue at f 1,f 2. The continuity of the other functionals are proved similarly. Lemma For any f 1,f 2 in DR +,R 2 that satisfy the hypothesis of lemma and such that the sizes of the jumps of f 1 are all distinct, F is continuous at f 1,f 2 in the J 1 topology. Proof : The proof follows mainly the steps of Lemma 2.4.5, we keep the same notation. The jump which takes place at the instant F f 1,f 2 has value h. With the additional hypothesis that the values of the jumps for f 1 are all different we have unicity for the value h. Let us define h, the second highest jump f 1 before 62

74 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS instant If 2 1. With the additional condition that α < 1 8 h h we have with the same arguments as in the proof of the continuity of J that for any n n, f 1 n effectuates at e n F f 1,f 2 a jump larger than h 2α, and larger than all the other jumps of f 1 n before e n If 2 1 = If 2 n1 which are smaller than h +2α. So for n n, the largest jump of f 1 before If 2 n1 is obtained for e n F f 1,f 2, that is to say for any n n, F f 1 n,f 2 n = e n F f 1,f 2, this implies F f 1 n,f 2 n n F f 1,f Supremum of the Local time - and other functionals Supremum of the local time proof of Theorem First, notice that since the diffusion X is almost surely transient to the right, the random variable sup x< L+,x is P-almost surely finite. So almost surely, lim sup t + x< Lt,x/t =. As a consequence, we only have to study the asymptotic behavior of sup x Lt,x/t as t +. We start with the proof of the following proposition, which makes a link between the supremum of the local time and the process Y 1,Y 2 t. Proposition Let α >. For any ε > and large t, P1 vε,t P suplt,x/t α P 1 + +vε,t, x where 1 P 1 ± ln 2ε := P[ l t N 2ε t 1 HN 2ε t 1 H N 2ε H t N 2ε t 1 α± t, and with H k := Y2 t ke κφt = 1 k t i=1 H i, lk := Y t k N, N 2ε l j max 1 j Nt 2ε 1 t α± t 1 ke κφt = 1 k t i=1 l i for any t := inf { m 1, H m > 1 2ε }, α t ± := α 1±loglogt 1/2, and v is a positive function such that lim t + vε,t const ε κ 1 κ. The proof of this proposition relies on the three following lemmata. The first one deals with the local time at the h t -minima for which the diffusion X already escaped before time t. The second deals with the local time at the last h t -minimum m Nt in the remaining time before time t. Finally the last one is a technical point. ], 63

75 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS Lemma For any large t >, 2 k n t, any x > and γ > possibly depending on t, define the repartition function F γ x := P max L H Lj k 1, mj γt, U i xt. 1 j k 1 Then for large t, for all 2 k n t, x > and γ >, i=1 F γ x e D 1h t F γ x F + γ x+e D 1h t, where F γ ± x := P max 1 j k 1 l j γ t ± t, k 1 i=1 H i x ± t t x ± t := x1±2ε t, ε t and D 1 are given in Proposition with γ ± t := γ1 ± 2ε t, Lemma For any t >, define for every γ > and < x < 1 possibly depending on t, [ ] f γ x := E P Wκ m 1 L X t1 x, m 1 γt,h L 1 > t1 x,h L 1 < H L 1. For such t, γ and x, we also introduce f γ x := E P Wκ m 1 sup L X [t1 x,y] γt,h L 1 > t1 x,h L 1 < H L 1, y D 1 with D 1 defined in Here X is an independent copy of X starting at m 1, and the definition of H for X is the same as the definition of H for X. Let ε,1/2. There exists c 2 > such that for large t, for every x [ε,1 ε], with f ± γ x := P fγ x ont 1 f γ x f γ x f γ + x+on 1 t, R 1 γ 1 x 1±ε t,h 1 > t1 x1 ε t and ε t = e c 2h t. Lemma For any < a < 1/4, we have for any t >, 1 k n t P [ Hk > 1 a/2, 1 2a < H k 1 1 3a/4 ] sa,t, with sa,t such that lim t + sa,t = const a 1 κ. For any ε,1/2, t >, P[εt Hm Nt 1 εt] 1 sε,t, with sε,t such that lim t + sε,t = const ε 1 κ κ. 64

76 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS We postpone the proof of these lemmata after the proof of Proposition Proof of Proposition : Recall from that N t is the largest index such that sup s t Xs m Nt. In particular, H Lj H mj+1 = Hm j+1 Hm Nt t for every 1 j N t 1 on V t {N t n t }. The main idea is to use the fact that the supremum of the local time at time t is achieved in the neighborhood of the h t -minima m i, 1 i n t. We start with the upper bound. Let α > and < ε < 1/2. Notice that PN t =,V t P[H m 1 > t] C 2 v t by Using 2.3.1, 2.3.2, 2.3.3, 2.5.3, Lemma and Remark 2.2.1, we have for t large enough, P sup x R E [P Wκ + sε,t. Lt,x αt E [P Wκ max Lt,m j αt 1 j N t max 1 j N t 1 L[ H Lj, m j ] αt,lt, mnt αt,q,b 1 n t,b 2 n t,v t ] ] with Q := {εt Hm Nt 1 εt, 1 N t n t } and s satisfying lim t + sε,t C + ε 1 κ κ. We will introduce in what follows different measures denoted by the letter ν ; they depend on k but we do not write k as a subscript to simplify the notation. First, define two measures ν Wκ 1 and ν Wκ 2 on,1 by, for every < y < 1, ν Wκ 1 y :=ν Wκ 1 [,y] :=P Wκ max L[ H Lj ] k 1, mj αt,h mk U i < ṽ t,h m k yt, 1 j k 1 ν Wκ 2 y :=ν Wκ 2 [,y] :=P Wκ m k L X [t1 y, m k ] αt, H m k+1 > t1 y, H m k+1 < H L k i=1, H m k+1 H Lk ṽt, with X a diffusion starting from m k independent of X conditionally on W κ, and H has the same definition as H see but for X. Partitioning on the values of N t, and H m k, we obtain by the strong markov property applied at time H m k under P Wκ, that the probability E [ P Wκ. ] in the line below is smaller than 1 k n t 1 ε ε E ν Wκ 2 xdν Wκ 1 x = [ 1 ε ] E ν Wκ 2 xdν Wκ 1 x k n ε t The next step is to prove that the previous expectation can be approximated by a product of expectations. First notice that both y ν Wκ 1 y and y ν Wκ 2 y are 65

77 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS positive increasing. So integrating by parts 1 ε ε ν Wκ 2 xdν Wκ 1 x = [ ν Wκ 2 xν Wκ 1 x ] 1 ε ε [ ν Wκ 2 xν Wκ 1 x ] 1 ε ε [ ν Wκ 2 x 1 ε ε 1 ε ε ν Wκ 1 xdν Wκ 2 x ν Wκ 1 xdν Wκ ] 1 ε = ν Wκ 1 x ν Wκ 1 x ε with ν Wκ 1 x := P Wκ G 1,H m k k 1 i=1 U i < ṽ t, k 1 i=1 U i +ṽ t xt G 1 := {max 1 j k 1 LH L j, m j αt} and ν Wκ 1 ε 1 ε 2 x I 1 := ν Wκ 2 xd ν 1 x ν Wκ 2 xdν Wκ 3 x =: I 1, ε ε k 1 3 x := P G Wκ 1, U i +ṽ t xt. i=1 +I 1, ν Wκ 1 x and First, we deal with what is going to be a negligible part, that is to say the first term in As ν Wκ 1 x P Wκ G 1,H m k k 1 i=1 U i < ṽ t, k 1 i=1 U i xt because by definition k 1 i=1 U i < H m k, we have, for ε < x < 1 ε, k 1 ν Wκ 1 x ν Wκ 1 x P xt ṽ Wκ t < U i xt =: h k x. so [ ν Wκ 2 x ν Wκ 1 x ν Wκ 1 x ] 1 ε ν Wκ ε 2 1 εh k 1 ε+ν Wκ 2 εh k ε. Notice that k 1 i=1 U i is measurable with respect to σ Xs, s H Lk 1 ;Wκ x,x L k 1 +, since L k 1 L + k 1, whereas the event in the definition of νwκ 2 belongs to σ X s, s min H k m L,H k+1 ;W κ x W κ m k, L + k 1 x L k+1 +, with X an independent copy of X starting at m k. So independence of X and X, and independence of the two portions of the environment involved see Lemma imply independence between ν Wκ 2 and h k. Hence, with for any x, [ ] 1 ε E ν Wκ 2 x ν Wκ 1 x ν Wκ 1 x ε [ ] E ν Wκ 2 1 ε E [ h k 1 ε ] [ +E ν Wκ 2 ε ]E [ h k ε ] [ ] = E ν Wκ 2 1 ε E [ h k 1 ε ] [ ] +E ν Wκ 2 ε E [ h k ε ] i=1 ν Wκ 2 x := P Wκ m 1 L X [t1 x, m 1 ] αt, H m 2 > t1 x, H m 2 < H L 1, H m2 H L1 ṽt. 66

78 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS Wκ As E ν 2 x P[U 1 > t1 x ṽ t ] and for every small ε > and t large enough h k x P Wκ x εt < k 1 i=1 U i xt we can apply Proposition 2.3.5, we get P E [ h k 1 ε ] [ E 1 2ε k 1 H i < 1+ε t t i=1 ] ν Wκ 2 1 ε 1 ε 1 ε t P H 1 > tε ṽ t +3e D 1h t. 1+ε t By and the first part of Lemma 2.6.2, for any < a < 1 and b >, lim P t + 1 k n t Therefore, we obtain 1 a < 1 k n t E k 1 i=1 H i t 1 PH 1 > tb = const b κ 1 [ ] ν Wκ 2 1 ε E [ h k 1 ε ] C + ut,ε 1 a y κ 1 dy const b κ 1 1 a κ with u a positive function such that lim t + ut,ε = maxε 1 κ,ε κ. A similar argument also works for the second term in 2.5.7, which yields 1 k n t E [ [ ν Wκ 2 x ν Wκ 1 x ν Wκ 1 x ] 1 ε ε ] 2C + ut,ε We now deal with I 1. By independence between X and X, and the independent parts of the potential W κ involved in ν Wκ 2 x and ν Wκ 3 x, EI 1 = 1 ε ε ν 2 xdν 3 x, with ν 2 x := E ν Wκ 2 x Wκ = E ν 2 x and ν 3 x := E ν Wκ 3 x. By the lower bound in Lemma 2.5.2, we have ν 3 x = F α x ṽ t /t Fα x ṽ t /t e D 1h t for every x > ε for large t. So, again since y ν 2 y is positive increasing and ν 3 is a repartition function, integrating by parts twice as in gives with the change of variables u = x ṽ t /t, 1 ε ε ν 2 xdν 3 x 1 ε ṽt/t ε ṽ t/t ν 2 x+ṽ t /tdf α x+e D 1h t [ + F α x Fα x ν 2 x+ṽ t /t ] 1 ε ṽt/t ε ṽ t/t

79 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS Recall see before Lemma that ṽ t /t = 2/logh t = o1 as t +. Then we can prove in a similar way we have obtained that : 1 k n t [ ] 1 ε ṽt/t F α x Fα x ν 2 x+ṽ t /t C + ut,ε, ε ṽ t/t with as usual a possibly enlarged C +. Indeed by Lemma 2.5.2, F α ε ṽ t /t Fα ε ṽ t /t ν 2 ε e D 1t = on 1 t for every 1 k n t for large t, and Fα Fα 1 ε ṽt /t F α + Fα 1 ε ṽt /t+e D1t P max 1 j k 1 l j [γt t,γ t + t] +P k 1 i=1 H i [x t t,x + t t] +e D1t for every k n t for large t, with γ = α and x = 1 ε ṽ t /t. The first probability is less than n t PS 1 e 1 [γt t,γ t + t] = n t E γt1+2ε t/s 1 γt1 2ε t/s 1 e u/2 du/2 8n t ε t sup v ve v = o1/n t, whereas the second one is treated as 2.5.8, which leads to So the important term in the right hand side of inequality comes from the integral. We now work on ν 2 x. We have, ν 2 x E P Wκ m 1 [ LX t1 x, m 1 αt, H L1 > t1 x ṽt, H L1 < H L 1 ] E P Wκ m 1 [ LX t1 x ṽ t, m 1 αt, H L1 > t1 x ṽt, H L1 < H L 1 ] = fα x+ṽ t /t, as defined in Lemma Then, as F α x is positive and increasing in x, using Lemma with γ = α, we obtain 1 ε ṽt/t ε ṽ t/t ν 2 x+ṽ t /tdf α x 1 ε ṽt/t ε ṽ t/t f + αx+2ṽ t /tdf α x+on 1 t Now, as f αx+2ṽ + t /t can be written since H k = l k R k, see Proposition 2.3.5, f αx+2ṽ + t /t = P 1 x 2ṽ t /t l k α1+ε H t,h k > t1 x 2ṽ t /t1 ε t, k we get by independence of the random variables l j,h j,j n t, 1 ε ṽt/t ε ṽ t/t f + αx+2ṽ t /tdf α x 1 lk P[ l k 1 Hk 1 H k H α+ ε t k, Hk 1 δ t, k 1 l j max 1 j k 1 t α, Hk 1 1 ε+δ t ], with δ t := 3ṽ t /t, ε t k := α+l k /H k δ t. The idea now is to make appear the event { N 2ε t = k } in the above probability recall 68

80 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS the definition of Nt 2ε given in Proposition and then sum over k. We first prove that the sum over k n t, of the above probability is small if we intersect its event with the event { Nt 2ε k }. In other words, let us prove that 1 := 1 k n t P [ Hk 1 δ t, Hk 1 1 ε+δ t, N 2ε t k ] is small. As { N 2ε t k } = { Hk 1 2ε } { Hk 1 > 1 2ε }, and since for t large enough, { Hk 1 δ t} { Hk 1 2ε } =, we have 1 P [ Hk 1 δ t, 1 2ε < H ] k 1 1 ε+δ t. 1 k n t Therefore, for t large enough, with sε,t defined in Lemma 2.5.4, 1 1 k n t P [ Hk > 1 ε/2, 1 2ε < H k 1 1 3ε/4 ] sε,t by Finally, combining equations from to leads to P E I 1 1 k n t [ 1 H N 2ε t 1 l N 2ε t H N 2ε t l N 2ε t 1 α+ ε t N 2ε t, l j max 1 j Nt 2ε 1 t α H N 2ε t 1 +sε,t+c + ut,ε+o To finish we have to deal with ε t Nt 2ε, a basic computation partitioning on the values of Nt 2ε, shows that P [ ε t Nt 2ε α δ t/6 ] C + P R 1 δ t = o1 as R 1 converges in distribution to R κ which is almost surely positive. Collecting this last fact, 2.5.4, 2.5.5, 2.5.6, and finish the proof of the upper bound. Proof of the lower bound : The proof here follows the same line as the upper bound. The main difference comes from the fact that we can no longer use the inequality sup x R Lt,x sup 1 j Nt Lt,m j. So for this part of the proof we stress on what is different from the upper bound, and refer to the previous computations when very few changes occur. Assume for the moment that { } P suplt,x 2 w t =: E 2 1 o1, x R with w t := te κ1+3δ 1φt, and recall that δ is chosen small enough such that κ1+ 3δ < 1 see Lemma This fact is a direct consequence of the upperbound of Psup x R Lt,x αt see at the beginning of the proof of Theorem ] 69

81 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS for a proof of Recall , and define for any l 1, E 3 l := E3l E 1 3l, 2 with { E 1 3l := E 2 3l := l 1 j=1 [ sup L H Lj,x L H mj ] },x t α t, x D j { [ sup Lt,x L H m ] } l,x t α t, x D l with α t := αt 2 w t /t. Recall the definitions of the events B i, 1 i 4 in Sections and We have for large t, { supx R+ Lt,x αt } V t E 2 {N t n t } 4 i=1b i n t E 3 N t V t E 2 {N t n t } 4 i=1b i n t. Indeed,Lt,x w t for everyx R + nt j=1 [ L j, L ] j onb2 n t B 3 n t V t {N t n t }, and on the same event intersected with B 4 n t, Lt,x w t +te 2φt < 2 w t for every x nt j=1 [ L j, L ] j Dj, whereas for x Dj, LH m j,x w t if j n t and Lt,x L H Lj,x wt if j < N t. Notice that by Lemmata 2.2.1, 2.3.1, 2.3.2, and the above assumption , P V t E 2 {N t n t } 4 i=1b i n t 1 o1. We now deal with PE 3 N t B 1 N t B 2 n t V t {N t n t }. Using Lemma 2.2.1, the fact that H L k H m k+1 and the strong Markov property with respect to P Wκ, we obtain with PE 3 N t B 1 N t B 2 n t V t Q n t 1 ε E ν Wκ 4 yp Wκ E3k,B 1 1 k,b 2 k 1,H m k /t dy o1 k=1 ν Wκ 4 y ε :=P Wκ m k sup x D k L X t1 y,x t α t, H L k > t1 y, H L k < H L k Now, by computations similar to the ones giving the upper bounds in and 2.5.1, we have PE 3 N t B 1 N t B 2 n t V t,q n t 1 ε E ν Wκ 4 ydν Wκ 5 y n t o1 = k=1 ε k=1 1 ε ε ν 4 ydν 5 y o1., 7

82 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS with ν Wκ 5 y := P Wκ E3k,B 1 1 k,b 2 k 1, k 1 i=1 U i/t y, ν 4 y := E ν Wκ 4 y and ν 5 y := E ν Wκ 5 y. The next step is to remove B 1 k in the above expression. For that, we only have to prove that n t 1 ε k=1 ε E 4 yp E Wκ 3k, 1 B k 1 1 k,b 2 k 1, U i /t dy ν Wκ is negligible, one can check that this quantity is smaller than n t 1 ε k=1 n t k=1 P ε E k 1 i=1 [ ] P Wκ m k H L k < H L k,h L k > t1 y U i /t 1, i=1 k 1 P B 1 k,b 2 k 1, U i /t dy k U i /t > 1, B1 k i=1 P B1 n t C 2 v t = o1, i=1 where the last inequality comes from Therefore, collecting the above computations yields P suplt,x α x R n t k=1 1 ε ε ν 4 yd ν 5 y o1, with ν 5 y := P E3k,B 1 2 k 1, k 1 i=1 U i/t y. We start with an estimation of the repartition function ν 5 y. Recall that like in the proof of Lemma 2.3.6, by the strong Markov property, the occupation time formula and the sequence U j,{lh L j,x LH m j,x,x D j },j n t under B 2 n t is equal to a sequence H j L j,{l j H j L j,x,x D j },j n t, with this time Lj H j L j := A j L j e Ṽ ju L B j[τ Bj 1,A j u/a j L j ]du, L j L j H j L j,x := A j L j e Ṽ j x L B j[τ Bj 1,A j x/a j L j ], where A j u = u m j eṽ j x dx. Using Remark 2.2.1, Lemma 2.2.1, Fact ii, and then and 2.7.6, we have for large t for any 1 j n t since φt = ologt, P [ τ j κr t /8 m j +r t τ j r t ] 1 C + e c r t, P [ τ j r t m j r t τ j κr t/8 ] 1 C + e c r t

83 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS with c >. Therefore for any j, P D j [ τ j r t, τ j r t ] 1 2C + e c r t. Then on {D j [ τ j r t, τ j r t ]}, for any x D j, L j H j L j,x A j L j L B j[τ Bj 1,A j x/a j L j ]. Also with probability 1 2C + e c r t, D j [ τ j r t, τ j r t ] so for any x D j, A j τ j r t A j x A j τ j r t With Remark 2.2.1, Lemma 2.2.1, Fact and 2.7.8, we obtain with a probability larger than 1 e c r t, e ht/4 e 2rt e 1 1/2ht Aj τ j r t Aj τ j r t A j Lj A j Lj e 2rt e 1 1/2ht e ht/ Therefore, applying with δ = e ht/4 and ε = δ 1/3, we obtain with a probability larger than 1 e c r t, sup A j Lj LB j τ B j 1,A j x/a j Lj A j Lj LB j τ B j 1, 1+e ht/12. x D j Collecting the different estimates we then obtain, ν 5 y P max L j Hj Lj, mj tᾱt, 1 j k 1 k 1 j=1 H j Lj t y C + e c r t, with ᾱ t := α t 1+e h t/12 1. We can then inverse the equality in law we have used above, and then obtain ν 5 y Fᾱt y C + e c r t, with Fᾱt defined in Lemma Then we can follow the same lines as for the upper bound especially computations after 2.5.9, and obtain via Lemma and by choosing C large enough in such a way that c r t /φt = c C > κ1+δ : 1 ε ε ν 4 yd ν 5 y 1 ε ε ν 4 ydf + ᾱ t y on 1 t. Remark also that implies the concentration of the local time at the h t - minima : with probability larger than 1 C + e c r t, sup L j Hj Lj,y Lj Hj Lj, mj y D j e ht/12 L j Hj Lj, mj

84 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS We now work on ν 4 y. By the second part of Lemma it is equal to E P Wκ m 1 sup L X t1 y,z t α t,h L1 > t1 y,h L1 < H L 1 z D 1 =: ν 4 y, and by Lemma 2.5.3, ν 4 y f α t y on 1 t. Therefore 1 ε ε ν 4 yd ν 5 y 1 ε ε f α t ydf + ᾱ t y on 1 t. From now on, the computations are very close from that of the upper bound see and below and we do not give more details. Proof of Lemmata 2.5.2, and Proof of Lemma : This is a direct consequence of Proposition Proof of Lemma : To obtain the result we use a similar method than in [5]. That is to say, we study the inverse of the local time at m 1, and use our knowledge about H L 1. From the definitions of f γ and f γ we have easily f γ x f γ x for all x. So, to prove 2.5.1, we only need to prove the upper bound for f γ and the lower bound for f γ. We fix ε,1/2. Upper bound for f γ x. Recall that σu, m 1 = inf{s >, Ls, m 1 u}, u. First, notice that for < x < 1, f γ x is equal to [ E P Wκ m 1 Lt1 x, m 1 γt,h L1 > t1 x,h L1 < H L ] 1 [ = E = E P Wκ m 1 σγt, m 1 t1 x,h L1 > t1 x,h L1 < H L 1 ] [ P Wκ m 1 H L1 > σγt, m1 t1 x,h L1 < H L ] [ +E P Wκ m 1 σγt, m 1 > H L1 > t1 x,h L1 < H L ] Let us first study the expectation in On { H L1 > σγt, m1,h L1 < H L } 1 under P W κ m 1, X remains between L 1 and L 1 until time σγt, m 1 which is finite. On this event and under P Wκ m 1, considering and as in [61] p. 248, the inverse of the local time can be written for X starting at m 1 as σγt, m 1 = L1 L 1 e Ṽ 1 z L B σb γt,,a 1 z dz =: I, where A 1 z = z m 1 eṽ 1 y dy and B is a standard Brownian motion independent of W κ, such that B starts at A 1 m 1 = and is killed when it first hits A 1 L 1. In 73

85 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS , we integrate only between L 1 and L 1 because under P Wκ m 1, e Ṽ 1 z L B σb γt,,a 1 z = Lσγt, m 1,z = for z / [ L 1, L 1 ] as explained after We have L1 I = γt e L B Ṽ 1z σ B1,,ãz dz, L 1 with ãz := γt 1 A 1 z = γt 1 z eṽ 1y m 1 dy and where B := Bγt 2./γt. By scale invariance B is also a standard Brownian motion that we still denote by B in the sequel. Also, recall that σ U r,y := inf{s >, L U s,y > r} for r >, y R is the inverse of the local time of the process U. Since we consider X starting at m 1, we have H L 1 = H L 1 H m 1 = U 1, for which Proposition gives { } E P Wκ m H L1 1 H 1 εt H 1 = P{ U 1 H 1 ε t H 1 } =: G 1 1 e D 1h t, with ε t := e d 1h t, if δ > is chosen small enough. This will explain the appearance of H 1 in f γ ± x. So, we now deal with I. Notice that γt 1 I can be split into two terms γt 1 I = I 1 +I 2, with I 1 := τ1 h t/2 τ 1 ht/2 e Ṽ 1 z L B σb 1,,ãz dz, and I 2 := γt 1 I I 1. We now prove that the main contribution in γt 1 I comes from I 1 and obtain its approximation in probability. Let ε,1/1. First, using the second part of Lemma 2.2.1, followed by Remark 2.2.1, Fact ii for which we need i 2, and finally the first part of Lemma 2.2.1, we get P [ A 1 τ 1 h t /2 e ht1+ε/2, A 1 τ 1 h t /2 e ht1+ε/2] = P [ A 2 τ 2 h t /2 e h t1+ε/2, A 2 τ 2 h t /2 e h t1+ε/2 ] 1 2P [ F + h t /2 > e ht1+ε/2] P [ V t ] 1 C+ e κεht/ Therefore, since ã τ 1 h t /2 ãz ã τ 1 h t /2 for all z [ τ 1 h t /2, τ 1 h t /2 ], P z [ τ 1 h t /2, τ 1 h t /2 ], ãz e logt1 3ε/2 1 C + e κεht/ Also, using and the second Ray-Knight theorem see before , we have P sup L B σ B 1,,u 1 ε t e tε / u e logt1 3ε/2 with ε t := t 1 5ε/4. So we obtain [ E P Wκ I1 m 1 R ] 1 εt R1 1 C + e κεht/4,

86 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS with R 1 := τ 1 h t/2 1z dz. We now prove that I τ 1 ht/2e Ṽ 2 is negligible compared to the integral R 1 which appears in the previous equation, and then compared to I 1. First thanks to and the second Ray-Knight theorem, we have [ [ E P Wκ m 1 sup L B σb 1,,ãz ] ] > e εlogt 2e εlogt. z [ L 1, L 1 ] So with probability larger than 1 2e εlogt, we have τ I 2 e εlogt 1 ht/2 L 1 e Ṽ 1 z dz + L1 τ 1 h t/2 e Ṽ 1 z dz =: e εlogt I 3. By Lemma 2.6.8, with a probability larger than 1 2e c εh t for large t, I 3 C + h 2 te 1 εht/2. Also, by Lemma 2.3.6, with probability larger 1 e D h t, R1 = R 1 which is the same R 1 as in , which law is given by the sum of two independent copies of F h t /2. So using 2.7.9, with a probability larger than 1 2e D h t, R 1 = R 1 e εht/2. We deduce from the last three inequalities that with a probability larger than 1 e c εh t, I 2 < R 1 e 1 5εht/2 = R 1 e 1 5εht/ Finally, using γt 1 I = I 1 +I 2 together with and , we get E [ P Wκ m 1 I γtr1 2t 1 5ε/4 γtr 1, H L1 > σγt, m1, H L1 < H L 1 ] C + e εc h t We recall that by , σγt, m 1 = I on { H L1 > σγt, m1,h L1 } < H L 1 under P Wκ m 1. Hence, combining with gives for large t for every x [ε,1 ε], {H L1 > σγt, m1 t1 x,h L1 < H L 1 } { 1 γ } R 1 1 x 1+ε t,h 1 > t1 x1 ε t,h L1 > σγt, m1 Eε, where E 1 ε is such that E [ P Wκ m 1 E 1 ε ] C + e εc h t +e D 1h t and where, as defined in the statement of the lemma, ε t = e c 2h t with c 2 > chosen small enough. 75

87 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS Now, let us study On the event inside the probability in , σγt, m 1 might be infinite. We work under P Wκ m 1. There exists a Brownian motion B such that, with T 1 playing under P Wκ m 1 the same role as T does under P see , H L1 = T 1 τ B A 1 L1 and σγt, m1 = T 1 σ B γt, as in and in [61] p Also by , notice for further use that under P Wκ m 1, L σyt, m 1,z = e Ṽ 1 z L B σb yt,,a 1 z, z R,y, So, we have σγt, m 1 > H L1 σb γt, > τ B A 1 L1 L B [ σb γt,, ] = γt > L B [ τ B A 1 L1, ]. Now, note that, as in in the proof of Lemma 2.3.6, L B [ τ B A 1 L 1, ] = A 1 L 1 L B τ B1,, where B := BA 1 L 1 2./A 1 L 1. Also, by definition of e 1 given in , we have L B τ B1, = e 1. As a consequence, σγt, m 1 > H L1 γt > A 1 L1 e1 γtr 1 > A 1 L1 e1 R 1. Then, according to , we have A 1 L1 1 e d h t S1 with probability greater than 1 e D h t. Moreover, according to and to the fact that under P Wκ m 1 the diffusion X starts at m 1, we have H 1 = e 1 S 1 R ε t 1 H L 1 with probability greater than 1 e D h t. As a consequence, σγt, m 1 > H L 1 γtr 1 > 1 e d h t 1+εt 1 H L 1, except on an event which probability E [ P Wκ m 1. ] is less than 2e D h t. Combining this with we get for large t for every x [ε,1 ε], { } σγt, m 1 > H L 1 > t1 x,h L 1 < H L 1 { 1 γ } R 1 1 x 1+ε t,h 1 > t1 x1 ε t,σγt, m 1 > H L 1 Eε, where Eε 2 is such that E [ P Wκ m 1 Eε] 2 2e D h t +e D 1h t and where, as before, ε t = e c 2h t with c 2 > possibly smaller than before. Combining , and with the strong Markov property, we get for large t for every x [ε,1 ε], since φt = ologt, 1 f γ x P γ R 1 1 x 1+ε t,h 1 > t1 x1 ε t +on 1 t. = f + γ x+on 1 t. 76

88 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS Lower bound for f γ. Let γ := γ 1+e ht/12 1 and y := 1 x/[r1 1 4 ε t ]. We have to distinguish the cases H L1 > σ γt, m1 and σ γt, m 1 > H L1. We work under P Wκ m 1. On { y γ,h L1 > σ γt, m1 t1 x,h L1 } < H L 1, we can express the local time of X at the inverse of its local time in m 1 at time yt in terms of the standard Brownian motion driving the diffusion. More precisely by and by scale invariance, there exists a Brownian motion B such that for any z D 1, L σyt, m 1,z = yte Ṽ 1 z L B σb 1,,âz with âz := yt 1 z eṽ 1u m 1 du = A 1 z/yt. Notice that by 2.1.7, F h t /2 τ W κ h t /2 in law, so P [ R 1 > 8h t /κ ] 2P [ F h t /2 > 4h t /κ ] 2P [ τ W κ h t /2 > 4h t /κ ] e c h t for large t. Moreover, we prove with the same method used to prove that τ h t /2 m 1 r t m 1 + r t τh t /2 with probability at least 1 C + e c h t. This and give e ht1+ε/2 A 1 [ τ h t /2] A 1 z A 1 [ τh t /2] e ht1+ε/2 for any z D 1 with probability 1 e c εh t. So, for large t for every x [ε,1 ε], âz e ht1+ε/2 R 1 /t1 x e logt1 3ε/2 for these z with such probability. Hence with the same method we used to prove from and , we get for large t for every x [ε,1 ε], E P Wκ LB m 1 sup σ B 1,,âz 1 ε t 1 2e c εh t. z D 1 The above inequality together with imply that for large t for every x [ε,1 ε], E P Wκ m 1 { z D 1, Lσyt, m 1,z yte Ṽ 1 z 2yte Ṽ 1 z ε t, y γ, H L1 > σ γt, m1 t1 x, H L1 < H L } 1 2e c εh t On { y γ,h L1 > σ γt, m1 t1 x,h L1 < H L 1 }, ift1 x > σyt, m1, then σyt, m 1 ytr 1 < 4tyR 1 ε t, and by applied with γ replaced by y, this has on the previous event a probabilitye P Wκ m 1. less thanc + e c εh t. Thus on the previous event, we have t1 x σyt, m 1, except on a sub event of probability smaller than C + e c εh t. This is true for every x [ε,1 ε] for large t. Then since the local time is increasing in time, we have on the previous event for any z D 1, Lt1 x,z Lσyt, m 1,z, which is according to less than yte V 1z 1+2 ε t yt1+2 ε t for every z D 1 with probability E P Wκ m 1. at least 1 2e c εh t. Combining this and the definition of our y gives for large t, for every x [ε,1 ε], E > 1 x } 1+2 ε t =: G 2 R ε t {y γ,h L1 > σ γt, m1 t1 x,h L1 < H L 1 } { supz D1 Lt1 x,z P Wκ m 1 t 2+C + e c εh t

89 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS As a consequence, for t large enough so that 1+2 ε t 1+e ht/12, we have for every x [ε,1 ε], {y γ,h L1 > σ γt, m1 t1 x,h L1 < H L 1 } { } sup Lt1 x,z y1+2 ε t t γt, H L1 > σ γt, m1 Eε z D 1 by definition of γ, where E 3 ε is such that E P Wκ m 1 E 3 ε 2+C + e c εh t. On the other hand, from the definition of σ., m 1, and the definition of γ, we have for large t for every x [ε,1 ε], {y γ,σ γt, m 1 > H L1 > t1 x,h L1 < H L 1 } {L H L1, m1 γt,σ γt, m1 > H L1 > t1 x,h L1 < H L 1 } { sup L } H L1,z γt,σ γt, m1 > H L1 > t1 x Eε 4 z D { 1 } sup Lt1 x,z γt, σ γt, m 1 > H L1 Eε, z D 1 where E 4 ε is the event where fails, it is such that E P Wκ m 1 E 4 ε C + e c r t. Combining and we get for large t for every x [ε,1 ε], under P Wκ m 1, { } y γ,h L 1 > t1 x,h L 1 < H L 1 { } sup Lt1 x,z γt Eε, 5 z D 1 where Eε 5 is such that E P Wκ m 1 Eε 5 C + e c r t = C + e c C φt = on 1 t as t + is we choose C large enough. Combining this with , and Proposition 2.3.5, we obtain for large t for every x [ε,1 ε], f γ x P 1 x R 1 γ1 ε t,e 1 S 1 R 1 > t1 x1+ε t = f γ x on 1 t, on 1 t where the constant c 2 in the definition of ε t = e c 2h t has been decreased if necessary. This proves the lower bound for f γ x and then finishes the proof of the lemma. Proof of Lemma : Let < a < 1/4. We start with By Proposition 2.3.5, the H i, i 1 are i.i.d., so H k 1 and H k H k 1 = H k are independent for k 1. Thus for t >, P [ Hk > 1 a/2, 1 2a < H k 1 1 3a/4 ] 1 k n t = 1 3a/4 1 2a dµ t xe κφt P[H 1 > 1 x a/2],

90 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS where the measure µ t is defined by x dµ ty := e κφt 1 k n t P [ Hk 1 x ]. We know that µ t converges vaguely as t + to the measure µ which has a density with respect to the Lebesgue measure equal to ΓκC κ 1 x κ 1 1 x>, with C κ > see Lemma Also thanks to Lemma 2.4.1, e κφt P [H 1 /t > x] converges uniformly on every compact subset of,+ to C κ x κ /Γ1 κ. Therefore, lim t + = 1 k n t P [ Hk > 1 a/2, 1 2a < H k 1 1 3a/4 ] 1 ΓκΓ1 κ const a 1 κ. 1 3a/4 1 2a x κ 1 1 x a/2 κ dx For 2.5.3, we apply with r = ε, 1/2 and s = 1 ε, which gives lim Pεt Hm N t 1 εt t + = 1 sinπκ ε x κ 1 1 x κ dx+ π 1 sinπκ 1 ε κ π κ which implies the result. 1 1 ε ε κ + 1 εκ 1 1 κ ε1 κ x κ 1 1 x κ dx, Proof of Theorem : The proof of this theorem is a direct consequence of Propositions and and of Lemmata and Notice that the proof of the upper bound does not use the proof of the lower bound, but we use the upper bound for the proof of the lower bound. In particular from the upper bound of Theorem which makes use of the upper bound of Proposition but not of its lower bound, we have limsup t + PL t < 2 w t P Y 1 Y ε for any ε > as lim t + w t /t =. From this, as Y 1 Y is positive, we obtain lim t + PL t < 2 w t =, which proves assertion at the beginning of the proof of the lower bound of Proposition Thanks to Proposition and to the remark before this proposition, we only need to study the convergence of P 1 ± the limit when t goes to infinity and then the limit when ε goes to. The latter can be written in term of functionals of Y 1,Y 2 t as follows. Let Y t := Y2 t 1 1 2ε ; we have Nt 2ε e κφt = Y t, and [ 1 Y P 1 ± t = P 2Y t Y1Y t t Y1Y t ] t Y2Y t t Y2Y t t α± t, Y1 t Y t α t ± [ 1 = P K I,1 2ε Y 1,Y 2 t K I,1 2εY 1,Y 2 t K I,1 2ε Y 1,Y 2 t K I,1 2ε Y 1,Y 2 t K I,1 2ε Y 1,Y 2 t α± t, J I,1 2ε Y 1,Y 2 t α ± t ], 79

91 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS with the notation K I,a, KI,a,... introduced in and before. The hypotheses of Lemma 4.5 are : finite number of large jumps on compact intervals, strictly increasing, starting at, and jumping over 1 without reaching it. These properties are naturally almost surely satisfied by a κ-stable subordinator so, almost surely, the paths of Y 1,Y 2 satisfy the hypotheses of Lemma see e.g. [8] III.2 p. 75. Therefore they are points of continuity for J I,1 2ε, K I,1 2ε, K I,1 2ε, K I,1 2ε and K I,1 2ε. Combining this continuity with Proposition 2.1.4, continuous mapping theorem, and replacing the functionals by their expressions, we obtain, when t goes to infinity, the convergence of P 1 ± to [ 1 Y2 P Y ε Y 1 Y ε Y 1 Y ε Y 2 Y ε Y 2 Y ε α, Y 1 Y ε ] α. Then, note that almost surely Y 2 Y < 1 so we have a.s. Y ε = Y2 1 1 for all ε small enough. We deduce that the above expression converges to the repartition function of maxi 1,I 2 see for definitions of I 1 and I 2 when ε goes to, and this yields Theorem Favorite site proof of Theorem Ê Thanks to Section 2.3, we know precisely the nature of the contribution of each h t -valley to the local time. The difficulty in proving Theorem was the need to consider only a part of the contribution of the last h t -valley. The proofs of the first two points and of Theorem are thus easier to obtain, since they do not require to "cut" the contribution of any valley. Let us prove the first point the second one, 2.1.1, is obtained similarly. We have, using 2.2.7, P[L Hm Nt+1 αt] P L H LNt αt, Q,Vt +P Q +P V t +P B 3 n t P sup l j /t 1 ε t 1 α, Q,V t +P Q +o1, 1 j N t where we fixed some ε > and Q := {εt Hm Nt 1 εt, 1 N t n t } as after from there we see thatlim ε lim t + P Q =. In the last inequality we used Proposition 2.3.5, Lemma and Lemma To lighten notation, let 8

92 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS α t := 1 ε t 1 α. We have P sup l j /t α t, Q, V t 1 j N t P sup l j /t α t, B 1 n t, Q, V t 1 j N t P +P B 1 n t sup 1 j N t l j /t α t, HNt 1 δ t, HNt 1 1 ε+δ t, Q +o1, with δ t = 3ṽ t /t and where we used together with Proposition Partitioning on the values of N t we get that the above is less than sup l j /t α t, Hk 1 δ t, Hk 1 1 ε+δ t, Q +o1. 1 j k 1 k n t P Since the sum 1 defined in the proof of the upper bound of Proposition see and below is smaller than sε,t satisfying lim ε lim t + sε,t =, we can intersect the event on the above probability with {k = Nt 2ε } and get P[L Hm Nt+1 αt] P sup l j /t α t +P Q +sε,t+o1. 1 j N 2ε t Then, as in the proof of Theorem we have that Y 1,Y 2 almost surely satisfies the hypothesis of Lemma 2.4.5, and is therefore almost surely a point of continuity for J I,12 ε defined just above From this continuity, Proposition and the continuous mapping theorem we get sup 1 j N 2ε t l j /t = J I,1 2ε Y1,Y 2 t L t + J I,1 2ε Y 1,Y 2 = Y 1 Y ε. Then, as in the proof of Theorem we have almost surely Y ε = Y2 1 1 for all ε small enough so Y 1 Y ε converges almost surely to Y 1 Y when ε goes to. Thus, we get lim supp[l Hm Nt+1 αt] P Y 1 Y α. t + A lower bound is proved similarly, so we get the following, proving : lim t + P[L Hm Nt+1 αt] = P Y 1 Y α. To obtain the result for the favorite site, we first argue that we essentially need to obtain the asymptotic behavior of N t/n t, where N t := min{j 1,Lm j,t = max 1 k Nt Lm k,t}. Indeed, define for any ε,1/2, K 1 := {1 εm Nt Xt 1+εm Nt }, K 2 := { 1 εm N t F t 1+εm N t }. 81

93 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS Then, we have, lim t + PK 1 = 1 by the localization result Theorem combined with the fact that Xt/t κ converges in law under P to a positive limit as t + by [45]. Let us now justify that lim t + PK 2 = 1. According to proved at the start of the proof of Theorem 2.1.3, to Lemma and 2.3.3, we have P suplt,x 2 w t,b 4 n t,n t n t x R 1. t + Notice that on the event inside the above probability, for t large enough so that 2 w t te 2φt, we have Ft D N t recall the definition of D j in Since D N t is centered at m N t and its half-length is deterministic and equal to r t = C φt we only need to justify that P εm N t C φt t + 1. We have m N t m 1 and Pm 1 C φt/ε P m 1 C φt/ε o1 by Lemma So using , we thus deduce that lim t + PK 2 = 1. We can now write for x >, [ ] [ ] [ F P t F Xt x = P t Xt x,k mn t 1,K 2 +vε,t P x 1+ε ] +vε,t. m Nt 1 ε where vε,t, satisfies lim ε lim t + vε,t =. Similarly, we have [ ] [ F P t Xt x mn t P x 1 ε ] vε,t. m Nt 1+ε Hence, we obtain [ mn t P x 1 ε ] m Nt 1+ε [ ] [ F vε,t P t Xt x mn t P x 1+ε ] +vε,t. m Nt 1 ε So, we observe that we only have to study the random variable m N t m Nt. For that we first remark that Nt and N t diverge when t goes to infinity. Indeed by Lemma 2.6.1, the correct normalisation for the convergence in law of N t is e κφt, so PN t e 1 εκφt = 1 o1. For Nt, we first notice that the previous result for N t also gives for t large, PN t e 1 ε/2κφt = 1 o1. Therefore P N t e 1 εκφt P max k e 1 εκφt Lm k,t max Lm k,t k<e 1 ε/2κφt +o1. Now, since Lm k,t = L m k,h Lk H mk + H mk L =: lk for k < N t on k V t {N t n t } B 2 n t which has probability 1 o1 by Lemmas and 2.3.1, P max Lm k,t max Lm k,t k e 1 εκφt k<e 1 ε/2κφt P max 1 εκφt lk 1 ε/2κφt lk max +o1, k e k<e 82

94 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS with lk,k e 1 ε/2κφt i.i.d. random variables underpby strong Markov property and the second part of Lemma 2.2.1, and with queue distributions given by and Proposition It is then clear that for large t, Pmax k e 1 εκφt l k max k<e 1 ε/2κφt l k = o1, and we therefore obtain that PN t e 1 εκφt = 1 o1. Then, following the work of [37], we know that m i m i 1,i 2 are i.i.d. random variables with a known Laplace transform given by 2.19 in [37], this allows to compute the first and fourth moments of m 1 := m 2 m 1 and therefore obtain after an elementary but tedious computation that for large t, E m 1 C 7 e κht C 7 >, see also 2.17 in [37] and E m 1 E m 1 4 C 8 e 4κht C 8 >, which yields as t + and k +, E [ m k /k E m 1 4] C 8 e 4κht /k 2. These facts allow us to write by a Markov inequality that P[ m Nt E m 1 N t > εe m 1 N t ] [ mj E m 1 j > εe m1 j] +o1 j e 1 εκφt P 2C 8 C 7 4 +o1 ε 4 j 2 j e 1 εκφt C + ε 4 e 1 εκφt +o1. This yields that { m Nt E m 1 N t εe m 1 N t } as well as with a similar computation { mn t E m 1 Nt εe m1 Nt} are realized with a probability close to one. Now including these events in the probability in , eventually enlarging vε, t we get [ ] [ ] [ ] N P t x 1 ε2 F vε,t P t N N t 1+ε 2 Xt x P t x 1+ε2 +vε,t. N t 1 ε 2 Notice that the random variables involved now Nt and N t only depend of what happens in the bottom of the h t -valleys, and we have to deal with [ ] N P t y = P [ [ ] ] N Nt = N t 1{y=1} +P t y, Nt < N t 1 {y 1} + 1 {y>1}, N t N t for any y >. We are now interested in the limit when t goes to infinity of the above two probabilities. We first use the same lines as for the proof of Section 5.1, that is to say we give a lower and an upper bound of this probability involving the i.i.d. sequences l j,j and H j,j. In the same way we have obtained Proposition 2.5.1, we then have for any ε > and large t, P vε,t PN t = N t P +vε,t 83

95 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS with P := P [ 1 H N 2ε t 1 l N 2ε t H N 2ε t l N 2ε t 1 H N 2ε t 1 > max 1 j N 2ε recall that H k = Y 2 ke κφt = 1 k t i=1 H i, l k = Y 1 ke κφt = 1 k t i=1 l i, Nt 2ε := inf{m 1, H m > 1 2ε}, and v is a positive function such that lim t + vε,t const ε κ 1 κ with an eventually larger const than in Proposition In the same way, for any y >, ε > and t large enough, ] N P 1 vε,t P[ t y, Nt < N t 1 y 1 N P 1 + +vε,t, t where t 1 l j t ], P ± 1 := P [ N t /N 2ε t y ±ε,1 H N 2ε t 1 l N 2ε t H N 2ε t l N 2ε t 1 H N 2ε t 1 max 1 j N 2ε t 1 l j t ] 1 y 1, with Nt := min{j 1,l j = max k N 2ε l t k }. This together with Lemma yields for large t, P[N t = N t ] P[I 1 < I 2 ] lim vε,t+o1 t + and [ ] N P t e κφt N t e κφt y,n t < N t lim vε,t,+o1, t + [ F Y 1,Y 2 P Y2 1 1 y, I 1 I 2 ] where F is defined at the beginning of Section Replacing y by x 1 ε2 for the 1+ε 2 lower bound and by x 1+ε2 for the upper bound and taking the limit when t goes 1 ε 2 to infinity and then ε we obtain for < x < 1, [ ] [ N lim P t F Y 1,Y 2 x = P t + N t Y2 1 1 x, I 1 I 2 ]. To finish the proof of the last result of Theorem we finally have to prove Lemma below. Lemma The random variable F Y 1,Y 2 Y follows a uniform law U [,1] and is independent of the couple I 1,I 2. Proof : 84

96 2.5. SUPREMUM OF THE LOCAL TIME - AND OTHER FUNCTIONALS For anys >, letg 1 s := inf{u s, Y 1 u Y 1 u = Y 1s}. The fact that for every s >, G 1 s/s follows a uniform distribution is basic. Since the independence that we seek is specific we give some details. The process of the jumps of Y 1,Y 2 in [,s] is a Poisson point process in [,s] R + 2 the coordinate in [,s] for the instant when the jump occurs and the other coordinate for the jump with intensity measure λ ν where λ is the Lebesgue measure on [,s] and ν, as defined in the introduction, is the Lévy measure of Y 1,Y 2. Let us give a particular construction of the process Y 1,Y 2 on [,s] : Let P n n 1 be a countable partition of R + 2 by Borelian sets such that n 1, < νp n < +. For each n we define an i.i.d. sequence S n k k 1 of random variables in R + 2, an i.i.d. sequence U n k k 1 of random variables in [,s] and a random variable T n such that n 1, S n 1 ν. P n /νp n, U n 1 U [,s], T n PsνP n, For any n 1, the variables S n k k 1, U n k k 1 and T n are independent, The triplets S n k k 1,U n k k 1,T n n 1 are independent, where U stands for uniform and P. for Poisson distribution. We know that the random set S n := {U n k,s n k, n 1, 1 k T n } is a Poisson point process in[,s] R + 2 with intensity measureλ ν. SinceY 1,Y 2 is pure jump, its restriction to [,s] is equal in law to the process Z 1,Z 2 defined by r [,s], Z 1,Z 2 r = n 1,1 k T n S n k 1 U n k r. In particular, withπ i x 1,x 2 := x i forx 1,x 2 R 2,i {1,2} andg Z 1 s := inf { u s, Z 1 u Z 1 u = Z 1s } L = G 1 s, we have Z1s = max{π 1 Sk, n n 1, 1 k T n }, { } G1 Z s = inf Uk, n n 1, 1 k T n, π 1 Sk n = Z1s, Z 1 s = π 1 Sk, n Z 2 s = π 2 Sk. n n 1,1 k T n n 1,1 k T n G1 Z s is the position of the highest jump of Z 1 on [,s] We thus have that G 1 s/s = L U [,1] and it is independent from Y1s,Y 1 s,y 2 s and from the sigma-field σy 1,Y 2 t+s Y 1,Y 2 s, t. We now have to replace s by Y For that we can consider for example the dyadic approximations of Y2 1 1, that is, t n := max { k N, k < Y 1 2 n 2 1 },n. Then, partitioning on the values of t n, using the independence we just proved and the fact that G 1 s/s follows a uniform distribution on [,1] we get that G 1 t n /t n follows a uniform distribution on [,1] and is independent from Y 1t n, Y 2 t n, Y 1 t n +2 n Y 1 t n, Y 2 t n +2 n Y 2 t n

97 2.6. RESULTS AND ADDITIONAL ARGUMENTS FROM THE PAPER [3] We let n goes to infinity, t n converges almost surely to Y from below. As a consequence, G 1 t n /t n converges almost surely to F Y 1,Y 2 Y while the quadruple in converges almost surely to Y 1Y 1 2 1, Y 2 Y 1 2 1, Y 1 Y Y 1 Y 1 2 1, Y 2 Y Y 2 Y As a consequence, F Y 1,Y 2 Y follows a uniform distribution on[,1] and is independent from the above quadruple for which I 1,I 2 is a measurable function, this yields the lemma. 2.6 Results and additional arguments from the paper [3] Some estimates on the diffusion X The first lemma below gives the right normalisation in law of the number of h t -valleys visited by X before time t. Lemma number of visited h t -valleys. Assume that < κ < 1. Then, under the annealed law P, N t e κφt t + N in law. The law of N is determined by its Laplace transform : u >, E e un = + j= 1 Γκj +1 j u, C κ where C κ is a positive constant. Moreover PN t > n t e φt. Proof : The convergence in distribution is exactly Proposition 1.6 of [3]. For the second fact we have PN t n t PÑt n t +PV t PÑt n t +e [ κ/2+o1]ht by Lemma 2.2.1, with Ñt := max{j 1, m j sup s t Xs}. Then equation 5.3 in [3] gives PÑt n t exp 2φt, which yields the result. The lemma below deals with the renewal structure we speak about on the introduction, and the consequence on the hitting time Hm Nt of the ultimate h t -valley visited by X before time t. Lemma Assume < κ < 1 and < δ < inf{2/27,κ 2 /2}. For t >, let µ t be the positive measure on R + such that x, n t µ t [,x] := e κφt P Hj x. j=1 86

98 2.6. RESULTS AND ADDITIONAL ARGUMENTS FROM THE PAPER [3] Recall that for any k, Hk := k j=1 H j/t, and H 1 = R 1 S 1 e 1 is defined in Proposition Then, µ t t converges vaguely as t + to µ defined by dµx := C κ Γκ 1 x κ 1 1,+ xdx, with C κ is the same constant as in Lemma For r < s 1, lim P 1 s Hm N t 1 r = sinπκ 1 r x κ 1 1 x κ dx t + t π Proof : The first part of the above lemma is very close to Lemma 5.1 of [3], indeed Proposition gives the proximity between the random variables U i,i n t and the random variables H i,i n t, moreover an important preliminary result in [3] Proposition 4.1 states that e κφt 1 Ee λu 1/t = C κ λ κ +o1 for large t. So we also know that 1 s e κφt 1 Ee λh 1/t = C κ λ κ +o1, notice that this result could also be deduced from with the help of a Tauberian theorem. Then by independence of the random variables H j and the fact that they are i.d., for any λ > + e λx dµ t x = 1 e κφt n t j=1 E e λ H 1 j t By as n t e κφt t + +, [E e λh 1/t ] nt+1 = o1. Hence, we get as t +, again by e λx dµ t x = e κφt 1+o1 1 Ee λh 1/t +o1 = 1 C κ λ +o1 κ = + e λx x κ 1 C κ Γκ dx+o1, which gives the vague convergence of measure µ t t. Also is equation 1.2 of Corollary 1.5 in [3]. In Lemma below, we approximate h j, the exit time of h t -valley number j if X leaves it on the right, by a product of 3 simpler random variables. To this aim, we recall that with the notation of Lemma and of its proof, for each 1 j n t, R j = τ j h t/2 jx dx, and A j u = u τ j ht/2e Ṽ eṽ jx m j dx, u R. Moreover, for some independent Brownian motions B j, 1 j n t, independent of W κ, h j = Lj L j e Ṽ j u L B j[τ Bj A j L j,a j u]du, e j = L B j[ τ B j A j L j, ] /A j L j. 87

99 2.6. RESULTS AND ADDITIONAL ARGUMENTS FROM THE PAPER [3] Lemma Let < ε < inf{2/27,κ 2 /2}. For large t, we have for every 1 j n t, P hj A j L j e j Rj > 2e 1 3εh t/6 A j L j e j Rj C + e c εh t Proof : We first notice that hj,a j L j,e j, R j is measurable with respect to the σ-field generated by Ṽ j x+ L + j 1, x L + j L + j 1 and B j, so, thanks to the second fact of Lemma 2.2.1, its law under P does not depend on j. Thus, the left hand side of does not depend on j. Hence we just have to prove for j = 2. This is actually already proved in [3], for which it is an important step. Indeed in this paper [3], our A j, B2 and h 2 are denoted respectively by Ãj, B and U, as defined in [3], eq and 3.18, and our R 2 and e 2 by I and e 1, as defined in [3], after eq Hence our for j = 2 is exactly [3], Lemma 4.7, which proves our lemma. The proof of [3], Lemma 4.7 is quite technical, however we can give a simple heuristic in order for the present paper to be more self-contained. The idea of the proof of [3], Lemma 4.7 is that, loosely speaking, for u close to m j, that is for u [ τ j h t/2, τ j h t /2 ], L B j[τ Bj A j L j,a j u] is nearly L B j[τ Bj A j L j,] = A j L j e j, whereas forufar from m j, that is foru [ L j, L j ] butu / [ τ j h t/2, τ j h t /2], e Ṽ jx is "nearly", with large probability. Finally, combining these heuristics gives h j A j L j e j Rj. The following lemma is used to prove Lemma and uses the notation of this lemma, and where the independent r.v. G + h t /2,h t, F + 1 h t, F 2 h t /2 and F 3 h t /2 defined before Proposition Lemma Assume < δ < inf{2/27,κ 2 /2}. For large t, possibly on an enlarged probability space, there exists R L 2 = F2 h t /2 + F3 h t /2 and S L 2 = F 1 + h t + G + h t /2,h t, such that R 2, S 2 and e 2 are independent and } L2 P{ eṽ 2x dx S 2 m 2 e d h t S 2, R2 = R 2 1 e D h t, where D >. Proof : Due to [3] Lemma 4.5 with its notation, we have I + := τ 2 h t m 2 e V 2x dx = L F + h t,i 2 + := L 2 τ 2 h ev 2x t dx = L G + h t /2,h t,i1 := τ 2 h t/2 m 2 e V 2x dx = L F h t /2 and finallyi2 := m 2 2x dx = L F h τ 2 ht/2e V t /2 withl 2 := inf{x > τ 2 h t, V 2 x = h t /2}. The problem is that I + is not independent of I1, so we would like to replace 88

100 2.6. RESULTS AND ADDITIONAL ARGUMENTS FROM THE PAPER [3] it by some I 1 + L = I + of it with better independence properties. It is proved in [3], at the top of page 32 that for large t, possibly in an enlarged probability space, there exists I 1 + such that I + I 1 + e 1 3δht/2 I 1 + with probability greater than 1 4e κδht/2 and where I 1 + L = F + h t by [3], eq LetS 2 := I 1 + +I 2 + I 1 +. Notice that onv t, by Remark 2.2.1, R 2 = I1 +I2 =: R 2 and L2 eṽ 2x m 2 dx = L 2 m 2 e V 2x dx = I + + I 2 +. The two previous inequalities give L2 eṽ 2x m 2 dx S 2 = I + I 1 + e 1 3δh t/2 S 2 and R 2 = R 2 with probability at least 1 5e κδht/2 thanks to Lemma This proves Moreover, by [3], Prop. 4.4 i, I 1 +, I 2 +, I1, I2 and e 2 which is denoted by e 1 in [3] are independent. So, e 2, S 2 = I 1 + +I 2 + and R 2 = I1 +I2 are independent, and R 2 L = F 2 h t /2+F 3 h t /2 and S 2 L = F + 1 h t +G + h t /2,h t. The last lemma of this section tells that with large probability, the diffusion X leaves every h t -valley [ L j, L j ], 1 j n t from its right. Recall that B j is defined after Lemma For large t, there exists c > such that [ { P nt j=1 maxl B j[τ Bj A j L j,a j u] = } ] 1 e c h t u< L j Proof : is essentially Lemma 3.2 in [3] : Indeed, recall the definition of A j := {max L u< L B j[τbj A j L j,a j u] = }, we j have nt j=1 A j = nt j=1 {H j L j < {H j L j }, with, for any L j x L j, H j x = inf{s >,B j s = x}, with B j a Brownian motion. Therefore P Wκ A j is equal to the probability P Wκ E j of Lemma 3.2 in [3]. It is proved in this lemma see 3.1 that for large t, PB := {P Wκ E j e κ/2ht } 1 3e κδht, so we obtain as PE j EP Wκ E j 1 B +PB e c h t /n t, for c > small enough Some estimates on the potential W κ and its functionals We start this section with the Laplace transform of the important functional R κ : Lemma Recall that < κ < 1. For any γ >, E e γrκ 2γ κ/2 2 = κγκi κ γ Moreover, R κ admits moments of any positive order. 89

101 2.6. RESULTS AND ADDITIONAL ARGUMENTS FROM THE PAPER [3] Proof : + e W κu du is the limit in law under P of τ W κx e W κu du as x +. This limit is given by [3], Lemma 4.2, which proves Note that in [3], Lemma κx 4.2, Wκ is denoted by R, and τ W e W κu du is denoted respectively by F x. Moreover the Laplace transform of R κ is of class C on a neighborhood of since x x κ /I κ x is C on such a neighborhood see e.g. [15] p Therefore R κ admits moments of any positive order. The following Lemma is a series of estimates concerning the different coordinates of valleys. Lemma For t large enough, for every 1 i n t, P < M < m 1 C + h t e κht, P τ i+1h t τ i+1 h t C + h t e κht, P inf Ṽ i < h t /2 e κht/8, [ τ i h+ t, τ i ht] P L + i L i 4h + t /κ e κht/8, P τ i h m i 8h/κ C + e κh/2 2, h h t, P m 1 r e r exp κ/2 2+κ 2 /4 h + t = o1, r = oh + t Proof : follows from eq. 2.8 of [3] ; is eq of [3] and are respectively eq and 2.32 of Lemma 2.7 of [3]. Moreover, is eq of the same reference. For , we know from definitions in that m 1 L 1 = τ Wκ h + t, where τ Wκ h + t is the first positive time the drifted Brownian motion W κ reaches h t. Using a Markov inequality together with 2..1 page 295 of [15] we obtain Pτ Wκ h + t r = Pe τwκ h + t e r e r e κ/2 2+κ 2 /4h + t, which is exactly The lemma below deals with two functionals involving coordinates far from the bottom m 1 of the first visited h t -valley [ L 1, L 1 ]. Lemma There exists c > such that for any ε > and t large enough, L1 P e Ṽ 1x dx C + h 2 te 1 εht/2 1 e c εh t, τ 1 h t/2 τ 1 ht/2 P e Ṽ 1x dx C + h 2 te 1 εht/2 1 e c εh t. L 1 9

102 2.7. APPENDIX Proof : The proof is inspired from steps 1 and 2 of Lemma 4.7 of [3]. For the first integral, let A 1 := { inf [ τ1 h t/2, τ 1 h t]ṽ1 > 1 εh t /2 }, A 2 := { L+ 1 L 1 4h + t /κ }. We have on A 1 A 2, L1 τ 1 h t/2 e Ṽ 1u du e 1 εht/2[ L1 τ 1 h t /2 ] 4h+ t h t e 1 εht/ κ Now, Fact 2.2.1, equation with α = 1/2, γ = 1 ε/2 and ω = 1, and Lemma give P A 1 P [ inf[τ1 h t/2,τ 1 h t]v 1 1 εh t /2,V t ] +PVt 3e κεht/2. Moreover, P A 2 e κh t/8 e κεht/2 by since we can take ε < 1/4. The second inequality, can be proved similarly. Lemma Recall that for h >, β h := E inf{u, W κ u inf [,u] W κ h}. For large h, τ 1 h e Wκu du, with τ 1h := β h C + e 1 κh Proof : is [3], eq. 3.38, since in [3], β h is defined at the top of page 23 and τ 1h in its Lemma Appendix Some estimates for Brownian motion, Bessel processes, W κ and their functionals We provide in this section some known formulas for some processes that appear in our study. The first lemma is about Laplace transforms of the exponential functionals defined in and Its proof can be found in [3], Lemma 4.2. Recall that C + respectively c is a positive constant that is as large resp. small as needed. Lemma There exist C 9 >, M > and η 1,1 such that y > M, γ,η 1 ], E e γf+ y/e y [1 2γ/κ+1] C 9 maxe κy,γ 3/2, E e γg+ y/2,y/e y [1 Γ1 κ2γ κ /Γ1+κ] C 9 maxγ κ e κy/2,γ. Moreover, there exists C 1 > such that for all y >, EF + y/e y C

103 2.7. APPENDIX Recall that W κ is a κ/2-drifted Brownian motion W κ Doob-conditioned to stay positive see above We have, Lemma Let < γ < α < ω. For all h large enough, we have P τ αh W κ γh < τ W κ ωh 2e κα γh, P τ W κ ωh τ W κ αh 1 4e [ω αh]2 /3, P τ W κ h > 8h/κ C + e κh/2 2, P τ W κ h h C + e c h, P τ W κ γh 1 C + e c [γh] 2, where P αh denotes the law of W κ starting from αh. Moreover the first inequality is still true if ω is a function of h such that lim h + ωh = +. Proof : The first 3 inequalities come from [3], Lemma 2.6. The fact that, in 2.7.3, ω can actually be taken as a function of h comes directly from eq of [3], which shows that the right hand side of is equivalent to e κα γh as h + if w = wh h is a consequence of with ω = γ and α = γ/2. We turn to By [3], eq. 2.7 and Fact 2.1, coming from [37], E e ατw κh h + const.e hκ/2 2α+κ 2 /4, in particular for α = 1 κ. Then a Markov inequality for P e ατw κh > e αh proves since 1 κ/2 21 κ+κ2 /4 <. We also need the following lemma, focusing only on some exponential functionals. Lemma Recall that F ± and G + are defined in and For all < ζ 1 and < ε < 1, for h large enough, P [ e 1 εζh F + ζh e 1+εζh] 1 4e κεζh/2, P [ F h e εh] 1 e c ε 2 h 2, P [ G + αh,h bhe h] 1 C + [bh] κ, < α < 1, bh > Proof : By Markov inequality and the last line of Lemma 2.7.1, P [ F + ζh > e 1+εζh] C 1 e εζh e κεζh/2 for large h. For the lower bound, we have by [3], eq for large h, P [ F + ζh e 1 εζh] 1 3e κεζh/2. 92

104 2.7. APPENDIX These two inequalities prove For 2.7.9, first F h e εh τ W κ εh, and using 2.7.7, τ W κ εh 1 with a probability larger than 1 e c ε 2 h 2, which proves Finally, notice that in law G + αh,h e h + e Wκx dx = e h A. By [32], 2/A is a gamma variable of parameter κ,1, and so has a density equal to e x x κ 1 1 R+ x/γκ, which leads to The following lemma is exactly Lemma 4.3 in [3] which proof can be found in that paper. Lemma Let Bs, s R be a standard two-sided Brownian motion. For every < ε < 1, < δ < 1 and x >, P sup LB τ B 1,u L B τ B 1, > εlb τ B 1, δ 1/6 C + u [ δ,δ] ε2/5, P sup L B τ B 1,u x 4e x/2, u [,1] P supl B τ B 1,u x 4/x u The next lemma says that with large probability, a 2-dimensional squared Bessel Process is bounded by some deterministic function. This lemma may be of independent interest. Lemma Let Q 2 u, u be a Bessel process of dimension 2, starting from, and two functions a. and k. from,+ to,+, having limit + on +. We have for large t, P u,kt], Q 2 2u 2e [ at+4loglog[ekt/u] ] u 1 C + exp[ at/2]. Proof : We consider for t > and i N, { A 1,i := sup Q kt } [at+4logi+1], A [kt/e i+1,kt/e i ] e i 2 := A 1,i. We recall that there exist two standard independent Brownian motions B 1 u, u andb 2 u, u such thatq 2 2u, u is equal in law tob1u+b 2 2u, 2 u. So for i N, P A 1,i 2P sup[kt/e i+1,kt/e ]B 2 i 1 > kte i [at+4logi+1] 4P sup [,kt/e ]B i 1 > kte i [at+4logi+1] = 4P B 1 1 > at+4logi+1 8exp[ at/2 2logi+1] i= 93

105 2.7. APPENDIX for large t so that at 1, by scaling, and since B 1 L = B 1, sup [,1] B 1 L = B 1 1 and PB 1 1 x e x2 /2 for x 1. Consequently for large t, P A 2 P 1 A 1,i 8exp[ at/2] i+1 = C +exp[ at/2] i= Now, let < u kt. There exists i N such that kt/e i+1 < u kt/e i. We have, e i kt/u, so e i+1 ekt/u and then logi + 1 loglog[ekt/u]. Consequently on A 2, Q 2 2u 2 kt/e i [at+4logi+1] 2eu [ at+4loglog[ekt/u] ]. This, combined with , proves the lemma. We also need some estimates on the local time of B at a given coordinate y R at the inverse of the local time of B at. Recall that σ B r,y = inf{s >, L B s,y > r} for r >, y R. By the second Ray-Knight Theorem, the processes L B σ B r,,y,y R + and L B σ B r,, y,y R + are two independent squared Bessel processes of dimension starting at r. The following lemma is proved in [67], Lemma 3.1 ; the results are stated for a Bessel process but are actually true for a squared Bessel process; see also [29], Lemma 2.3. i= Lemma We denote by Q y, y the square of a -dimensional Bessel process starting at 1. Let M >, u > and v >. Then, P sup Q y 1 1+uv u 4 exp [ u 2 /81+uv ], y v u P supq y M y = 1/M

106 Chapitre 3 Exponential functionals of spectrally one-sided Lévy processes conditioned to stay positive This work has been the object of an article [72] currently being reviewed. 3.1 Introduction We consider a spectrally negative Lévy process V which is not the opposite of a subordinator. We denote its Laplace exponent by Ψ V : t,λ, E [ e λvt] = e tψ V λ. In the case where V drifts to, it is well known that its Laplace exponent admits a non trivial zero that we denote here by κ, κ := inf{λ >, Ψ V λ = }. If V does not drift to, then is the only zero of Ψ V so we put κ := in this case. We denote by Q,γ,ν the generating triplet of V so Ψ V can be expressed as Ψ V λ = Q 2 λ2 γλ+ e λx 1 λx1 x <1 νdx In the end of the paper, we also consider Z, a spectrally positive Lévy process drifting to +. We are interested in the basic exponential functionals of V and Z conditioned to stay positive, IV := + e V t dt and IZ := 95 + e Z t dt.

107 3.1. INTRODUCTION For both we study finiteness, exponential moments and the asymptotic tail at. For IV, we also get self-decomposability, more precise estimates on the asymptotic tail at and a condition for smoothness of the density. Our first motivation is to extend to spectrally one-sided Lévy processes conditioned to stay positive the general study of the exponential functionals of Lévy processes. Those functionals have been widely studied because of their importance in probability theory. For example they are fundamental to the study of diffusions in random environments and appear in many applications such as mathematical finance, see [12] for a survey on those functionals and their applications. For a general Lévy process, equivalent conditions for the finiteness of the exponential functional are given in [12], the asymptotic tail at + of the functional is studied in [52], the absolute continuity is proved in [9] and properties of the density such as regularity are studied in [2] under some hypothesis on the jumps of the Lévy process and in [54]. In this paper we also obtain, as a by-product of our approach, some results on the exponential functionals of spectrally one-sided Lévy processes. Our second motivation is the possibility to apply our results to the study of diffusions in a spectrally negative Lévy environment. Such processes, introduced by Brox [17] when the environment is given by a brownian motion have been specifically studied for the spectrally negative Lévy case by Singh [66]. In [4], they prove that the supremum of local time L X of a diffusion in a drifted brownian environment converges in law and they express the limit law in term of a subordinator and an exponential functional of the environment conditioned to stay positive. In order to generalize their result to a diffusion in a spectrally negative Lévy environment, knowledge on the exponential functionals involved is needed. These are precisely exponential functionals of the environment which is spectrally negative and its dual which is spectrally positive conditioned to stay positive. Finally, we have hints that the almost sure asymptotic behavior of L X, for a diffusion in the spectrally negative Lévy environment V, is crucially linked to the right and left tails of the distribution of IV. This is why we study these tails here and give for the left tail a precise asymptotic estimate when it is possible, in particular, when Ψ V λ cλ α, for some constant c and α ]1,2]. For the right tail, we are only interested in the existence of some finite exponential moments. The application of the present work to diffusions in random environment is a work in preparation by the author [74], [73]. For A a process and S a borelian set, we denote τa,s := inf{t, At S}, RA,S := sup{t, At S}. We shall only write τa, x respectively RA, x instead of τa,{x} respectively RA,{x} and τa,x+ instead of τa,[x,+ [. For example, since V has no positive jumps, we see that it reaches each positive level continuously : x >, τv,x+ = τv,x and since moreover V converges to + we have x >, RV,[,x] = RV,x. 96

108 3.1. INTRODUCTION Also let At := inf{as, s [,t]} be the infimum process of A. If A is Markovian and x R we denote A x for the process A starting from x. For A we shall only write A. For any possibly random time T >, we write A T for the process A shifted and centered at time T : s, A T s := AT +s AT. We now recall some facts about V, that is, V conditioned to stay positive. For a spectrally negative Lévy process V, the Markov family Vx,x may be defined as in [8], Section VII.3. For any x, the process Vx must be seen as V conditioned to stay positive and starting from x. We denote V for the process V. It is known that Vx converges in the Skorokhod space to V when x goes to. For X a positive random variable, we denote V X for the process V conditioned to stay positive and starting from the random variable X. More rigorously, V X is the Markov process that conditionally on {X = x} has law Vx. For any positivex, we have from the Markov property and the absence of positive jumps that the process V, shifted at τv,x, its first passage time at x, is equal in law to Vx. In the case where V drifts to +, it is known from [8], Section VII.3, that Vx has the same law as V x conditioned in the usual sense to remain positive. This property, interesting for our study, is unfortunately not true when V oscillates or drifts to in these cases we have to do the conditioning until an hitting time. In the case where V drifts to, we define V to be "V conditioned to drift to + ", as in [8], Section VII.1. The Laplace exponent Ψ V of V satisfies Ψ V = Ψ V κ +. where κ is the non-trivial zero of Ψ V. As a consequence Ψ V >, so V drifts to infinity this is deduced thanks to Corollary VII.2 in [8] and it is also proven that V = V. In order to do our proofs in a systematic way, we often work with V which is defined to be "V conditioned to drift to + " in the case where V drifts to and "only V " in the other cases when V oscillates or drifts to +. As a consequence, V always denotes a spectrally negative Lévy process that does not drifts to it oscillates if V does and it drifts to + if V drifts to + or. In any case we have that for all < x < y, Vxt, t τvx,y is equal in law to Vxt, t τvx,y conditionally on {τvx,y < τvx,],]}. Note that the same identity is true with V instead of V, but the advantage of dealing with V is that τvx,y is always finite while τv x,y can possibly be infinite when V drifts to which simplifies the argumentation. Let W be the scale function of V, defined as in Section VII.2 of [8]. It satisfies < x < y, PτV x,y < τv x,],] = Wx/Wx+y. According to Theorem VII.8 in [8], this function is continus, increasing, and for any λ > Φ V, + e λx Wxdx = 1 Ψ V λ < +. 97

109 3.1. INTRODUCTION Results In the special case of the exponential functional of a drifted brownian motion conditioned to stay positive, all the properties that are established here are already known and sometimes more explicitly. We discuss this case in the next subsection. Our first result is the finiteness of IV and the fact that it admits exponential moments. Theorem The random variable IV is almost surely finite, has finite expectation E[IV ] and [ ] λ < 1/E[IV ], E e λiv < Then, a fundamental point of our study is Proposition which says that for any positive y, IV satisfies the random affine equation IV L = A y +e y IV, where A y is independent of the second term and will be specified later. We see that IV is a positive self-decomposable random variable and is therefore absolutely continuous and unimodal. It is well known see for example expression 1.1 in [55] that the exponential functional IV of a spectrally negative Lévy process V is also self-decomposable as long as it is finite, it can be seen by splitting the trajectory at τv,y, the first passage time at y. Another consequence of is that for any positive y, IV can be written as the random series IV L = k e ky A y k, where the random variables A y k are iid and have the same law as Ay. This decomposition is a very useful tool for the study of the random variable IV and is also the base of the proofs of the results we present below. Our next results make a link between the asymptotic behavior of Ψ V and the properties of IV. Theorem Assume that there is α > 1 and a positive constant C such that for all λ large enough we have Ψ V λ Cλ α. Then for all δ ],1[ and x small enough we have P IV x exp δα 1/Cx 1/α Assume that there is α > 1 and a positive constant c such that for all λ large enough we have Ψ V λ cλ α. Then for all δ > 1 and x small enough we have P IV x exp δα α/α 1 /cx 1/α

110 3.1. INTRODUCTION Let us now recall how is usually quantified the asymptotic behavior of Ψ V. We define, as in [8], page 94, { } σ := sup α, lim λ + λ α Ψ V λ =, { } β := inf α, lim λ + λ α Ψ V λ =. Recall that Ψ V. = Ψ V κ+., so σ and β are identical whether they are defined from Ψ V or Ψ V. If Ψ V has α-regular variation for α [1,2] for example if V is a drifted α-stable Lévy process with no positive jumps, we have σ = β = α. Recall that Q is the brownian component of V. It is well known that Ψ V λ/λ 2 converges to Q/2 when λ goes to infinity so, when Q >, Ψ V has 2-regular variation, and when Q =, 1 σ β 2, where 1 σ comes from the convexity of Ψ V. Remark When V has bounded variation, we know see for example [8] Section I.1 that the brownian component of V is null, the Lévy measure ν of V satisfies x νdx < + and γ xνdx, the factor of λ in the expression of Ψ 1 1 Vλ, is positive otherwise V would be the opposite of a subordinator. It is thus easy to see that in this case Ψ V λ/λ converges to γ xνdx when λ goes to infinity, 1 so σ = β = 1. In the remaining, we sometimes assume that σ > 1, the reader should be aware that it excludes the case where V has bounded variation. However, this case is quite easy and shall be treated in the remarks. We are now ready to state our general results on the asymptotic tails at of IV : Theorem We have β > β, lim x x 1/β 1 log P IV x =, if σ > 1, σ ]1,σ[, lim x x 1/σ 1 log P IV x = Theorem gives for PIV x a lower bound involving σ and an upper bound involving β. In the case of α-regular variation we can expect, under some extra hypothesis, to get a stronger result. We indeed have : Theorem We assume that there is a positive constant C and α ]1,2] such that Ψ V λ λ + Cλ α, then log P IV x α 1 x Cx 1 α 1. 99

111 3.1. INTRODUCTION The above theorem is true in particular when, for some α ]1,2], V is an α-stable spectrally negative Lévy process with adjonction or not of a drift. In particular, it agrees exactly with the tail given in the next subsection for the particular case of a drifted brownian motion. Remark Since Ψ V λ/λ 2 has always a finite limit at + we get, from Theorem 3.1.2, that there is always a positive constant K depending on V such that for x small enough P IV x e K/x. Remark Note that Theorem holds when β = 1 and 1/β 1 can then equal any number in ],+ [. When V has bounded variation, we even have a stronger result : PIV x is null for x small enough. Remark Recall that IV is unimodal. If was a mode, then we would have PIV x cx for some positive constant c and x small enough, which is incompatible with As a consequence the density of IV is non-decreasing on a neighborhood of. This allows to remark that Theorems 3.1.2, 3.1.4, and Remarks 3.1.6, are true for the density of IV in place of the repartition function PIV.. When V drifts to + we prove Proposition which says that the left tail of IV is the same as the left tail of IV. This implies that all the results we prove for the left tail of IV are true for the left tail of IV : Proposition If V drifts to +, then Theorems 3.1.2, 3.1.4, and Remarks 3.1.6, are true for IV in place of IV. Proposition is an example of how the study of the exponential functional of the Lévy process conditioned to stay positive can be useful for the study of the exponential functional of the corresponding Lévy process. We already mentioned that the law of IV is absolutely continuous but we do not know how smooth the density is in general. The following theorem provides a condition for smoothness : Theorem If σ > 1 and β are such that 2β 2 3σβ +σ +β 1 <, then the density of IV belongs to the Schwartz space. All its derivatives converge to at + and. This theorem admits the following corollary : Corollary If V drifts to +, is such that σ > 1 and is satisfied, then the density of IV is of class C and all its derivatives converge to at + and. 1

112 3.1. INTRODUCTION Here again, the study of the exponential functional of the Lévy process conditioned to stay positive implies results about the exponential functional of the corresponding Lévy process. Remark If Ψ V has α-regular variation with α > 1, then σ = β = α so the condition becomes α 1 2 <, but this is always true for α > 1, so Theorem and Corollary apply. In other words, α-regular variation for the Laplace exponent of V implies smoothness of the density for IV and IV if it is finite when α > 1. In the spectrally positive case, the finiteness of the exponential functional is quite easy to obtain, but our argument also yields the existence of some finite exponential moments. We can state the result as follows : Theorem The random variable IZ is almost surely finite and admits some finite exponential moments. We also obtain a lower bound for the asymptotic tail atof bothiz andiz. This tail is heavier than the one given for IV and this comes from the positive jumps. Theorem If Z has unbounded variation and non-zero Lévy measure then, there is a positive constant c such that e clogx2 PIZ x P IZ x. The lower bound for PIZ x does not require the hypothesis of unbounded variation. Remark If the Lévy measure of Z is the zero measure then it is known, from the Lévy-Khintchine formula, that Z is a drifted brownian motion. The exact asymptotic tail at of IZ is then given by Theorem and it is thinner than the one provided by Theorem The existence of jumps thus plays an important role for the asymptotic tail at of the exponential functional and the proof of Theorem indeed crucially relies on this hypothesis. The study of the spectrally positive case does not go as far as the study of the spectrally negative case. The reason for this is twofold. First, we do not have, in the spectrally positive case, a decomposition of the law of IZ as in 3.1.3, which deprives us of an important tool for the study. Secondly we do not need, in the applications, the results on the exponential functional to be as precise, in the spectrally positive case, as in the spectrally negative case. Indeed, in the study of a diffusion in a spectrally negative Lévy environment V drifting to, a random variable R appears. Its law is the convolution of the laws of IV and IˆV, where ˆV := V is the dual process of V and is thus spectrally positive. The combination of the above theorems shows that for some things the behavior of IV is dominant in the study of R when V has jumps. In particular, the asymptotic tail at of R is the same as the one of IV. 11

113 3.1. INTRODUCTION The rest of the paper is organized as follows. In Section 3.2 we prove some preliminary results on V. In Section 3.3 we prove Theorem and establish Proposition about the self-decomposability of IV. In Section 3.4 we prove Theorems 3.1.2, and by studying the asymptotic behavior of the Laplace transform of IV, and in the case where V drifts to +, we establish a connection between the tails at of the exponential functionals IV and IV. In Section 3.5 we prove Theorem and Corollary via a study of excursions. Section 3.6 is devoted to the spectrally positive case and the proofs of Theorems and The example of drifted brownian motion conditioned to stay positive The most simple case is the intersection of the spectrally positive and the spectrally negative case, that is, when V is a drifted brownian motion. All the results mentioned here are already known in this case. We define the κ-drifted brownian motion by W κ t := Wt κ 2 t. It is known that the two processes W κ and W κ are equal in law. This follows, for example, from the expression of the generator of W κ, or from the fact that for positive κ, the Laplace exponent of W κ is equal to the Laplace exponent of W κ, so the processes conditioned to stay positive have the same law. We thus only consider positive κ. It is known see 4.6 in [3], see also Lemma 6.6 in [4] that IW k is almost surely finite and has Laplace transform E[e λiw k ] = 1 2 2λ κ 2 κ Γ1+κ I κ 2, 2λ where I κ is a modified Bessel function. This expression can also be written E[e λiw k ] = 1 Γ1+κ + j= 1 2λ j j!γ1+j+κ, and it is easy to see that it can be analytically extended in a neighborhood of, so the random variable IW k admits some finite exponential moments. An easy calculation on the asymptotic of this expression when λ goes to infinity yields log E[e λiw k ] 2 2λ λ + Combining and De Bruijn s Theorem see Theorem in [14] we get log P IW k x 2 x x This estimate can be seen as a particular case of Theorem when it is applied in the case of a drifted brownian motion. 12

114 3.2. PRELIMINARY RESULTS ON V AND FINITENESS OF IV As the expression extends to a neighborhood of, we get the expression of the characteristic function of IW k which can be proved, using estimates on modified Bessel functions, to belong to the Schwartz space. Therefore, the density of IW k, which is the Fourier transform of its characteristic function, belongs to the Schwartz space, but this is already included in Theorem Preliminary results on V and finiteness of IV Exponential functionals and excursions theory We fix y >. In this subsection, we use excursions to prove that the integral of exponential V τv,y+. or V stopped at there last passage time at y and respectively are equal in law to some subordinators stopped at independent exponential random variables. It is easy to see that regularity of {y} for the markovian processes Vy and Vy is equivalent to the regularity of{} forv orv which in turn, according to Corollary VII.5 in [8], is equivalent to the fact that V has unbounded variation. The property of {y} being instantaneous for Vy and Vy is equivalent to the same property of {} for V, but this is a well known property of spectrally negative Lévy processes. {y} is thus always instantaneous for Vy and Vy and the only alternative is whether it is regular or not, which corresponds to the fact that V has or not unbounded variation. We apply excursions theory away from y see [8]. Let us denote by L y respectively L y a local time at y of the process Vy respectively Vy and ηy respectively ηy the associated excursions measure. We denote η for η. The inverse of the local time Ly, 1 respectively L, 1 y is a subordinator and Vy respectively Vy can be represented as a Poisson point process on the set of excursions, with intensity measure ηy respectively ηy. Note that this is also true in the irregular case when V has bounded variation if the local time L y respectively L y is defined artificially as in [8], Section IV.5. In this case, the excursion measure is proportional to the law of the first excursion and in particular the total mass of the excursion measure is finite. In the case where V drifts to +, we also consider the excursions of V away from. Then, L denotes a local time at of V and η the associated excursion measure. Given ξ : [,ζ] R an excursion away from y, we define ζξ to be its life-time, H y ξ := sup [,ζξ] ξ y its height and Gξ := ζξ e ξt dt. For any h >, we consider IP h, FP h and N three subsets that make a partition of the excursions of V away from y. These three subsets are respectively : the set of excursions higher than h that stay positive, the set of excursions of height smaller 13

115 3.2. PRELIMINARY RESULTS ON V AND FINITENESS OF IV than h that stay positive, the set of excursions that reach ],] : IP h := {ξ, t [,ζξ],ξt >, H y ξ h}, FP h := {ξ, t [,ζξ],ξt >, H y ξ < h}, N := {ξ, τξ,],] < ζξ}. N does not depend on h. IP and FP are defined as the monotone limits of the sets IP h and FP h : IP is the set of infinite excursions that stay positive and FP is the set of finite excursions that stay positive. η yip and η yn are always finite whereas η yfp is infinite in the regular case when V has unbounded variation. Also, note that η yip = if V oscillates. Lemma Let y be positive and let S be a pure jump subordinator with Lévy measure Gη y. FP, the image measure of η y. FP by G. Let T be an exponential random variable with parameter η yip +η yn, independent of S. We have RV,y τv,y where τ.,. and R.,. are defined in the introduction. e V t dt L = S T, Démonstration. V τv,y+. has the same law as Vy, from the Markov property applied to V at time τv,y and the absence of positive jumps. As a consequence, RV,y e V t dt is equal in law to RVy,y e V τv y t dt and we are left to prove the,y result for the latter. Then, let us fix h >. As it is mentioned in the introduction, Vy t, t,y + h is equal in law to Vyt, t τvy,y + h conditionally on {τvy,y +h < τvy,],]}. Vyt, t τvy,y + h can be built from the Poisson point process on the set of excursions with intensity measure ηy. Vy t, t τvy,y +h can be built from this same process, conditioned not to have jumps in N before its first jump in IP h. In other words, we build Vy t, t τvy,y + h from the process of jumps in FP h stopped at the exponential time that has parameter ηyip h + ηyn at which occurs the first jump in IP h N and conditionally to the fact that this jump belongs to IP h. Then, the process of jumps in FP h and in IP h N are independent and by a property of Poisson point processes, the fact that the first jump in IP h N belongs to IP h is independent of the time when this jump occurs. As a consequence, Vy t, t τvy,y+h is built from a Poisson point process with intensity measure ηy. FP h, until an independent exponential time, T h, of parameter ηyip h + ηyn where we pick, independently, a jump following the law ηy. IP h /ηyip h and we only keep the part of this excursion that is before its hitting time of y +h. 14

116 3.2. PRELIMINARY RESULTS ON V AND FINITENESS OF IV Let R h V y,y be the last passage time of V y at y before τv y,y +h : R h V y,y := sup{t [,τv y,y +h], V y t = y}. From above, if p s s is a Poisson point process in FP with measure ηy. FP and if T h is an independent exponential random variable with parameter ηyip h + ηyn, thenvy t, t R h Vy,y is built by putting aside the excursions of the processp s 1 ps FPh, s T h. SinceVy converges almost surely to+,r h Vy,y converges almost surely to RVy,y, the last passage time at y, when h goes to infinity. On the other hand, IP h N decreases to IP N when h goes to infinity. As a consequence T h increases to an exponential random variable T with parameter ηyip +ηyn >. Also, FP h increases to FP when h goes to infinity. Then, identifying the limits when h goes to infinity of both Vy t, t R h Vy,y and p s 1 ps FPh, s T h, we get that Vy t, t RVy,y is built by putting aside the excursions of the process p s, s T, where T is an exponential random variable with parameter ηyip + ηyn and is independent from p s, s. Now, remark that RVy,y e V y t dt is the sum of the images by G of the excursions of Vy t, t RVy,y away from y. We thus have RV y,y e V y t dt L = <s<t Gp s By properties of Poisson point processes, the process in the right hand side, <s<. Gp s, is the sum of the jumps of a Poisson point process on R +, with intensity measure Gηy. FP. Thus, from the Lévy-Ito decomposition, it has the same law as the subordinator S, which yields the result. Remark In the case where V has bounded variation, the total mass of η y is finite so S is only a compound Poisson process. In particular S T can then be null with positive probability. y > is still fixed and arbitrary, let R y V, be the last passage time of V at before τv,y : R y V, := sup{t [,τv,y], V t = }. In order to study the trajectory of V before R y V,, we now consider excursions away from. Let I y and F y denote respectively the subset of excursions higher than y and lower than y : I y := {ξ, H ξ y}, F y := {ξ, H ξ < y}. A similar proof as for Lemma gives the following lemma. 15

117 3.2. PRELIMINARY RESULTS ON V AND FINITENESS OF IV Lemma Let S be a pure jump subordinator with Lévy measure Gη. F y, the image measure of η. F y by G. Let T be an exponential random variable with parameter η I y which is independent of S. We have R y V, e V t dt L = S T In the case where V drifts to + we need to study the trajectory before RV,, the last passage time of V at. We still consider excursions away from. Let I and F denote respectively the subsets of infinite and finite excursions : I := {ξ, ζξ = + }, F := {ξ, ζξ < + }. A similar proof as for Lemma gives the following lemma. Lemma We assume that V drifts to +. Let S be a pure jump subordinator with Lévy measure Gη. F, the image measure of η. F by G. Let T be an exponential random variable with parameter ηi which is independent of S. We have RV, e Vt dt L = S T V and V shifted at a last passage time To obtain decomposition of the law ofiv, we splitv at its last passage time at a point y and obtain two independent trajectories that we can identify. Lemma Corollary VII.19 of [8] For any positive y, the two trajectories V t, t RV,y and V t+rv,y y, t are independent and the second is equal in law to V. Lemma The two trajectories V t, t R y V, and V t+r y V,, t τv,y R y V, are independent and the second is equal in law to V t, t τv,y. As a consequence we have τv,y L = τv,y R y V, τv,y. We assume that V drifts to +. The two trajectories Vt, t RV, and Vt+RV,, t are independent and the second is equal in law to V. Démonstration. We fix y > and a ],y[. Let us denote by es, s the excursions process of V away from. Recall the notations I y and F y, T y := inf{s, es I y } is the time when occurs the first excursion higher than y and ξ y is this excursion. 16

118 3.2. PRELIMINARY RESULTS ON V AND FINITENESS OF IV Decomposing V as its excursions away from, we see that R y V, is the instant when begins the first excursion higher than y, so V t+r y V,, t τv,y R y V, = ξ y t, t τξ y,y V t, t R y V, is thus a function of es1 es Fy, s T y while V t+r y V,, t τv,y R y V, is a function of ξ y. By properties of Poisson point processes, T y is an exponential random variable independent of ξ y and the process of finite excursions es1 es Fy, s is also independent of ξ y. Therefore the objects es1 es Fy, s T y and ξ y are independent. From this independence we deduce that V t, t R y V, V t+r y V,, t τv,y R y V,, which is the required independence. It only remains to prove that the right hand side in has the same law as V t, t τv,y. Using the Markov property at time τξ a,a, for an excursion ξ a I a, we have that ξ a.+τξ a,a equals in law V a killed when it ever reaches. Since I y I a we can apply this to an excursion ξ y I y and get that ξ y t + τξ y,a, t τξ y,y τξ y,a is equal in law to Vat, t τva,y conditioned to reach y before. Since Va has no positive jumps, reaching y before is the same as reaching y before ],]. As we mentioned in the introduction, Vat, t τva,y conditioned to reach y before ],] is equal in law to Vat, t τva,y. Putting all this together we get ξ y t+τξ y,a, t τξ y,y τξ y,a L = V at, t τv a,y. Since τξ y,a converges almost surely to when a goes to and V a converges in law to V according to Proposition VII.14 in [8], we can let a go to in both members and get ξ y t, t τξ y,y L = V t, t τv,y As a consequence the right hand side in has the same law as V t, t τv,y, which concludes the proof of the first point of the lemma. We now assume that V drifts to + and prove the second point. For the independence, the arguments of the proof of the first point can be repeated, just replacing y by + we consider ξ, the infinite excursion away from, instead of the first excursion higher than y. To prove that Vt+RV,, t is equal in law to V, it suffices to prove that ξ is equal in law to V. Let y be finite, we know from the proof of the first point that is true for any excursion in I y. Since ξ I y we have ξ t, t τξ,y L = V t, t τv,y. 17

119 3.3. FINITENESS, EXPONENTIAL MOMENTS, AND SELF-DECOMPOSABILITY Since y is arbitrary and τv,y converges almost surely to + when y goes to +, we get ξ t, t L = V t, t, which gives the result. 3.3 Finiteness, exponential moments, and self-decomposability Finiteness and exponential moments : Proof of Theorem We are grateful to an anonymous referee for the following proof that is considerably simpler than the proof given by the author in the previous versions of this paper. Démonstration. of Theorem The idea of the proof is to provide finite upper bounds for the moments of IV. The first step is to prove that E[IV ] < +. Using Fubini s Theorem and Corollary VII.16 of [8] we have E[IV ] = + E[e V t ]dt = Now, using Corollary VII.3 of [8] we get E[IV ] = + + = e y WyPτV,y dtdy = e y PV t dydt e yywy PVt dydt. t + Since PτV,y < + = Psup [,+ [ V y = e κy we obtain E[IV ] = + e 1+κy Wydy < +, e y WyPτV,y < + dy. where the finiteness comes from the fact that + e λy Wydy < + for λ > κ. As a consequence, the exponential functional IV is almost surely finite and has finite expectation. We now turn to the proof of the finiteness of the Laplace transform. We proceed by bounding the moments of the exponential functional. For any x let us define hx := E[IVx]. For any x > we have IV = + e V t dt + τv,x e V t dt L = 18 + e V x t dt = IV x,

120 3.3. FINITENESS, EXPONENTIAL MOMENTS, AND SELF-DECOMPOSABILITY where, for the equality in law, we used the Markov property for V at time τv,x. As a consequence we have x >, hx = E[IV x] E[IV ] < Now, note that for any k 1, IV E[ ] [ + + ] k = E... e V t 1... e V t k dt 1...dt k [ ] = k! E e V t 1... e V t k dt 1...dt k, t 1 <...<t k so that IV k 1, E[ ] [ ] k /k! = E e V t 1... e V t k dt 1...dt k t 1 <...<t k Let us prove by induction that for any k 1, E[ IV k ] k! E[IV ] k < is clearly true for k = 1. Let us assume that it is true for some arbitrary rank k. According to 3.3.2, E[IV k+1 ]/k +1! equals [ ] E e V t 1... e V t k e V t k+1 dt 1...dt k dt k+1 t 1 <...<t k+1 [ + ] =E e V t 1... e V t k e V s ds dt 1...dt k t 1 <...<t k t [ [ k + =E e V t 1... e V t k E e V s ds ] σv u, u t k ]dt 1...dt k. t 1 <...<t k t k From the Markov property at time t k, the conditional expectation in the above expression equals hv t k which, according to 3.3.1, is almost surely less than E[IV ]. We thus get [ t1 <...<t k ] E[IV k+1 ]/k +1! E[IV ] E e V t 1... e V t k dt 1...dt k = E[IV ] E[ IV k ] /k! E[IV ] k+1, where we used and the induction hypothesis. Thus the induction is proved. As a consequence for all λ ],1/E[IV ][ we have [ ] E e λiv = [ IV k λ k E ] k /k! λe[iv ] k < +. The finiteness of the Laplace transform is obvious for λ, so the result is proved. k 19

121 3.3. FINITENESS, EXPONENTIAL MOMENTS, AND SELF-DECOMPOSABILITY Remark If V does not oscillate, the finiteness of IV can be derived as a consequence of Lemma Indeed, from the second statement of Lemma applied to V, we have IV L = + e V t+rv, dt = + RV, e V t dt + e V t dt = IV, and V drifts to +, so Theorem 1 in [12] ensures that IV is almost surely finite which yields the result Decomposition of the law of IV In this subsection, we prove that the law of IV is solution of the random affine equation and we give a decomposition of its non-trivial coefficient A y. This is a key point of our analysis of the law of IV. Proposition For any y >, the law of IV satisfies the random affine equation IV L = τv,y e V t dt+s T +e y IṼ, where the three terms of the right hand side are independent, S T is as in Lemma 3.2.1, and Ṽ is an independent copy of V. We define to lighten notations. A y := τv,y e V t dt+s T As a consequence, IV has the same law as the sum of a random series : IV L = k e ky A y k, where the random variables A y k are iid and have the same law as Ay. Remark The almost sure convergence of the random series in is a consequence of the almost sure finiteness, given by Theorem 3.1.1, of the positive random variable IV. Also, it is a well known fact on random power series with iid coefficients that their radius of convergence is almost surely equal to a constant belonging to {,1}. Since this constant, in the case of the power series in 3.3.5, has been proved to be greater that e y, we deduce that it equals 1. Démonstration. of Proposition We fixy >. AsV has no positive jumps and goes to infinity we haveτv,y RV,y < + and V τv,y = V RV,y = y. 11

122 3.3. FINITENESS, EXPONENTIAL MOMENTS, AND SELF-DECOMPOSABILITY We write : IV = = RV,y RV,y RV,y L = e V t dt+ + RV,y e V t dt + e V t dt+e y e V t+rv,y y dt e V t dt+e y IṼ, where we used Lemma for the last equality in which Ṽ is an independent copy of V. We now decompose : RV,y e V t dt = τv,y e V t dt+ RV,y τv,y e V t dt Since V τv,y = y, combining with the Markov property at time τv,y, the two terms in the right hand side of are independent : τv,y e V t dt RV,y τv,y e V t dt. Now, thanks to Lemma 3.2.1, the second term has the same law as S T with S T as in the lemma. This achieves the proof. We have two remarks here : Remark It is possible to prove Theorem the fact that IV is finite and admits some finite exponential moments by invoking and proving that each of the two terms composing A y admit some finite exponential moments. Remark Let S and T be as in Proposition and ǫ ],E[S 1 ][ where we do not bother with the fact that E[S 1 ] is finite or not. Then we have PS T t P S t/ǫ t PT t/ǫ. The first factor in the right hand side converges to 1 thanks to the law of large numbers for Lévy processes see for example Theorem 36.5 in [59] and the second is equal to e pt/ǫ, where p is the parameter of the exponential random variable T. Therefore, the Laplace transform of S T is not finite everywhere, so neither is the Laplace transform of IV because of This is why we can not say better than "IV admits some finite exponential moments". 111

123 3.4. ASYMPTOTIC TAIL AT : PROOF OF THEOREMS 3.1.2, AND Asymptotic tail at : Proof of Theorems 3.1.2, and First, let us prove that Theorem easily implies Theorem and prove Remark Démonstration. of Theorem Assume that Theorem is proved. We first prove Let us fix β > β and ǫ >. From the definition of β we have that Ψ V λ ǫλ β for all λ large enough. Using the first point of Theorem we deduce that lim sup x x 1/β 1 log P IV x β 1/ǫ 1/β 1. Since ǫ can be chosen as small as we want we obtain We now assume that σ > 1 and prove Let us fix σ ]1,σ[ and M >. From the definition of σ we have that Ψ V λ Mλ σ for all λ large enough. Using the second point of Theorem we deduce that liminf x x 1/σ 1 log P IV x σ σ /σ 1 /M 1/σ 1. Since M can be chosen as large as we want we obtain Démonstration. of Remark Let us assume that V has bounded variation. As it can be seen from Remark 3.1.3, it is the difference of a positive drift dt and a pure jump subordinator S t : t >, Vt = dt S t dt. Let us fix y >, we have almost surely τv,y e Vt dt τv,y e dt dt = 1 d 1 e τv,y Since V has bounded variation, we have PVt >, t τv,y > see for example 47.1 in [59] and we can see that V t, t τv,y is equal in law to V, t τv,y conditioned in the usual sense to remain positive. Combining with 3.4.1, we see that almost surely IV τv,y e V t dt 1 d 1 e τv,y. Then, since τv,y converges almost surely to + when y goes to +, we deduce that IV is more than the positive constant 1/d almost surely. As a consequence PIV x is null for x 1/d. In the next Subsection we prepare the proofs of Theorems and

124 3.4. ASYMPTOTIC TAIL AT : PROOF OF THEOREMS 3.1.2, AND Laplace transform of IV In order to prove asymptotic estimates on PIV x, we first study the Laplace transform of IV via the decomposition given by Proposition It is thus natural that we need first to study the Laplace transform of A y. First, let us define a notation. V is a spectrally negative Lévy process, so, according to Theorem VII.1 in [8], the process τv,. is a subordinator which Laplace exponent Φ V is defined for λ by [ Φ V λ := log E e λτv,1 ], and we have Φ V = Ψ 1 V. Proposition We fix y >. Let A y be as in Proposition 3.3.2, then, for all ǫ > and λ large enough we have 1 ǫyφ V e y λ log E [ e λay ] 1+ǫyΦV λ Démonstration. of Proposition According to the definition of A y in the Proposition 3.3.2, A y can be decomposed as the sum of two independent random variables, one having the same law as τv,y e V t dt and another having the same law as S T, defined as in Lemma Let Φ S be the Laplace exponent of the subordinator S : λ >, Φ S λ := log E [ e λs 1 ] We can see that the Laplace transform of the random variable S T is given by We thus have log E [ e λay ] = log E λ >, E [ ] η e λs T yip +ηyn = ηyip +ηyn+φ S λ. = log E [ [ exp exp λ λ τv,y τv,y e V t dt e V t dt ] ] log E [ ] e λs T log p p+φ S λ, where we denoted p for the constant η yip + η yn. Using the fact that V is non-negative and the first point of Lemma we have τv,y e V t dt τv,y sto τv,y,

125 3.4. ASYMPTOTIC TAIL AT : PROOF OF THEOREMS 3.1.2, AND where sto denotes a stochastic inequality. As a consequence of and of the definition of Φ V we have log E [ exp λ τv,y e V t dt ] Combining this inequality with we obtain log E [ ] e λay yφv λ log [ ] log E e λτv,y = yφ V λ. p p+φ S λ Using the first point of Lemma and Lemma we have [ ] τv,y log E exp λ e V t dt = log E [ e λ τv,y e V t dt ] [ λ S T] /E e, where S T is as S T from Lemma Here again, if Φ S denotes the Laplace exponent of the subordinator S as in we have Moreover we have [ ] λ >, E e λ S T = τv,y η I y η I y +Φ Sλ. e V t dt e y τv,y. Putting together the above three expressions and the definition of Φ V we obtain [ ] τv,y log E exp λ e V t dt yφ V e y η I y λ+log η I y +Φ Sλ. Combining the above inequality with and the fact that the term logp/p+ Φ S λ is non-negative we get log E [ ] e λay yφv e y η I y λ+log η I y +Φ Sλ According to the Lévy-Khintchine formula for subordinators. The Laplace exponent Φ S can be written Φ S λ = γ S λ+ + 1 e λx ν S dx, so by dominated convergence, there exists a positive constant C S such that for large λ, Φ S λ C S λ. Similarly, there is a positive constant C S such that for large λ, Φ Sλ C Sλ. On the other hand, sinceψ V λ/λ 2 is bounded whenλgoes to infinity, there is a positive constant c such that Φ V λ cλ 1/2 for large λ. Combining all this with and we get for any fixed ǫ > and λ large enough. 114

126 3.4. ASYMPTOTIC TAIL AT : PROOF OF THEOREMS 3.1.2, AND Proposition Assume that there is α 1 and a positive constant C such that for all λ large enough we have Ψ V λ Cλ α, then we have [ ] lim inf log E e λiv /λ 1/α α/c 1/α λ + Assume that there is α 1 and a positive constant c such that for all λ large enough we have Ψ V λ cλ α, then we have [ ] lim sup log E e λiv λ + /λ 1/α α/c 1/α Démonstration. Let us fix y > for which we apply the decomposition Let us denote [ ] Mλ := log E e λiv = [ E e k log λe ky A y], where the second equality comes from and from the fact that the sequence A y k k is iid. To establish the left tail of IV we study the asymptotic behavior of Mλ and the latter, thanks to the above expression, is related to the asymptotic behavior of loge[e λay ]. Unfortunately, Proposition can not be applied simultaneously to all the terms of the sum defining Mλ. We separate this sum into three parts : a sum over a finite number of small indices for which we can apply to each term, a sum over an infinite number of large indices that can be neglected, and a sum over the remaining indices in finite number that can be neglected. We now prove the second point of the proposition. We assume that there is α 1 and a positive constant c such that for all λ large enough we have Ψ V λ cλ α, and prove Let us fix δ > 1. Since Ψ V. = Ψ V κ+. and Φ V = Ψ 1, we have V for all λ large enough that Φ V λ δλ 1/α /c 1/α. According to Proposition there exists λ δ > 1 such that is satisfied with ǫ = δ 1 for all λ λ δ. By increasing λ δ if necessary, we can also assume that Φ V λ δλ 1/α /c 1/α for all λ λ δ. Putting all this together we get λ λ δ log E [ e λay ] δ 2 yλ 1/α /c 1/α Also, let us choose M ],1[ small enough so that λ [,M], log E [ e λay ] 2λE[A y ]. For any λ > λ δ, we define n 1 λ := logλ/λ δ /y and n 2 λ := logλ/m/y. From the definition of Mλ we can write λ > λ δ, Mλ = T 1 λ+t 2 λ+t 3 λ,

127 3.4. ASYMPTOTIC TAIL AT : PROOF OF THEOREMS 3.1.2, AND with T 1 λ := T 3 λ := n 1 λ k= + k=n 2 λ+1 [ log E e λe ky A y], T 2 λ := [ log E e λe ky A y]. n 2 λ k=n 1 λ+1 [ log E e λe ky A y], From the definition of n 1 λ, can be applied to each term of the sum defining T 1 λ, we thus have n 1 λ T 1 λ δ 2 yλ/c 1/α We get that for λ large enough k= e ky/α = δ 2 yλ/c 1/α1 e yn 1λ+1/α 1 e y/α. T 1 λ δ 2 yλ/c 1/α 1 1 e y/α Using the monotony of the Laplace transform and the definitions of n 1 λ and n 2 λ we get T 2 λ n 2 λ n 1 λlog E [e λe n 1 λ+1y A y] y +logλ δ/m y From the definitions of n 2 λ and M we have T 3 λ 2λE[A y ] + k=n 2 λ+1 log E [ e λ δa y ] < e ky = 2λe yk 2λ+1 E[A y ]/1 e y 2ME[A y ]/1 e y < Putting , , and into we obtain that for λ large enough lim supmλ/λ 1/α δ 2 y λ + c 1/α 1 e y/α. Since, from the definition Mλ = loge[e λiv ] which does not depend on δ nor on y, we can let δ go to 1 and then y go to in the above expression. We obtain [ ] lim sup log E e λiv /λ 1/α α/c 1/α. λ + For the first point of the proposition, we proceed exactly as for the second point, using the lower bound of instead of the upper bound and noticing that n 1 λ converges toward + as λ goes to infinity, so that the factor 1 e yn 1λ+1/α /1 e y/α converges to 1/1 e y/α. 116

128 3.4. ASYMPTOTIC TAIL AT : PROOF OF THEOREMS 3.1.2, AND Tail at of IV : proof of Theorems and We now prove Theorems and by using Proposition together with the deep link that exists between the left tail of a random variable and the asymptotic behavior of its Laplace transform. Démonstration. of Theorem We assume that there is α > 1 and a positive constant C such that for all λ large enough we have Ψ V λ Cλ α. We fix δ ],1[. According to the first point of Proposition 3.4.2, there exists λ δ > such that for all λ > λ δ we have [ ] E e λiv exp δ 1 1/α αλ 1/α /C 1/α Let us fix x ],δ α 1/α C 1/α λ 1 α/α δ [. Using Markov inequality and we get that for any λ > λ δ, P IV x [ ] e λx E e λiv exp λx δ 1 1/α αλ 1/α /C 1/α. Sincex ],δ α 1/α C 1/α λ 1 α/α δ [ we haveδc 1/α 1 x α/α 1 λ δ so, in the above inequality, we can replace λ by δc 1/α 1 x α/α 1. We obtain which is P IV x exp δα 1/Cx 1/α 1, We now prove the second point. Assume that there is α > 1 and a positive constant c such that for all λ large enough we have Ψ V λ cλ α. We fix δ > 1 and r ],1[. According to the second point of Proposition 3.4.2, there exists λ δ > such that for all λ > λ δ we have [ ] E e λiv exp δ rα 1/α αλ 1/α /c 1/α Let us fix x ],δ α 1/α αc 1/α λ 1 α/α δ [. Using we get that for any λ > λ δ, exp δ rα 1/α αλ 1/α /c 1/α [ ] E e λiv [ ] [ ] = E e λiv 1 {IV x} +E e λiv 1 {IV >x} so we get P IV x +exp λx, P IV x exp δ rα 1/α αλ 1/α /c 1/α exp λx. Since x ],δ α 1/α αc 1/α λ 1 α/α δ [ we have δα α/α 1 c 1/α 1 x α/α 1 λ δ so, in the above inequality, we can replace λ by δα α/α 1 c 1/α 1 x α/α 1. We obtain P IV x exp δ 1+rα 1/α α α/α 1 /cx 1/α 1 exp δα α/α 1 /cx 1/α

129 3.4. ASYMPTOTIC TAIL AT : PROOF OF THEOREMS 3.1.2, AND Since δ > δ 1+rα 1/α, the second term converges to faster than the first one when goes to so we get, for x large enough, P IV x 1 2 exp δ 1+rα 1/α α α/α 1 /cx 1/α 1 exp δα α/α 1 /cx 1/α 1, where the last inequality is true for x small enough and comes from the fact that r ],1[. This is precisely Démonstration. of Theorem We assume that there is a positive constantc andα ]1,2] such thatψ V λ λ + Cλ α. From Proposition we deduce that log E [ e λiv ] λ + αλ1/α /C 1/α, and the application of De Bruijn s Theorem see Theorem in [14] yields the result Connection between IV and IV : proof of Proposition In this subsection, we assume that V drifts to + so that IV < + and we prove a simple connection between the asymptotic tails at of IV and IV. Proposition If V drifts to +, there is a positive constant c such that for all positive ǫ and x small enough, PIV x P IV x PIV 1+ǫx/cǫx. As in the proof of Proposition 3.3.2, we decompose IV as the sum of two independent random variables, one having the same law as a subordinator stopped at an independent exponential time and the other having the same law as IV. We first need an easy lemma about the asymptotic tail at of a subordinator stopped at an independent exponential time. Lemma Let S be a subordinator and T an independent exponential random variable, there exists a positive constant c such that for all x small enough PS T < x cx. Démonstration. We prove in fact a stronger result : the function x PS T < x is sub-additive, that is x,y, PS T < x+y PS T < x+ps T < y

130 3.4. ASYMPTOTIC TAIL AT : PROOF OF THEOREMS 3.1.2, AND and the lemma follows easily. Recall from the introduction the notation τs, h+ for τs,[h,+ [. Let x,y > the case when x = or y = is obvious, we have PS T < x+y = PS T < x+pt τs,x+, T < τs,x+y+ PS T < x+p T τs,x+, T < τs,[s τs,x+ +y,+ [ because S τs,x+ x almost surely, = PS T < x+pt τs,x+ PT < τs,y+, from the characteristic property of the exponential distribution and the Markov property applied to S at time τs,x+. Since PT τs,x+ 1 and PT < τs,y+ = PS T < y we obtain We now prove the proposition Démonstration. of Proposition We write : IV = = RV, RV, RV, L = e Vt dt+ e Vt dt+ + RV, + e Vt dt e Vt+RV, dt e Vt dt+iv, where we used the second point of Lemma for the last inequality in which the two terms RV, e Vt dt and IV are independent. We thus have PIV x P IV x According to Lemma 3.2.4, the term RV, e Vt dt has the same law ass T where S is a pure jump subordinator with Lévy measure Gη. F, the image measure by G of η. F, and T an independent exponential random variable with parameter ηi. For ǫ > and x, combining the equality in law of lemma with we obtain PIV 1+ǫx P IV x PS T ǫx cǫxp IV x, for an appropriate constant c >, when x is small enough, according to lemma Combining with we get the result. Now, if V drifts to +, Proposition easily implies that the results of Theorems 3.1.2, 3.1.4, and of Remarks 3.1.6, are also true for IV, as long as they are for IV. Proposition is thus proved. 119

131 3.5. SMOOTHNESS OF THE DENSITY : PROOF OF THEOREM Smoothness of the density : Proof of Theorem According to Proposition 3.3.2, IV contains, as a convolution factor, the sum of infinitely many independent multiples of random variables having the same law as S T S T being as in Lemma We can thus use a condition on the Lévy mesure of S to have the existence of the smooth density for IV. Actually, the condition that we check for S is the one of Proposition 28.3 in [59], which is a condition on the Lévy measure of a Lévy process for it to have ac density with bounded derivatives. As a jump of S is the image by the mapping G of an excursion of V, we start by lemmas on the excursions of V. Lemma Assume σ > 1 and choose σ such that 1 < σ < σ. For all h small enough we have η ξ, ζξ < +, H ξ > h h σ 1. Démonstration. We consider excursions away from. Let M > be a fixed level and, for any h ],M[, p h denotes the probability that V has no finite excursion of height in ]h,m[ before its first excursion higher than M. Since the set of finite excursions of height in ]h,m[ and the set of excursions higher than M are disjoint, we have, by a property of Poisson point processes p h = η ξ, H ξ > M η ξ, H ξ > M+η ξ, ζξ < +, H ξ ]h,m[, so we only need to give an upper bound for p h. Now, note that p h is only the probability that τv,h and τv,m belong to the same excursion of V away from, so p h = P s ]τv,h,τv,m[, V s = P s ],τv h,m[, V h s, where we used the Markov property at time τv,h. = P τv h,m < τvh,],] = W h/w M +h W h/w M, h where W is the scale function of V, and where the last equivalence comes from the continuity of the scale function see the Introduction. Recall that W has a Laplace transform given by the expression + W xe λx dx = 1, λ Ψ V λ 12

132 3.5. SMOOTHNESS OF THE DENSITY : PROOF OF THEOREM Now, for any h >, by increases of W we have 2h 2h W h h 1 W xdx e 2 h 1 W xe x/h dx + e 2 h 1 W xe x/h dx = e 2 hψ V h 1, because of From the definition of σ and the fact that σ < σ, we know that Ψ V h 1 /e 2 h σ provided h is small enough. We deduce that whenever h is small enough, W h h σ 1. We thus get for h small enough, and taking the inverse in 3.5.1, p h 2h σ 1 /W M, η ξ, ζξ < +, H ξ ]h,m[ η ξ, H ξ > M p h Since η ξ, ζξ < +, H ξ > h η ξ, ζξ < +, H ξ ]h,m[, the combination of and yields the result. We got rid of the constants since the same result with the same constant is true for σ increased a little bit. We now need the following lemma which states that for a spectrally negative Lévy process, the excursions of a given height can be split into two independents parts of which the law are known. Lemma For any h >, assume that the processx follows the lawη. H. > h. Then we have : Xs, s τx,h V s, s τv,h, Xs, τx,h s τx.+τx,h, V h s, s τvh,, Xs, s τx,h Xs, τx,h s τx.+τx,h,. Note that the time τv h, may possibly be infinite, but this is unlikely when h is small. Démonstration. of Lemma The first point is a consequence of the first point of Lemma The second and third points come from the Markov property. We can now prove the main lemma of this section, it will allow us to check the condition on the Lévy mesure of S. 121

133 3.5. SMOOTHNESS OF THE DENSITY : PROOF OF THEOREM Lemma Assume σ > 1 and choose σ and β such that 1 < σ < σ β < β. We also choose ǫ > β 1 1 and fix C > an arbitrary constant. Then, for r small σ 1 enough we have η ξ, ζξ ]x,r] x σ 1/β, x ],Cr 1+ǫ ]. Démonstration. Let us fix r > and x ],Cr 1+ǫ ], then η ξ, ζξ ]x,r] η ξ, ζξ [x,r], H ξ > x 1/β = η ξ, ζξ ]x,r] H ξ > x 1/β η ξ, H ξ > x 1/β, where η. H. > x 1/β is the measure of excursions conditioned to be higher than x 1/β. The last quantity thus equals [η ξ,ζξ > x H ξ > x 1/β η ξ,ζξ > r H ξ > x 1/β ] η ξ, H ξ > x 1/β We now study the three quantities appearing in and show that this expression is of the same order as η ξ, H ξ > x 1/β for which Lemma provides a lower bound. We start by proving that η ξ, ζξ > r H ξ > x 1/β converges to uniformly in x ],Cr 1+ǫ ] when r goes to. First, Lemma gives for all x in ],Cr 1+ǫ ] that η ξ, ζξ > r H ξ > x 1/β P τv, > r/2 +P τv,x 1/β > r/2. x 1/β The first thing is thus to prove that sup P τv, > r/ x ],Cr 1+ǫ x ] 1/β r Before reaching if it does the process V reaches ],] not necessarily at x 1/β. We define τ N x1/β := τv,],]. For all x in ],Cr 1+ǫ ] we have x 1/β P τv, > r/2 P τ N > r/4 +P τ N x 1/β x 1/β x1/β r/4, τv.+τ N x 1/β x 1/β, > r/ Since x τ N x1/β is stochastically increasing in the variable x we have sup P τ N > r/4 = P τ N > r/4 = P V r/4 > C 1/β r 1+ǫ/β. x ],Cr 1+ǫ x 1/β C 1/β r 1+ǫ/β ] 122

134 3.5. SMOOTHNESS OF THE DENSITY : PROOF OF THEOREM We introduce T, an exponential random variable with parameter 1 independent of the process V, and a decreasing function q :],+ [ ],+ [ that converges to + as r go to and will be specified latter. We have P V r/4 > C 1/β r 1+ǫ/β P V T/qr > C 1/β r 1+ǫ/β +PT/qr > r/4 V T/qr = P exp > exp C 1/β +e rqr/4 r [ 1+ǫ/β ] V T/qr e C1/β E exp +e rqr/4, from Markov inequality, r 1+ǫ/β = 1/Φ V 1 qrr1+ǫ/β ec1/β +e rqr/4, Ψ V r 1+ǫ/β /qr 1 from the expression of the Laplace transform of the random variable V T/qr that can be found page 192 of [8], and where Φ V = Ψ 1 as in Section The V last expression goes to if these three conditions are satisfied : qrr r +, Φ V qrr 1+ǫ/β r, qr/ψ V r 1+ǫ/β Φ V qrr 1+ǫ/β r. From the definition of σ and β and the fact that σ < σ β < β, we have Φ V u u 1/σ, Ψ V u u σ and Φ V u u 1/β, provided u is large enough. The three conditions can thus be simplified and we only need to have : qrr r +, qr 1/σ r 1+ǫ/β r, qr 1 1/β r σ 11+ǫ/β r. Elevating qr to the right power so it makes its exponent disappear, these three conditions become qrr r +, qrr σ 1+ǫ/β r, qrr σ 11+ǫ/β 1 r. Since σ < β we can check that σ /β > σ 1/β 1, so the third condition implies the second. We then only need to verify the first and the third condition, 123

135 3.5. SMOOTHNESS OF THE DENSITY : PROOF OF THEOREM but from the choice of ǫ, we have that σ 11+ǫ/β 1 > 1, so qr can be chosen such that the first and the third conditions are satisfied. This yields sup P τ N > r/ x ],Cr 1+ǫ x 1/β ] r We now turn to the second term of It is known that the jumps of V is a Poisson process with intensity measure ν, the Lévy measure of V. The probability that V has a jump smaller than r before time r/4 is thus 1 e rν ], r]/4 rν ], r]/4 := γr and this goes to for any Lévy measure. As a consequence, on {τ N x1/β r/4} we have V τ N x 1/β x1/β [ r,], exepted on some event having probability less than γr. We thus get sup P τ N x ],Cr 1+ǫ x1/β r/4, τv.+τ N x ] 1/β x 1/β, > r/4 P τv,r > r/4 +γr, and we now only need to show that τv,r/r converges to in probability. Now recall that we know from Theorem VII.1 in [8] that the Laplace transform of this random variable is given by [ E exp λ τv ],r = e rφ V r λ/r. Since Φ V u u 1/σ for u large enough, the last quantity converges to 1 when r goes to so we indeed have the convergence to in probability of τv,r/r and as a consequence sup P τ N x ],Cr 1+ǫ x1/β r/4, τv.+τ N x ] 1/β x 1/β, > r/4. r Putting this, together with 3.5.9, in we get We now deal with the second term of 3.5.6, more precisely we prove that sup P τv,x 1/β > r/ x ],Cr 1+ǫ ] r Because of the increases of the quantity PτV,x 1/β > r/2 we can write sup P τv,x 1/β > r/2 = P τv,c 1/β r 1+ǫ/β > r/2 x ],Cr 1+ǫ ] P τv,c 1/β r 1+ǫ/β > r/2, according to the first point of Lemma applied with y = C 1/β r 1+ǫ/β. 124

136 3.5. SMOOTHNESS OF THE DENSITY : PROOF OF THEOREM Therefore, will follow if we prove that the random variable τv,c 1/β r 1+ǫ/β /r converge to in probability as r goes to. Again, we see from Theorem VII.1 in [8] that the Laplace transform of this random variable is given by E [ exp λ τv,c 1/β r 1+ǫ/β r ] = e C1/β r 1+ǫ/β Φ V λr 1, but, since Φ V u u 1/σ provided λ is large enough, we have that, for small r, r 1+ǫ/β Φ V λr 1 r 1+ǫ/β 1/σ λ 1/σ. By the choice of ǫ, we have 1 + ǫ > β 1σ 1 > β /σ, so the last quantity converges to as r goes to. This shows that the Laplace transform of τv,c 1/β r 1+ǫ/β /r converges to1asr goes toso we get the asserted convergence to in probability as r goes to and follows. Putting and in we get sup η ξ, ζξ > r H ξ > x 1/β, x ],Cr 1+ǫ ] r which means that among the excursions of heigh greater than x 1/β, those of length greater than r are in negligible proportion, and it thus remains to show that those of length greater than x are in non-negligible proportion. We want to show that lim inf r inf x ],Cr 1+ǫ ] η ξ, ζξ > x H ξ > x 1/β > From the second point of Lemma we have for all x in ],Cr 1+ǫ ], η ξ, ζξ > x H ξ > x 1/β P τv, > x P τ N > x x 1/β x 1/β As we did before, we introduce T, an exponential random variable with parameter 1 independent of the process V, and a decreasing function q :],+ [ ],+ [ that converges to + at and will be specified latter. For any x in ],Cr 1+ǫ ] we have P τ N > x = P V x > x 1/β x 1/β P V T/qx > x 1/β PT/qx < x V T/qx = P > 1 1+e xqx x 1/β We choose qx = 1/x and, because of , will follow if we prove that the random variable V T/qx/x 1/β converges in probability to as x goes to. According to [8] p 192, the Laplace transform of this random variable is given by E [ exp λ V T/qx x 1/β ] = qx Φ V qx λ/x 1/β This last quantity converges to 1 when x goes to if Φ V qxqx Ψ V λ/x 1/β. 125

137 3.5. SMOOTHNESS OF THE DENSITY : PROOF OF THEOREM λ/x 1/β is negligible compared to Φ V qx when x goes to, Ψ V λ/x 1/β is negligible compared to qx when x goes to. These two conditions can be written x 1/β Φ V qx r +, Ψ V λ/x 1/β /qx r. Now, because of the definition of β, because β > β, and because qx = 1/x, it is easy to see that these two conditions are satisfied. This shows that the Laplace transform of the random variable V T/qx/x 1/β converges to 1 as x goes to, so this random variable converges to in probability and follows. For the factor η ξ, H ξ > x 1/β of we note that it is trivially more than η ξ, ζξ < +, H ξ > x 1/β and we can use Lemma which yields η ξ, H ξ > x 1/β x σ 1/β, for x small enough. Putting , and in 3.5.5, we get the asserted result. Here again, we actually obtain the result up to a multiplicative constant, but since the result is still true if, for example, we increase σ a little bit, we can get rid of the constant. We can now prove Theorem Démonstration. of Theorem We make the assumption that is satisfied so, by continuity of the left hand side of in σ and β, we can choose σ and β to be as in Lemma 3.5.3, but close enough to respectively σ and β so that they also satisfy We also choose ǫ as in Lemma We fix y >. Let S is a pure jump subordinator with Lévy mesure µ. := Gη y. FP, the image measure of η y. FP by the mapping G. From the Lévy-Khintchine formula, the characteristic function of St is where so, taking the real part, ξ R, Φ S ξ := ξ R, RΦ S ξ = E [ e iξst] = e tφ Sξ, + + π/ ξ 2ξ2 π 2 e iξx 1µdx, 1 cosξx µdx 1 cosξx µdx π/ ξ x 2 µdx

138 3.5. SMOOTHNESS OF THE DENSITY : PROOF OF THEOREM We now prove that the measure µ satisfies the hypothesis of Proposition 28.3 of [59]. We have for any r ],1[ : r r r 1+ǫ x 2 µdx = 2 xµ[x,r[dx 2 xµ[x,r[dx We thus need to minorate µ[x,r[ for x ],r 1+ǫ ] : µ[x,r[ = ηy{ξ, Gξ [x,r[} FP { } FP = η y ξ, ζξ e ξt dt [x,r[ from the definition of G, { ηy ξ, ζξ [e 2y x,r[, supξ 2y } FP { ηy ξ, ζξ [e 2y x,r[ } ηy{ξ, supξ > 2y} ηyfp c = η { ξ, ζξ [e 2y x,r[ } ηy{ξ, supξ > 2y} ηyip ηyn = η { ξ, ζξ [e 2y x,r[ } c, where we have put c := η y{supξ > 2y} + η yip + η yn. Note that c is well defined because the quantities η y{supξ > 2y}, η yip and η yn are finite. We now apply Lemma taking C = e 2y and we deduce that for r small enough, x ],r 1+ǫ ], µ[x,r[ e 2yσ 1/β x σ 1/β c. Combining this with , we get that whenever r is small enough : If r r 1+ǫ x 2 µdx 2e 2yσ 1/β x 1 σ 1/β dx 2c, r 1+ǫ 2e 2yσ 1/β 2 σ 1/β r1+ǫ2 σ 1/β cr 21+ǫ. xdx 1+ǫ2 σ 1/3β 1 < 2, then, choosing δ ],2 1+ǫ2 σ 1/β [ and combining the above estimate with , we get that Φ S ξ RΦ S ξ ξ δ, whenever ξ is large enough. As the only assumption on ǫ is that it is greater than β 1/σ 1 1, we can choose ǫ such that is satisfied if and only if β 1 σ 1 2 σ 1/3β 1 < 2, which is equivalent to the fact that σ and β satisfy Therefore, we have proved that there exists δ > such that is true. 127

139 3.5. SMOOTHNESS OF THE DENSITY : PROOF OF THEOREM Let T be an exponential random variable with parameter p := η yip +η yn which is independent of S, the Fourier transform of S T is E [ ] + e iξs T = pe tφsξ e pt dt = because of Proposition gives the decomposition IV = k p p Φ S ξ = O ξ δ, ξ + e ky B y k + k e ky C y k where all the random variables in the two series are mutually independent, and where each term B y k has the same law as S T. Therefore, the characteristic function of IV is the product of a characteristic function bounded by 1 and of k E[eie ky ξs T ] which, thanks to , goes to faster than any negative power of ξ. This proves that the density of IV is of class C and that all its derivatives converge to when x goes to +. The derivatives of the density of IV also converge to when x goes to since this density is of class C and null on ],[. To prove that φ IV actually belongs to the Schwartz space, we have to study a little more deeply the infinite product. Let us denote by ψ the characteristic function of k e ky C y k. Then φ IV ξ = ψξ ] [e k E ie ky ξs T The random variable S T admits moments of any positive order because it is a convolution factor of IV which admits moments of any positive order thanks to Theorem As a consequence the derivatives at any order of functions of the kind of ξ E[e ie ky ξs T ], for integers k, are defined and bounded. For any n N and m > n, we can see by induction that the n th derivative P m,n := ξ m k= ] E [e n ie ky ξs T is a finite sum of products. In each of these products, there are at least m n factors of the form E[e ie ky ξs T ] for some integers k and the other factors are derivatives at some orders of the functions ξ E[e ie ky ξs T ] for some integers k. Therefore, from , we deduce that P m,n ξ = O ξ + ξ m nδ We decompose into φ IV ξ = m k= [ ] E e ie ky ξs T R m ξ, 128

140 3.6. THE SPECTRALLY POSITIVE CASE where R m ξ := ψξ k m+1 ] E [e ie ky ξs T. From the Leibniz formula applied to the product, we have φ n IV ξ = n k= CnP k m,k ξ R m n k ξ. R m is the Fourier transform of a random variable that admits moments of any positive order because it is a convolution factors of IV, so its derivatives at any order are defined and bounded. From we thus get that φ n IV ξ = O ξ + ξ m nδ. As m is arbitrary, φ n goes to faster than any negative power of ξ. Therefore, IV φ IV belongs to the Schwartz space and so does the density of IV, since the Schwartz space is stable by Fourier transform. Remark The case where V has bounded variation is not contained in Theorem Moreover, Remark shows that the law of S T S T being as in Lemma has an atom at if V has bounded variation, so there is no hope to generalize our proof of Theorem to this case. We also prove Corollary Démonstration. of Corollary Since V drifts to + we have V = V so the expression in the proof of Proposition tells us that IV is a convolution factor of IV. Now, under the assumptions of the corollary, Theorem applies and we get the regularity of the density of IV thanks to the boundedness of the derivatives of the density of IV and the differentiation under the integral sign theorem. We get the convergence to at + of the derivatives of the density of IV thanks to the boundedness of the derivatives of the density of IV and the dominated convergence theorem. The convergence to at of the derivatives of the density of IV comes from the fact that this density is of class C and null on ],[. 3.6 The spectrally positive case We now make a brief study of the exponential functional of Z where Z is a spectrally positive Lévy process drifting to +. If Z is a subordinator, then it stays positive and IZ is only IZ which is already known to be finite and have some finite exponential moments see for example Theorem 2 in [12], so Theorem is already known in this case. 129

141 3.6. THE SPECTRALLY POSITIVE CASE We thus assume that Z is not a subordinator. Since, in this case, Z is spectrally negative and not the opposite of a subordinator then, we denote by κ the non-trivial zero of Ψ Z, it is regular for ],+ [ according to Theorem VII.1 in [8], so Z is regular for],[. Moreover,Z drifts to+. We can thus define the Markov family Z x, x as in [31], Chapter 8. It can be seen from there that the processes such defined are Markov, have infinite life-time this is where we need the hypothesis that Z drifts to + and that Z, that we denote by Z, is indeed well defined. Here again, for any x, the process Z x must be seen as Z conditioned to stay positive and starting from x. Note that, since Z converges almost surely to infinity, for x >, Z x is only Z x conditioned in the usual sense to remain positive Finiteness, exponential moments : Proof of Theorem The idea is that adding a small term of negative drift to Z does not change its convergence to +. It makes Z ultimately greater than a deterministic linear function for which the exponential functional is defined and deterministically bounded. The key point is thus to control the time taken by Z to become greater than the linear function once and for good. We start with the following lemma. Lemma For any y >, there exists ǫ > and positive constants c 1 and c 2 such that s >, P RZ y. y +ǫ.,],] > s c 1 e c 2s. Démonstration. We fix y >. From Corollary VII.2 in [8], a spectrally negative Lévy process X drifts to if and only if E[X1] <. Z is a spectrally positive Lévy process drifting to + so taking the dual in the theorem we get E[Z1] >. Now E[Z ǫ.1] = E[Z1] ǫ which is positive for ǫ chosen small enough. Still taking the dual in Corollary VII.2 in [8], this implies that Z ǫ. is also a spectrally positive Lévy process that dirfts to + and which is not a subordinator. We have P RZy. y +ǫ.,],] > s = P inf = C P t [s,+ [ Zyt y +ǫt inf t [s,+ [ Z yt y +ǫt, inf [,+ [ Z y >, where C := 1/Pinf [,+ [ Z y > = 1/1 e κy. This comes from the fact that Z y is only Z y conditioned to stay positive in the usual sense. Now, noting that Z y = y+z, 13

142 3.6. THE SPECTRALLY POSITIVE CASE we bound the above quantity by C P inf t [s,+ [ Zt ǫt C P = C P sup t [s,+ [ = C P Zt+ǫt, Zs+ǫs sup Zt+ǫt t [s,+ [ +C P Zs+ǫs sup Z s t+ǫt Zs+ǫs, Zs+ǫs t [,+ [ +C P Zs+ǫs > From the independence of the increments, the process Z s + ǫ. is equal in law to Z + ǫ. and independent from Zs + ǫs. From Corollary VII.2 in [8], the supremum over [,+ [ of the process Z s +ǫ. follows an exponential distribution with parameter α, where α is the non-trivial zero of Ψ Z+ǫ.. From this, combined with the independence from Zs+ǫs, becomes C E e α Zs+ǫs 1 { Zs+ǫs } +C P Zs+ǫs > C E e α Zs+ǫs/2 1 { Zs+ǫs } +C P e α Zs+ǫs/2 > 1, from the decreases of negative exponential and composing by function x exp α 2 x in the probability of the second term, C E e α/2 Zs+ǫs +C P e α Zs+ǫs/2 > 1 2C E e α/2 Zs+ǫs, where we used Markov inequality in the second term, = 2C e sψ Z+ǫ.α/2. As Ψ Z+ǫ. is negative on ],α[, we get the result with c 1 = 2C and c 2 = Ψ Z+ǫ. α/2. We can now prove Theorem Démonstration. of Theorem We fixy >. Letm y be the point where the processz y reaches its infimum,m y := sup{s, Z ys Z ys = inf [,+ [ Z y}. Note that from the absence of negative jumps the infimum is always reached at least at m y so Z ym y = inf [,+ [ Z y. In order to get Z from Z y, we use the decomposition given by Theorem 24 in [31], that is : The two processes Z ym y +s Z ym y, s and Z ys, s < m y are independent, 131

143 3.6. THE SPECTRALLY POSITIVE CASE Zym y +s Zym y, s is equal in law to Z. Now, IZ y = = my my e Z yu du+ + m y e Z yu du+e Z ym y e Zyu du + + e y e Z ym y+u Zym y du, because almost surely Z ym y y, L = e y IZ, because of the above decomposition. We thus get e Z ym y+u Z ym y du IZ sto e y IZ y, where sto denotes a stochastic inequality. As a consequence we only need to prove the result for IZ y. We now choose ǫ > as in Lemma We have IZ y = RZ y. y+ǫ.,],] e Z yt dt+ R Z y. y +ǫ.,],] + + RZy. y+ǫ.,],] e Z yt dt + RZy. y+ǫ.,],] e Z yt dt, but for t RZ y. y +ǫ.,],] we have Z yt y +ǫt, so IZ y R Z y. y +ǫ.,],] + R Z y. y +ǫ.,],] + + = R Z y. y +ǫ.,],] +e y /ǫ. RZy. y+ǫ.,],] e y ǫt dt + e y ǫt dt From Lemma 3.6.1, this is almost surely finite and admits some finite exponential moments. Thanks to3.6.2, we have the same foriz, which is the expected result Tails at of IZ : Proof of Theorem We need an analogous of Lemma in order to compare, as we did in subsection 3.4.3, the exponential functionals IZ and IZ. We define m, the point where the process Z reaches its infimum : m := sup{s, Zs Zs = inf [,+ [ Z}. Here again, from the absence of negative jumps, the infimum is always reached at least at m so Zm = inf [,+ [ Z. 132

144 3.6. THE SPECTRALLY POSITIVE CASE Lemma If Z has unbounded variation, then Zm+. Zm has the same law as Z. Démonstration. Zm +. Zm is only the infinite excursion of the post-infimum process Z Z, so we only need to prove that this infinite excursion has the same law as Z, and for this we want to apply Proposition 4.7 of [33]. We already know that, because it is spectrally positive, Z is regular for ],[. Taking the dual of the process in Corollary VII.5 in [8], we get the regularity of {} and ],+ [ for Z, thanks to the hypothesis of unbounded variation. The hypothesis of the proposition in [33] are thus fulfilled. Let N denote the excursion measure of the Markov process Z Z and L 1,U the ladder process of Z : L 1 is the inverse of the local time at of Z Z and for any positive t, Ut = ZL 1 t. Recall from the introduction the notation τa,h+ for τa,[h,+ [. We denote by U the potential measure of U and, since Z is spectrally negative, the formula page 191 in [8] applies and yields that U] x,] = 1 e κx /κ for any x. Proposition 4.7 of [33] tells us that for any positive measurable function G defined on the space of càd-làg functions from [,+ [ to R with finite lifetime, we have E [ G ] Z s s τz,h+ = N G ξs s τξ,h+ U] ξτξ,h+,] τξ,h+ < N ξ,τξ,h+ < = c h N 1 e κξτξ,h+ G ξs s τξ,h+ τξ,h+ <, replacing U by its expression and where we set c h := N ξ,τξ,h+ < /κ. Let ξ denote the infinite excursion of Z Z, then, for any positive measurable function F, we get that E [ F ] ξ s s τξ,h+ = N P inf Z ξτξ,h+ > F ξs s τξ,h+ τξ,h+ < [,+ [ [ P inf[,+ [ Z Z τz,h+ > where we used with = E c h 1 e κz τz,h+ G ξs s τξ,h+ := P inf [,+ [ Z ξτξ,h+ > c h 1 e κξτξ,h+ F Z s s τz,h+ ], F ξ s s τξ,h+. Il follows from and from Z τz,h+ h that, Pinf [,+ [ Z ξτξ,h+ > /c h 1 e κξτξ,h+ is a bounded martingale with respect to the filtration F h := σz s, s τz,h+ and that it converges almost surely to some constant. As a consequence, this quantity is almost surely equal to 1 for any positive h, hence, h >, E [ F ξ s s τξ,h+] = E [ F Z s s τz,h+], 133

145 3.6. THE SPECTRALLY POSITIVE CASE so the infinite excursion of Z Z indeed has the same law as Z. We can now prove Theorem Démonstration. of Theorem We have IZ = = m m + m e Zt dt+ + m e Zt dt + e Zt dt+e Zm e Zm+t Zm dt e Zm+t Zm dt, because almost surely Zm. Since Z has unbounded variation, we can use Lemma which tells us that the last term is equal in law to IZ. We thus get m PIZ x PIZ x, so we only need to prove the result for IZ. Obtaining is the only thing for which we need the hypothesis of unbounded variation in this proof. The result that we now prove for IZ is thus true without this hypothesis. LetQ, γ, ν be the generating triplet of Z in the Lévy-Khintchine representation. Since ν is non zero, there exist < γ 1 < γ 2 < + such that ν[γ 1,γ 2 [ >. Then, for η ], 1[ we define ν η,1 := ην. [γ 1,γ 2 [ and ν η,2 := ν ην. [γ 1,γ 2 [, and setz η,1 andz η,2 to be two independent Lévy processes which generating triplets are respectively,,ν η,1 and Q,γ,ν η,2. We have ν = ν η,1 +ν η,2 so according to the Lévy-Khintchine formula, Z L = Z η,1 +Z η,2. According to Corollary VII.2 in [8], a spectrally negative Lévy process X drifts to if and only if E[X1] <. Z is a spectrally positive Lévy process drifting to + so taking the dual in the theorem we get E[Z1] >. Now since E[Z1] = E[Z η,1 1] + E[Z η,2 1] and E[Z η,1 1] < γ 2 ην[γ 1,γ 2 [, we have that E[Z η,2 1] is positive for η small enough. Still taking the dual in Corollary VII.2 in [8], this implies that Z η,2 drifts to + for η small enough. We thus choose such an η ],1[ and denote by m 2 the point at which Z η,2 reaches its minimum. 134

146 3.6. THE SPECTRALLY POSITIVE CASE Z η,1 is a compound Poisson process, we define N the counting process of its jumps :Nt := {s [,t], Z η,1 s Z η,1 s > }.N is thus a standard Poisson process with parameter c := η ν[γ 1,γ 2 [ and it is independent of Z η,2. We have : IZ = + e Zη,2 m 2 e Zη,2 m 2 e Zη,1 t+z η,2 t dt + + = e Zη,2 m 2 Iγ 1 N, e Zη,1 t dt e γ 1Nt dt so, from the independence between the two factors : PIγ 1 N x/2 P e Zη,2 m 2 2 PIZ x We put c 1 := Pe Zη,2 m 2 2 >. Now by a property of standard Poisson processes, it is easy to see that Iγ 1 N has the same law as 1 c + k= e γ 1k e k, where e k k N is a sequence of iid exponential random variable with parameter 1. This allows us to compute the Laplace transform of Iγ 1 N : λ, E [ e λiγ 1N ] = + k= 1 1+ λ c e γ 1k. We put Kλ := min{k N, λe γ 1k 1} and taking the logarithm we get log E [ e λiγ 1N ] = + k= log 1+ λ e γ 1k c Kλlog 1+ λc k Kλ Kλlog 1+ λc log k log 1+ λ e γ 1k c 1+ 1 e γ 1k. c Now, since Kλ λ + logλ/γ 1, we get that log E [ e λiγ 1N ] 2logλ 2 /γ 1, 135

147 3.6. THE SPECTRALLY POSITIVE CASE for λ large enough. Now, reasoning as in the proof of where, from a lower bound on the Laplace transform of IV, we deduced a lower bound for its asymptotic tail at we get e c 2logx 2 PIγ 1 N x, for some positive constant c 2. Combining with and 3.6.6, we get the sought result. 136

148 Chapitre 4 Path decomposition of a spectrally negative Lévy process, and local time of a diffusion in this environment This work has been the object of an article [74] that will be shortly submitted. 4.1 Introduction Let V be a two-sided spectrally negative Lévy process which is not the opposite of a subordinator in particular, V can not be a compound Poisson process, drifts to at +, and such that V =. We denote its Laplace exponent by Ψ V : t,λ, E [ e λvt] = e tψ V λ. It is well-known, for such V, that Ψ V admits a non trivial zero that we denote here by κ, κ := inf{λ >, Ψ V λ = } >. We are here interested in a diffusion in this potential V. Let us recall that such a diffusion Xt, t in a random càd-làg potential V is defined informally by X = and dxt = dβt 1 2 V Xtdt, where β is a Brownian motion independent from V. Rigorously, X is defined by its conditional generator given V, 1 d 2 evx e Vx d. dx dx The fact that V drifts to puts us in the case where the diffusion X is a.s. transient to the right. In [66], Singh makes the study of the asymptotic behavior of 137

149 4.1. INTRODUCTION X. When < κ < 1, he proves in particular that Xt/t κ L t + C1/S κ κ, where C is some explicit positive constant depending on V and S κ follows a completely asymmetric κ-stable distribution. Putting this in relation with the results of Kawazu and Tanaka [45], we see that κ plays a similar role as the drift of the brownian environment, at least for the asymptotic behavior of the diffusion. In this paper, we prove that the same is true for the behavior of the local time of X. We denote by L X t,x,t >,x R the version of the local time that is continus in time and càd-làg in space, and we define respectively the supremum of the local time and the favorite site until instant t as L Xt = supl X t,x and x R F t := inf{x R, L X t,x L X t,x = L Xt}. We study the convergence in distribution of L X t and F t when κ > 1, and of L X t when < κ < 1. When V is a drifted brownian motion, the case where < κ < 1 has already been studied by Andreoletti, Devulder and Véchambre in [4]. In this case, there is a useful renewal structure obtained from a valleys decomposition of the potential and the Markov property for the diffusion. The limit distribution they obtain involves a κ-stable subordinator and an exponential functional of the environment conditioned to stay positive. When < κ < 1, we extend their result to the diffusion in V and obtain a limit distribution in terms of the exponential functionals of V and its dual conditioned to stay positive : IV := + e V t dt and IˆV := + e ˆV t dt, where ˆV, the dual of V, is equal in law to V. It is proved in Theorems 1.1 and 1.13 of Véchambre [72] that these functionals are indeed finite and well-defined. The fact, proved in [72], that IV and IˆV admit some finite exponential moments is of fundamental interest for our generalization of the results of [4]. The almost sure asymptotic behavior of the local time is studied, in the discrete transient case, by Gantert and Shi [41]. We believe that the present work will allow us, in the future, to link the asymptotic almost sure behavior of L X t with the left tail of IV. This tail is given in [72] and can be very different, when V is a general spectrally negative Lévy process, than when V is a drifted brownian motion. As a consequence we can expect, in the Lévy case, many possible behaviors for the almost sure asymptotic of L X t, and this is our main motivation to generalize, here, the study of [4] to the Lévy case. When κ > 1, we adopt the point of view of [66] and link the local time to a generalized Ornstein-Uhlenbeck process. This approach also provides the convergence of the favorite site and can certainly be used to study the almost sure behavior of the supremum of the local time. 138

150 4.1. INTRODUCTION Main results Our main results are the convergences in distribution for the supremum of the local time. Let us define the constants K and m similarly as in [66] : [ + κ 1 ] K := E e Vt dt and m := 2 Ψ V 1 >, and for any α,s >, let Fα,s denote the Fréchet distribution with parameters α and s, that is, the distribution with repartition function Fα,s[,t] = e s/tα. When κ > 1, the limit distribution of the supremum of the local time can be expressed as follows : Theorem If κ > 1, L Xt/t 1/κ L t + Fκ,2Γκκ2 K/m 1/κ. Examples : In some cases the parameters of the limit distribution are more explicit. Let W κ be the κ-drifted brownian motion : W κ t := Wt κ t. If we choose 2 V = W κ for κ > 1, then K = 2 κ 1 /Γκ see Example 1.1 in [66] and m = 4/κ 1. The limit distribution of the supremum of the local time is therefore Fκ,4κ 2 κ 1/8 1/κ. This is precisely the second point of Theorem 1.6 of Devulder [28]. If V is such that κ = 2 then [ + ] + K = E e Vt dt = E [ e Vt] + dt = e tψ V 1 dt = 1 Ψ V 1 = m 2, and the limit distribution of L X t/t1/2 is therefore F2,2 2. We also prove that the distribution of the favorite site is asymptotically uniform : Theorem If κ > 1, mf t/t L t + U, where U denotes the uniform distribution on [,1]. When < κ < 1, we have to introduce some notations in order to express the limit distribution. Let G 1 and G 2 be two independent random variables with G L 1 = IV and G L 2 = IˆV. We define R := G 1 + G 2. R is the analogue of R κ defined in [4] if V if the κ-drifted brownian motion, then R = L R κ, indeed, it is known that Wκ L = Ŵ κ, so R is, as R κ, the sum of two independent copies of IWκ. Let also C be the constant of Corollary 5 of Bertoin, Yor [12] applied to V which 139

151 4.1. INTRODUCTION gives the right tail of the exponential functional of the Lévy process V, the fact that this corollary can be applied is the object of Lemma in Section 4.5. We put C := C/2 + u κ e u/2 du. Now, let Y 1 be the κ-stable subordinator with Laplace exponent C Γ1 κλ κ : t,λ, E [ e λy 1t ] = e tc Γ1 κλ κ. We now consider the pure jump Lévy process Y 1,Y 2 where the component Y 2 is defined multiplying each jump of Y 1 by an independent copy of R. We can also define Y 1,Y 2 from its κ-stable Lévy measure ν supported on ],+ [ ],+ [ and defined by x >,y >, ν[x,+ [ [y,+ [ = C y κe [ R κ 1 R y x ]+ R C x κp > y. x It is easy to see that the first definition implies the second and also that the Lévy process Y 1,Y 2 has Laplace transform t,α,β, E [ e αy 1t βy 2 t ] = e tc Γ1 κe[α+βr κ]. As in [4], our limit distribution for the supremum of the local time is a function of Y 1,Y 2. For Z an increasing càd-làg process and s, we put respectively Zs, Z s and Z 1 s for respectively the left-limit of Z at s, the largest jump of Z before s and the generalized inverse of Z at s : Zs = lim Zr, r < s Z s := sup Zr Zr, Z 1 s := inf{u, Zu > s}, r s where inf = + by convention. We now define the couple of random variables I 1,I 2 : I 1 := Y1Y 2 1 1, I 2 := 1 Y 2 Y2 1 1 Y 1Y2 1 1 Y 1 Y2 1 1 Y 2 Y2 1 1 Y 2 Y2 1 1, and we have Theorem If < κ < 1, V has unbounded variations and V1 L p for some p > 1 then L L Xt/t I := maxi 1,I 2. t + This result is a generalization, for more general environments, of Theorem 1.3 of [4]. A key point in its proof is the fact that the contributions to the local time and to the time spent by the diffusion between the bottoms of two consecutive valleys are negligible, so the local maxima of the local time are localized at the bottom of the valleys, where most of the time is spent. This is where appears the pure jump subordinator Y 1,Y 2 with two correlated components, one for the local time, and one for the time spent. Each jump of Y 1,Y 2 represents the contribution of the bottom of a valley to these two quantities. 14

152 4.1. INTRODUCTION Our generalization of the results known in the brownian case also yields some other results such as the convergence of the supremum of the local time before and after the last valley the first two points of Theorem 1.5 in [4]. When < κ < 1, our study relies deeply on the decomposition of the potential into h-valleys and on the localization of the contributions to the local time and to the time spent near the h-minima. In fact, we have to make h going to infinity when t goes to infinity, this is why we briefly study the asymptotic of the h-valleys. The definition of the h-extrema and h-valleys are given in the next section but we already note the following result about the asymptotic of m 1,m 2,..., the sequence of h-minima. Theorem When h goes to infinity, the renormalized random sequence e κh m 1,m 2,... converges in distribution to the jumping times sequence of a standard Poisson process with parameter q which depends explicitly on the law of V. If V = W κ, the κ-drifted brownian motion, then q = κ 2 /2. Some of the estimates used to prove this theorem will also be useful to establish the negligibility of the local time between the bottoms of two consecutive valleys when < κ < 1. However, the main interest of this theorem is that it informs us on the typical distance between two consecutive minima, and this provides an heuristic explanation for why the method based on valleys fails when κ > 1. As we can see in Theorem 4.1.4, the distance between two consecutive h-minima is of order e κh. In the case < κ < 1, what happens between the bottoms of the h- valleys can be neglected, so the main contributions to the local time and to the time spent by the diffusion are localized at the h-minima and are highly correlated, this explains the form of the limit distribution for the favorite site given in [4]. When κ > 1, this distance between two consecutive h-minima is so large that the time spent by the diffusion between the h-minima is no longer negligible compared to the time spent in the bottoms of the h-valleys, and there are extreme values taken by the local time between the h-valleys. In fact, it is impossible to use the valleys to localize the large values of the local time as we do when < κ < 1, this explains the asymptotic uniform distribution for the favorite site. Moreover, the case κ > 1 is the case where X has positive speed. The local time at time t is then close to the local time at the hitting time of t/m and the latter can be expressed thanks to a generalized Ornstein-Uhlenbeck process. The supremum of the local time is therefore similar to the supremum of the heights of iid excursions of a Markov process. This explains the appearance of a Fréchet distribution in the limit distribution. The rest of the paper is organized as follows. In section 4.2 we study the case where κ > 1 and prove Theorems and In section 4.3 we recall the definitions of the h-extrema and h-valleys for V and establish some usual properties such as the independence of the consecutive slopes and the law of the bottoms of the valleys. We end this section by proving Theorem In section 4.4 we assume < κ < 1 and prove Theorem In section 4.5 we prove some fundamental estimates on V, V and ˆV and some technical results. 141

153 4.1. INTRODUCTION Facts and notations We denote by Q,γ,ν the generating triplet of V so Ψ V can be expressed as Ψ V λ = Q 2 λ2 γλ+ e λx 1 λx1 x <1 νdx If V jumps at instant u we denote Vu := Vu Vu, the jump of V at u. For r >, we define V < r to be the sum of the jumps of V that are less than r : s, V < r s := Vu1 { Vu< r}. u s V is the sum of the processes V V < r and V < r that can be seen to be independent spectrally negative Lévy processes, thanks to the Lévy-Khintchine formula According to Corollary VI.2 of Bertoin [8] we have E[V1] < so for r chosen large enough we have E[V V < r 1] < which, thanks to Corollary VI.2 of [8], implies that V V < r drifts to. In other words, removing the very large jumps of V does not change its convergence to. For Y a process and S a borelian set, we denote τy,s := inf{t, Yt S}, KY,S := sup{t, Yt S}. We shall only writeτy,x instead ofτy,{x} andτy,x+ instead ofτy,[x,+ [. Since V has no positive jumps we see that it reaches each positive level continously : x >,τv,x+ = τv,x. Moreover, the law of the supremum of V is known, it is an exponential distribution with parameter κ see Corollary V.II in [8]. Y denotes the infimum process of Y : t, Yt := inf [,t] Y. The process V V is known as the process V reflected at its infimum, note that Proposition VI.1 of [8] tells that it is a càd-làg Markov process. The same holds for ˆV ˆV. If Y is Markovian and x R we denote Y x for the process Y starting from x. For Y we shall only write Y. For any possibly random time T >, we write Y T for the process Y shifted and centered at time T : s, Y T s := YT +s YT. Let B be a brownian motion starting at and independent from V. A diffusion in potential V can be defined via the formula Xt := A 1 V BT 1 V t, where A V x := T V s := x s e Vu du and for e 2VA 1 V Bu du. s τ B, + e Vu du, 142

154 4.2. SUPREMUM OF THE LOCAL TIME WHEN κ > 1 It is known that the local time of X at x until instant t has the following expression : L X t,x = e Vx L B T 1 V t,a Vx For the hitting times of r R by the diffusion X we shall use the frequent notation Hr instead of τx,r. We denote by H + r respectively H r the total time spent by the diffusion in [,+ [ respectively ],] before Hr. The quenched probability measure P V is the probability measure conditionally on the potential V. When we deal we events relative to the diffusion X, P represents the annealed probability measure, it is defined as P := P ω.pv dω. We often denote by d VT the total variation distance between two probability distributions on the same space. 4.2 Supremum of the local time when κ > 1 We now treat the case κ > 1. Since some of the lemmas we state in this section are true in a general context we do not assume κ > 1 yet. We thus have κ ],+ [ unless mentioned otherwise. As we mentioned in the introduction, the valleys are of no use in this section so we have to study directly the expression of the local time. It is given by According to [66], if m := 2/ψ V 1, then t is close to Ht/m when κ > 1 and t is large. It is then convenient to look at the local time until the hitting times. It has a simpler expression : L X Hr,x = e Vx L B τb,a V r,a V x The supremum of the local time until instant Hr can thus be written where and L XHr = max{m 1 r,m 2 r}, M 1 r := supl X Hr,x = supe Vx L B τb,a V r,a V x x< x< supe Vx supl B τb,a V +,A V x < x< x< M 2 r := sup L X Hr,x = sup e Vx L B τb,a V r,a V x. x [,r] x [,r] We also define the favorite positive site until time Hr by F +Hr := argmax [,r] L X Hr,. := inf{x [,r], L X Hr,x L X Hr,x = M 2 r}. 143

155 4.2. SUPREMUM OF THE LOCAL TIME WHEN κ > 1 To lighten the notations, we often use the notation argmax in this section. Now, note that M 2 r is the same as J 2 r of [66] with a supremum on [,r] in place of an integral on this interval. In particular, if, similarly to [66], we define x Zx := e Vx R e Vy dy, where R is a two-dimensional squared Bessel process independent from V, we get M 2 r = L sup Zx and F+Hr = L r argmax [,r] Z x [,r] We can therefore prove our results using some precise properties of Z. Recall from [66] the definition of L, the local time of Z for the position 1, of n the associated excursion measure, and of L 1 the right continus inverse of L. We denote by ξ a generic excursion The local time at hitting times We now prove a lemma to justify rigorously that the local time of the diffusion at some instant can be approximated by the local time at an hitting time. Lemma Let us denote by Q the positive constant denoted by n[ζ] in [66], for r large enough we have P L 1 r/q r 3/4 r L 1 r/q+r 3/4 1 r 1/ Assume κ > 1. For any α ]max{3/4,1/κ},1[ their exists ǫ > such that for r large enough we have PHr/m r α r Hr/m+r α 1 r ǫ, PL XHr/m r α L Xr L XHr/m+r α 1 r ǫ Démonstration. According to Lemma 5.1 in [66], E[L 1 1] = Q and the subordinator L 1 admits some finite positive exponential moments, so, in particular, it has moments of the second order. Let us define the Lévy process Ut := L 1 t Qt. U has finite mean equal to and moments of the second order, this implies that E[ Ut 2 ] = te[ U1 2 ]. Using Markov s inequality we get Then, t,s >, P Ut > s s 2 E [ Ut 2] = s 2 te [ U1 2] P r > L 1 r/q+r 3/4 P L 1 r/q+r 3/4 r+qr 3/4 > Qr 3/4 = P Ur/Q+r 3/4 > Qr 3/4 r/q+r 3/4 E [ U1 2] /Q 2 r 3/2, 144

156 4.2. SUPREMUM OF THE LOCAL TIME WHEN κ > 1 where we used with t = r/q+r 3/4 and s = Qr 3/4. We get a similar estimate for P r < L 1 r/q r 3/4 so follows. We now turn to Until the end of this proof we assume κ > 1. We have PHr/m r α > r PH r/m r α +H + r/m r α r mr α > mr α PH + > mr α /2 r/m r α +P Zxdx r mr α > mr α /2, because, as we can see from [66], t Zxdx has the same law as H +t denoted by J 2 t there. The second term in the right hand side of is less than τz,1 r/m r α P Zxdx > mr α /4 +P Zxdx r mr α > mr α /4 τz,1 r/m r α PτZ,1 > mr α /4+P Z 1 xdx r mr α > mr α /4, where we put Z 1 := ZτZ,1+.. It is well defined since Z has no positive jumps and thus reaches [1,+ [ continuously. Since Z is a Markov process, Z 1 has indeed the same law as Z starting from 1. According to Proposition 4.3 of [66] : PτZ,1 > t e ct, for some constant c > when t is large enough. We now put this in where we also use Lemma for the term PH + > mr α /2, we obtain for r large enough PHr/m r α > r P r/m r α Z 1 xdx r mr α > mrα /4 +c 1 r ακ 2+κ, for some positive constant c 1. Let us define Ũt := L 1 t Z 1 xdx ml 1 t. Note that, from Section 7 of [66] we can see that Ũ is a Lévy process with finite mean equal to and such that PŨ1 > x x + cx κ for some positive constant c. As a consequence, E[ Ũ1 γ ] < + for γ ]1/α,κ[. We choose such a γ and use successively Markov s and Von Barh-Esseen s inequalities : Ũt [ Ũt γ ] t,s >, P > s s γ E 2s t E [ Ũ1 γ ] [ ] γ + sup E Ũs γ. s [,1]

157 4.2. SUPREMUM OF THE LOCAL TIME WHEN κ > 1 We now use and get that for y large enough, y L 1 y/q+y 3/4 with probability greater than 1 y 1/4 so y Z 1 xdx my L 1 y/q+y 3/4 Z 1 xdx my = Ũy/Q+y3/4 +muy/q+y 3/4 +mqy 3/4. Let C be an arbitrary positive constant. Applying and with t = y/q+y 3/4 and s = Cy α we get for y large enough y P Z 1 xdx my > Cy α c 2 y 1 αγ +y 1 2α +y 1/4, for some positive constant c 2, depending on C. We prove a similar inequality for my y Z 1xdx so we have actually, for c 3 a positive constant depending on C and y large enough, y P Z 1 xdx my > Cyα c 3 y 1 αγ +y 1 2α +y 1/ Applying with a good choice of C, y = r/m r α and putting into we get PHr/m r α > r r ǫ /2, if ǫ > is chosen to be less than max{αγ 1,2α 1,ακ/2+κ,1/4} and r is large enough. Then, r/m+r α PHr/m+r α < r PH + r/m+r α < r P τz,1 Zxdx < r r/m+r α /2 P Z 1 xdx < r +PτZ,1 > r α /2, so, using and , we conclude the same way as forphr/m r α > r : we getphr/m+r α < r r ǫ /2 forr large enough, and combining with we get Finally, is only a consequence of and of the increases of L X. We also need an almost sure version of : Lemma Almost surely, for all t large enough we have Qt t 3/4 L 1 t Qt+t 3/4. 146

158 4.2. SUPREMUM OF THE LOCAL TIME WHEN κ > 1 Démonstration. Since L 1 1 admits some finite exponential moments we have an iterated logarithme law. n. Now, we study the supremum of the process Z in terms of its excursion measure Lemma There is ǫ > and r > such that for all r r and h > 1 we have e r/q+r7/8 nsupξ>h r ǫ P sup Zx h x [,r] e r/q r7/8 nsupξ>h +r ǫ. Démonstration. Recall the definition of Z 1 in the proof of Lemma For h > 1 we have P sup Zx h x [,r] = P sup Zx h x [τz,1,r] P sup Z 1 x h x [,r], because of the Markov property and the fact that the length of [τz,1,r] is less than r. Let us choose η ],3/4[. We have P sup Zx h x [τz,1,r] P sup Z 1 x h x [,r r η ] +PτZ,1 > r η From , and 4.2.1, we see that we only need to prove the lemma with Z 1 instead of Z and r 3/4 instead of r 7/8. We only prove the lower bound, since the proof of the upper bound is similar. For r large enough so that is true we have P sup Z 1 x h x [,r] P sup Z 1 x h x [,L 1 r/q+r 3/4 ] r 1/4, and from the point of view of excursions, the probability in the right hand side is only the probability that no excursion higher than h occurs before L exceeds r/q+r 3/4. It is known, from properties of Poisson point processes, that this probability equals e r/q+r3/4 nsupξ>h and the result follows Proof of Theorems and In this subsection we assume that κ > 1. Démonstration. of Theorem According to 4.2.7, we only need to prove that L X Hr rm 1/κ L r + Fκ,2Γκκ2 K/m 1/κ 147

159 4.2. SUPREMUM OF THE LOCAL TIME WHEN κ > 1 or, equivalently, L X Hr r 1/κ L r + Fκ,2Γκκ2 K 1/κ Now, recall Combining it with and 4.2.4, we are left to prove that a >, P sup Zx ar 1/κ Γκκ 2 K/a κ x [,r] r + e 2κ According to Proposition 5.1 of [66] we have nsupξ > h h + Q2κ Γκκ 2 K/h κ Applying this with h = ar 1/κ and putting it into Lemma we get and the convergence of L X t/t1/κ follows. Démonstration. of Theorem We now study the asymptotic of the favorite site. We first prove that F Hr/r L U r + L X Hr converges in probability to + because of Therefore, the combination of 4.2.2, and tells us that will follow if we prove that argmax [,r] Z/r L U r + Let us choose γ ],1/8κ[. Similarly as in the proof of Lemma 4.2.3, we have P sup Z 1 > r 1/κ γ = 1 e r/q r7/8 nsupξ>r 1/κ γ, [,L 1 r/q r 7/8 ] and using with h = r 1/κ γ, we get that this probability converges to 1 when r goes to infinity : P sup Z 1 x > r 1/κ γ x [,L 1 r/q r 7/8 ] r + Since L 1 r/q r 7/8 is a stopping time for Z 1 and is such that Z 1 L 1 r/q r 7/8 = 1, we have that P sup Z 1 < r 1/κ γ = P sup Z 1 < r 1/κ γ, [L 1 r/q r 7/8,L 1 r/q+r 7/8 ] [,L 1 2r 7/8 ] 148

160 4.2. SUPREMUM OF THE LOCAL TIME WHEN κ > 1 and we prove, similarly as for , that the probability in the right hand side converges to 1. We thus have P sup Z 1 < r 1/κ γ [L 1 r/q r 7/8,L 1 r/q+r 7/8 ] r + Putting and together, we get that, with a probability converging to 1, argmax [,.] Z 1 stays constant on [L 1 r/q r 7/8,L 1 r/q+r 7/8 ] : P x [L 1 r/q r 7/8,L 1 r/q+r 7/8 ], argmax [,x] Z 1 = argmax [,L 1 r/q]z 1 r Now, note that, on {τz,1 < r}, we have argmax [,r] Z = argmax [,r τz,1] Z 1. Then, choose η ],3/4[. According to 4.2.1, τz,1 < r η < r with a probability converging to 1 and, applying we get L 1 r/q r 7/8 L 1 r r η /Q r r η 3/4 r r η r L 1 r/q+r 7/8, with a probability converging to 1. Combining with we get P argmax [,r] Z = argmax [,L 1 r/q]z 1 r Let us denote ˆL := Largmax [,L 1 r/q]z 1. Considering the Poisson point process of excursions of Z 1 associated with the local time L, ˆL is the instant when occurs the highest excursion before the instant r/q. It is a well-known property of Poisson point processes that it follows a uniform distribution on [,r/q] : QˆL/r L = U For the process Z 1, this excursion begins at L 1 ˆL and ends at L 1 ˆL so L 1 ˆL argmax [,L 1 r/q]z 1 L 1 ˆL Then, note that arbitrary high excursions of Z arise if we wait long enough. As a consequence, L 1 ˆL converges almost surely to + when r goes to infinity. We can thus apply Lemma and get that almost surely, for r large enough : QˆL ˆL 3/4 L 1 ˆL argmax [,L 1 r/q]z 1 L 1 ˆL QˆL+ ˆL 3/4. Combining with , we deduce that argmax [,L 1 r/q]z 1 /r L t + U, and putting it together with , we get and follows. We now have to prove the result for the favorite site until a deterministic time, instead of the 149

161 4.2. SUPREMUM OF THE LOCAL TIME WHEN κ > 1 hitting time Hr. For this, let us choose α as in Lemma and γ ],1 α/κ[. We prove that, with high probability, the favorite site remains constant between Hr/m r α and Hr/m+r α, that is, P x [Hr/m r α,hr/m+r α ], F x = F Hr/m r First, note that, as a consequence of 4.2.2, PF Hr/m r α r/m 2r α r + 1, and, as a consequence of , we have P L XHr/m r α > r 1/κ γ r Then, since Hr/m r α is a stopping time for the diffusion X we have P inf X > r/m 2r α = P [Hr/m r α,+ [ inf X > r α 1, [,+ [ r + and using similarly the Markov property together with , we get P =P sup [r/m 2r α,r/m+r α ] L X Hr/m+r α,. r 1/κ γ sup L X H3r α,. r 1/κ γ [,3r α ] r + The four estimates , 4.2.3, and tell us that with high probability when r is large, at time Hr/m r α, the supremum of the local time has been reached before r/m 2r α and is larger than r 1/κ γ, moreover, the diffusion will never reach back ],r/m 2r α ], and at time Hr/m+r α, the supremum of the local time on [r/m 2r α,+ [ is less than r 1/κ γ. As a consequence, with a probability converging to 1, the favorite site does not move between Hr/m r α and Hr/m+r α, this proves Then, together with give PF r = F Hr/m r + 1, and this together with prove the sought convergence in distribution for the favorite site. 15

162 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS 4.3 Path decomposition of a spectrally negative Lévy process h-extrema, h-valleys and some processes conditioned to stay positive Let us recall the notion of h-extrema which was first introduced by Neveu et al. [53], and studied in the case of drifted Brownian motion by Faggionato [37]. For h >, we say that x R is an h-minimum for V if there exist u < x < v such that Vy Vy Vx Vx for all y [u,v], Vu Vx Vx +h and Vv Vx Vx +h. Moreover, x is an h-maximum for V if x is an h-minimum for V, and x is an h-extremum for V if it is an h-maximum or an h-minimum for V. Since V is not a compound Poisson process, it is known see Proposition VI.4, in [8] that it takes pairwise distinct values in its local extrema. Combining this with the fact that V has almost surely càd-làg paths and drifts to without being the opposite of a subordinator, we can check that the set of h-extrema is discrete, forms a sequence indexed by Z, unbounded from below and above, and that the h-minima and h-maxima alternate. We denote respectively by m i, i Z and M i, i Z the increasing sequences of h-minima and of h-maxima of V, such that m < m 1 and m i < M i < m i+1 for every i Z. As in [37], we define the classical h-valleys as the fragments of the trajectory of V between two h-maxima, translated at the h-minima between them : the i th classical h-valley is the process V i x, M i 1 x M i where V i := Vx Vm i, x R. In order to state the law of these valleys, we need to recall somme definitions about V and ˆV conditioned to stay positive but first, let us recall a useful fact. Fact Let Y be a spectrally negative Lévy process which is not the opposite of a subordinator, then it is regular for ],+ [ and the regularity for ],[ is equivalent with Y being of unbounded variations. Démonstration. The regularity for ], + [ and the condition for the regularity for ],[ are stated respectively in Theorem VII.1 and in Corollary VII.5 of [8]. V being spectrally negative, the Markov family Vx,x may be defined as in [8], Section VII.3. For any x, the process Vx must be seen as V conditioned to stay positive and starting from x. We denote V for the process V. It is known that Vx converges in the Skorokhod space to V when x goes to. Also, as well as V, V has no positive jumps and reaches every positive level continuously. 151

163 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS According to Fact 4.3.1, V is regular for ],+ [, so ˆV is for ],[. Moreover, ˆV drifts to +. We can thus define the Markov family ˆV x, x as in Doney [31], Chapter 8. It can be seen from there that the processes such defined are Markov and have infinite life-time. If moreover V has unbounded variations then ˆV is regular for ],+ [, and from Theorem 24 of [31], we have that ˆV, that we shall denote by ˆV, is well defined. Here again, for any x, the process ˆV x must be seen as ˆV conditioned to stay positive and starting from x. Note that, since ˆV converges almost surely to infinity, for x >, ˆV x is only ˆV x conditioned in the usual sense to remain positive Law of the valleys We now prove some facts about the law of the consecutive valleys near their bottom. In order to delimit the bottom of the valleys we define τ i h := sup{x < m i, V i x h}, τ i h := inf{x > m i, V i x = h}. For any i N, let P i 1 be the truncated process V i m i x, x m i τ i h, and P 2 i the truncated process V i m i +x, x τ i h m i. We have, Proposition valleys decomposition Assume V has unbounded variations. All the processes of the family P i j, i 1, j {1,2}, are independent and : For all i 2, the law of P i 1 is absolutely continus with respect to the law of the process ˆV x x τˆv,h+ and has density c h/1 e κˆv τˆv,h+ with respect to this law, where c h is a constant depending on h. For all i 1, P i 2 is equal in law to V x x τv,h this statement is true even if V has bounded variations. Moreover, the density c h /1 e κˆv τˆv,h+ is bounded by 2 for h deterministically large enough and it converges to 1 when h goes to infinity. Remark P 1 1 may be a part of the so-called central slope, so its law is different. Démonstration. We assume V has unbounded variations, recall from Fact that this implies a regularity condition. In Lemma 4 of Cheliotis [21], the assertion that lim t + X t =,lim t + X t = + can be dropped. Indeed, because of the regularity condition, the stopping times τ k and τ k are almost surely finite. This lemma is therefore true in our context. As a consequence, the proof of Lemma 1 of [21] also applies in our context. We thus get the fact that the slopes are independent and that except the central slope, all descending respectively ascending slopes have the same law. 152

164 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS For the law of the ascending slopes not covering the origin until their hitting time of h, we use the classical argument that can be found, for example, in the proof of Theorem 2 in [37] : the law of P i 2 is the law, before its hitting time of h, of the first excursion higher than h of V V, and using Proposition VII.15 of [8] which does not require V to have unbounded variations, we prove that the latter is the law of V x x τv,h, that is, V killed when hitting h. Then, by the time-reverse property the descending slopes not covering the origin have the same law as the ascending slopes of ˆV so, here again, we get that the law of P i 1 is the law, before its hitting time of [h,+ [, of the first excursion higher than h of ˆV ˆV. Unfortunately, since ˆV is not spectrally negative, we can no longer apply the previous argument to determine the law of the excursion. However, since ],[ and ],+ [ are regular for ˆV because they are for V and ˆV does not drift to, we can apply Proposition 4.7 of Duquesne [33] but we first need to introduce some notations. Let ˆL 1,Î denote the ladder process of ˆV, of which the definition has been adapted to our setting : ˆL 1 is the inverse of a local time at, denoted by ˆL, of ˆV ˆV, and for any positive t, Ît = ˆVˆL 1 t. We denote by ˆN the excursion measure of ˆV ˆV associated with ˆL, and for ξ an excursion, let ζξ := inf{s >, ξs = } denote its life-time. We denote by Î the potential measure of Î, and, since ˆV is spectrally positive, it can be seen from [8], Section VII.1, that Î[,x] = 1 e κx /κ for any x. Proposition 4.7 of [33] tells us that for any measurable function G defined on the space of càd-làg functions from [,+ [ to R with possibly finite life-time, we have that E[GˆV s s τˆv,h+] equals ˆN G Î[,ξτξ,h+[ ξs s τξ,h+ τξ,h+ < ζξ ˆN ξ,τξ,h+ < ζξ = ˆN 1 e κξτξ,h+ G ξs s τξ,h+ τξ,h+ < ζξ c h where we have set c h := κ/ ˆN ξ,τξ,h+ < ζξ. So, for any measurable function F, we get that ˆN F ξs s τξ,h+ τξ,h+ < ζξ [ =E c h 1 e F ˆV s κˆv τˆv,h+ s τˆv,h+ ]. That is, the law of the first excursion higher than h, and killed when reaching [h,+ [, of the process ˆV ˆV is absolutely continus with respect to the law of the process ˆV s s τˆv,h+ and has density c h/1 e κˆv τˆv,h+ with respect to this law. For the last assertion of the proposition, note that c h = κ/ ˆNξ,τξ,h+ < ζξ increases, when h goes to infinity, to c := κ/ ˆNξ,ζξ = ζξ, where ˆNξ,ζξ = 153

165 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS + is the measure of infinite excursions above for ˆV ˆV, which is strictly positive, because ˆV almost surely converges to infinity. Also, we have almost surely that ˆV τˆv,h+ h. Hence, for h deterministically large enough we have Then, c h /1 e κˆv τˆv,h+ 2c. P c h /1 e κˆv τˆv,h+ c = 1, h + so by the dominated convergence theorem we get c = 1 and the last assertion of the proposition follows Standard valleys Since V drifts to, the descending phases of V between two h-minima are quite important and have to be taken in consideration for the study of the diffusion in V. Therefore, as in [4], we here define a new sequence m i i 1 of h-minima that are separated by descending phases of V and we then show that for a large number of indices, this sequence coincides with the sequence m i i 1 with an overwhelming probability. Our definition of the standard valleys are similar to the one given in [4] but we have to improve it to get a definition more adapted to our context. We first introduce some notations. δ > is defined once and for all in the paper and can be chosen as small as we want. In this subsection h is a fixed positive number such that e 1 δκh h. We define τ h = L := and recursively for i 1, L i := inf{x > L i 1, Vx V L i 1 e 1 δκh }, τ i h := inf { x L i, Vx inf [ L i,x]v = h}, m i := inf { x L i, Vx = inf [ L i, τ ih] V}, L i := inf{x > τ i h, Vx h/2}, τ i a := sup{x < m i, Vx V m i a}, a [,h], τ + i a := inf{x > m i, Vx V m i = a}, a [,h]. Note that all these random variables depend on h, even if this does not appear in the notations. We also introduce the equivalent of V i for the m i,i N as follows : Ṽ i x := Vx V m i, x R. We call i th standard valley the re-centered truncated potential Ṽ i x, Li 1 x L i. Remark The random times L i, τ ih, and L i are stopping times. As a consequence, the sequence Ṽ i x+ m i, Li 1 m i x L i m i i 1 is iid. 154

166 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS Our definitions take in consideration the absence of positive jumps for V, in particular τ i h < + and V τ i h = V τ i h = V m i +h. We can see that the m i, i N, are h-minima. The next lemma, which is the analogue of Lemma 2.3 in Andreoletti, Devulder [3], shows that, with hight probability, the sequence m i i 1 coincides with the sequence m i i 1 for indices i n when n does not grow too fast with h. Lemma For all n 1 and h large enough, P V n,h := n {m i = m i } 1 ne δκh/3. i=1 Démonstration. As we said before, { m i, i N } {m i,i N }. Hence on the complementary V c n,h of V n,h, considering the smallest 1 i n such that m i m i, we have m i 1 = m i 1 < m i < m i. For this i, we can not have m i [ L i, m i[. Indeed, in this case, m i L i would be the starting point of an excursion higher than h for V L i V L i, but from the definitions, m i L i is the starting point of the first excursion higher than h for V L i V L i, which contradicts mi < m i. Hence, m i ] m i 1, L i [. m i 1 is an h-minimum and there cannot be any h-maximum belonging to [ m i 1, τ i 1 h[ so, as the h minima and the h maxima alternate, necessarily we have m i > τ i 1 h, so m i ] τ i 1 h, L i [. Since m i is an h-minimum, there is v i > m i such that Vm i = inf [mi,v i ]V and Vv i = Vm i + h. Since by definition of L i we have V L i = inf [ τ i 1 h, L i ]V, we cannot have L i ]m i,v i ] so we must have τ i 1 h < m i < v i < L i. For similar reasons we can neither have L i 1 ]m i,v i ] so τ i 1 h < m i < v i < L i 1 or Li 1 m i < v i < L i. We have proved that n c Vn,h c n = {m i = m i } E 1 i Ei 2, i=1 i=1 where { } Ei 1 := τ V τ i 1h V τ i 1h,h < τ V τ i 1h,], h/2], { } Ei 2 := τ V L i 1 V L i 1,h < τ V L i 1,], e 1 δκh ]. 155

167 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS Since τ i 1 h and L i 1 are stopping times, we get for h large enough : n 1 P {m i = m i } 2nP τ V V,h < τ V,], e 1 δκh ], i=1 and the result follows for h large enough according to Lemma 4.5.5, applied with a = h, b = e 1 δκh and η = δ/2. We now make use of the preceding lemma to precise the law of the bottoms of the standard valleys. First, for any i N, let us define P i 1 := Ṽ i m i x, x m i τ i h, P i 2 := Ṽ i m i +x, x τ i h m i, P i 3 := Ṽ i τ i h+x, x L i τ i h. Note that for any index i, m i = m i implies P i 1 = P i 1 and P i 2 = P i 2. Proposition Assume V has unbounded variations. All the processes of the family P i j, i 1, j {1,2,3}, are independent and for all i 1, Pi d VT 1,P 2 1 2e δκh/3, Pi L 2 = V x x τv,h, P i 3 L = h+vx x τv,], h/2]. P i The statements about the laws of 2 and 3 do not require the hypothesis of unbounded variations. The statements about the laws of P 1 is true for h large i enough. Démonstration. For the independence, we use Remark 4.3.4, so it only remains to i i i i prove that for any i 1, P 1, P 2 and P 3 are independent. P 2 is, after a stopping time, the first excursion of V V greater than h, considered up to its hitting time of h. It is therefore independent from the previous slopes and, using again Proposition VII.15 of [8] which does not require V to have unbounded variations, we see that it has the same law as V x x τv,h. Also, the Markov property applied at time i τ i h gives the asserted law of P 3 and its independence from P i i 1, P 2. i For the assertion on P 1, note that for h large enough, P P 2 1 P 2 1 2e δκh/3 i according to Lemma applied with n := 2 and that for any i 1, P 1 is equal 2 in law to P 1 according to Remark The assertion follows. P i We now consider the first ascend of h from the minimum, after τ i h, { } τ i+1h := inf u τ i h, Vu inf V = h, [ τ i h,u] { } m i+1 := inf u τ i h, Vu = inf V. [ τ i h, τ i+1 h] We prove that these coincide with τ i+1 h and m i+1 with a good probability. 156

168 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS Lemma There is a positive constant c such that for h large enough, i 1, P τ i h = τ i h, m i = m i 1 e ch. Démonstration. Note that m i = m i whenever τ i h = τ i h. We thus only prove the latter. Fix i 2 and recall the definitions of τ i h and τ i h. We have V L i = inf [ τi 1 h, L i ]V so, if L i < τ i h, we must have τ i h = τ i h. Therefore, { τ i h τ i h} { τ i h L } i = { τ i h L } i 1 { Li 1 < τ i h L } i Then, from the Markov property at time τ i 1 h and the definitions of L i 1 and τ i h we get P τ i h L i 1 = PτV V,h < τv,], h/2] 2e κh/2, where the last inequality comes from Lemma applied with a = h,b = h/2 and η = 1/4. Then,V L i 1 = inf [ τi 1 h, L i 1 ] V so, if τ i h > L i 1 we have τ i h L i 1 = τv L i 1 V L i 1,h. Combining this with the definition of L i and the Markov property at time L i 1 we get P Li 1 < τ i h L i P τ V V,h < τ V,], e 1 δκh ] e δκh/3, where the last inequality is true for h large enough, according to Lemma applied with a = h, b = e 1 δκh and η = δ/2. Putting together 4.3.1, and we get the result when i 2. If i = 1, and are true but recall that τ h = L =, so { τ 1h L } =. The result is therefore also true for i = Exponential functionals of the bottom of a standard valley Let Jh be the exponential functional of the bottom of the first standard valley : Jh := τ + 1 h/2 τ 1 h/2 e Ṽ 1 u du. We now make use of Propositions and and of the existence, proved in [72], of some finite exponential moments for IV and IˆV to prove a very tight convergence of Jh to R. This result is crucial in Section 4.4 where we make R appear in the limit distribution of the supremum of the local time. 157

169 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS Proposition Assume V has unbounded variations. The family of random variables Jh h> converges in distribution to R and there exists a positive λ such that λ < λ, E [ e λjh] h + E [ e λr], and the above quantities are all finite. As a consequence, the moments of any positive order of Jh converge to those of R when h goes to infinity. Démonstration. We consider a probability space on which are defined two independent processes Z 1 and Z 2 with Z L 1 = ˆV and Z L 2 = V, and we define Ĩh := R := τz1,h/2+ + e Z 1x dx+ e Z 1x dx+ + τz2,h/2 e Z 2x dx. e Z 2x dx, h > We have trivially the equality in law R = L R and the almost sure increase of the family Ĩh h> to R. 1 1 Then, from the definitions of P 1 and P 2, Jh = τ P1 1,h/2+ τ P1 1 P e 1 u 2,h/2 1 P du+ e 2 u du. According to Proposition 4.3.6, the two terms are independent and the second is equal in law to the second term of Ĩh. It is easy to see that, having four random variables A, B, C A and C B where A and C A respectively B and C B are defined on the same probability space and independent, and such that C A and C B have the same law, then we have this inequality for the total variation distance : In our case, we thus have d VT Jh,Ĩh d VT C A +A,C B +B d VT A,B. 1 τ P 1,h/2+ d VT 1 P e 1 u du, τz1,h/2+ e Z 1x dx Then, we see that if A and B are random variables valued on a metric space E, and f is a mesurable mapping from E to R, we have In our case, this yields 1 τ P 1,h/2+ d VT 1 P e 1 u du, d VT fa,fb d VT A,B. τz1,h/2+ P1 e Z1x dx d VT 1,Z 1 x x τz1,h

170 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS From the triangular inequality and Proposition 4.3.6, we have for h large enough, P1 P1 d VT 1,Z 1 x x τz1,h+ d VT 1,P 2 1 +d VT P 2 1,Z 1 x x τz1,h+ 2e δκh/3 +d VT P 2 1,Z 1 x x τz1,h Now, using Proposition 4.3.2, P 2 1 is absolutely continus with respect to the law of the process ˆV s s τˆv,h+ and has density c h/1 e κˆv τˆv,h+ with respect to this law. It is known that, if a random variable B has a density d B with respect to a random variable A both valued in a metric space, then, their total variation distance is expressed as follow : d VT A,B = 1 1 d B dl A. 2 That is, in our case, d VT P 2 1,Z 1 x x τz1,h+ = 1 2 E [ 1 c h 1 e κˆv τˆv,h+ Now combining 4.3.5, 4.3.6, and we get for large h, d VT Jh,Ĩh 2e δκh/3 + 1 [ ] 2 E c h 1, 1 e κˆv τˆv,h+ ] and because of the last assertion in Proposition and dominated convergence, the right-hand-side converges to when h goes to infinity. From this and the convergence of Ĩh h> to R we deduce that Jh h> converges in distribution to R and therefore to R. In order to prove 4.3.4, it only remain to prove a uniform integrability condition. In particular, will follow if we prove the existence of a positive λ and a positive finite constant C such that [ ] h >, E e λ Jh C Thanks to Theorems 1.1 and 1.13 of [72], the positive random variables IV and IˆV admit some finit exponential moments. We can therefore choose a positive λ such that [ ] [ ] E e λ IV < + and E e 2λ IˆV < For any such choice of λ and h >, [ [ ] τ P E e λ Jh = E exp λ E [ exp λ τ P 1 1,h/2+ 1 2,h/2 1 P e 1 u du 1 P e 2 u du ] ]

171 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS Then, according to Proposition 4.3.6, [ ] [ 1 τ P 2,h/2 ] 1 τv,h/2 P E exp λ e 2 u du = E exp λ e V u du [ ] E e λ IV From Remark and the fact that m i i 1 is a subsequence of m i i 1 we have, E =E =E [ [ [ exp exp exp + i>2 E E [ + i>2 [ exp λ τ P λ τ P λ τp exp λ τp 1 1,h/2+ 2 1,h/2+ 2 1,h/2+ λ τp 2 1,h/2+ i 1,h/2+ τp E [exp 2λ 1 P e 1 u du 2 P e 1 u du e P2 1 u du ] ] e Pi 1 u du e P2 1 u du i 1,h/2+ ] 1 P2 1 =P 2 1 e Pi 1 u du ] 1 P2 1 =P i 1 ] P ] P2 1 = P i 1. Now, according to Proposition 4.3.2, the two expectations are for large h less than respectively 2E[e λ IˆV ] and 2E[e 2λ IˆV ]. Using the arguments of the proof of Lemma we can prove that for h large enough P P 2 1 = P i 1 e δκi 2h/3. This proves that E [ exp λ τ P 1 1,h/2+ 1 P e 1 u du ] 3 E [ ] e 2λ IˆV, for h large enough. Now, combining , , and 4.3.1, we get so is proved as well Asymptotic of the h-minima sequence In this subsection, we are interested in the asymptotic distance between the h- minima when h goes to infinity. This leads to estimates that are useful to study the local time of the diffusion in V outside the bottoms of the valleys, and Theorem 16

172 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS is also proved in the end of the subsection. First, we define the first ascend of h for V V : τ h := inf{u, V Vu = h}, m h := inf{u, Vu = Vτ h}. Note that, since τ h =, τ h and m h coincide with respectively τ 1h and m 1, defined in Subsection We study m h by the mean of excursions theory. Let F denote the space of excursions, that is, càd-làg functions from [,+ [ to R, starting at zero and killed at the first positive instant when they reach. Note that this instant can possibly be infinite. For ξ F, recall the notation ζξ := inf{s >, ξs = } for the length of the excursion ξ. Also, let F h, and F h,+ denote respectively the set of excursions which height is strictly less than h and the set of excursions higher than h : { } { } F h, := ξ F, supξ < h [,ζ], F h,+ := ξ F, supξ h [,ζ] With the help of Fact 4.3.1, we see that {} is instantaneous for V V and it is regular if and only if V has unbounded variations, excursion theory above can thus be applied for V V see [8]. Let L be a local time at of V V, N its associated excursion measure and L 1 its right continus inverse. Then, the excursions above of V V form a Poisson point process on F with intensity measure N. In the irregular case when V has bounded variations the local time L has to be defined artificially as in [8], Section IV.5. In this case, the excursion measure is proportional to the law of the first excursion and in particular the total mass of the excursion measure is finite. Let us defines h, ands h,+ to be two independent pure jumps subordinators with Lévy measure respectively ζnf h,. and ζnf h,+., the image measures of respectivelynf h,. andnf h,+. byζ. SinceζNF h,.+ζnf h,+. = ζn, the Lévy-Khintchine representation yields that S := S h, + S h,+ is a pure jumps subordinator with Lévy measure ζn, it is therefore equal in law to L 1. We also define T h to be an exponential random variable with parameter NF h,+, independent from S h, and S h,+. We can now express m h in term of these objects. Lemma m h L = S h, T h, and τ h m h L = τv,h. Démonstration. Considering e t t, the excursion process of V V, we have that Lτ h is the instant when occurs the first jumps belonging to F h,+, and this jump corresponds to the excursions having m h as starting point. We can thus write m h = ζe t 1 et Fh, t Lτ h 161

173 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS Also,Lτ h follows an exponential distribution with parameternf h,+ and is independent from the process e t 1 et F h, t. By properties of Poisson point processes, the process in the right hand side of , t. ζe t1 et F h,, is the sum of the jumps of a Poisson point process on R +, with intensity measure ζnf h,.. Thus, from the Lévy-Khintchine representation, it has the same law as the subordinator S h,, which yields the result for m h. Then, Vx Vm h, x τ h m h is, considered up to its hitting time of h, the first excursion higher than h of V V. As we sais in the proof of Propositions and 4.3.6, the latter is equal in law to V x x τv,h. The result about τ h m h follows. We are now left to study S and NF h,+, the parameter of T h. For S, we have the following lemma : Lemma The random variable L 1 1 or equivalently S1 admits some finite exponential moments. Démonstration. We fix an arbitrary t > 1. N ξ F, ζξ > t is equal to N ξ F, ζξ > t, τξ,1 1+N ξ F, ζξ > t, τξ,1 > The first term is less than N ξ F, τξ,1 < +, inf{s, ξτξ,1+s = } > t 1 =N ξ F, τξ,1 < + PτV 1,],] > t 1 N ξ F, τξ,1 < + PVt 1 > 1 N ξ F, τξ,1 < + e λ E [ e λvt 1], where we used the Markov property in the excursions, chose some λ ],κ[ and used Markov s inequality. We thus get that Nξ F, ζξ > t, τξ,1 1 is less than N ξ F, τξ,1 < + e λ e t 1Ψ V λ, and Ψ V λ < since λ ],κ[. Then, the second term of is less than 1 PτV z,],] > t 1 N ζξ > 1, ξ1 dz, where we used the Markov property at time 1 in the excursions, 1 1 PVt 1 > z N ζξ > 1, ξ1 dz PVt 1 > 1 N ζξ > 1, ξ1 dz N ζξ > 1 e λ E [ e λvt 1], 162

174 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS for the same λ ],κ[ and again we used Markov s inequality. We thus get that Nξ F, ζξ > t, τξ,1 > 1 is less than N ξ F, ζξ > 1 e λ e t 1Ψ V λ Now putting and in , we get the existence of positive constants c 1,c 2 such that which proves that for some c >, t > 1, N ξ F, ζξ > t c 1 e c 2t + 1 e cζx Ndx < +. Using Theorem 25.3 in Sato [59], we deduce that L 1 1 admits some finite exponential moments. The next lemma deals with the asymptotic behavior of NF h,+. Lemma NF h,+ = e κh NF 1,+ e κ E [ e κv 1τV 1,],] ] +Oe 2κh. Démonstration. We fix h > 1. From the strong Markov property applied at the hitting time of 1 in the excursions belonging to F 1,+ we get Then, e κh 1 = P NF h,+ /NF 1,+ = Pτ V 1,h < τ V 1,],] =: p h sup [,+ [ = p h +P V 1 > h τv 1,],] < τv 1,h, sup V 1 τv 1,],]+. > h [,+ [ = p h +E [ e κh V 1τV 1,],] 1 τv1,],]<τv 1,h], where we used the strong Markov property at the stopping time τv 1,],]. We get p h = e κh e κ E [ e κv 1τV 1 ],],] 1 τv1,],]<τv 1,h = e κh e κ E [ e κv 1τV 1,],] ] +E [ e κv 1τV 1,],] 1 τv1,],]>τv 1,h]. According to , it only remains to bound the last term. Since e κv 1τV 1,],] is almost surely less than 1 we have E [ e κv 1τV 1,],] 1 τv1,],]>τv 1,h] PτV1,h <= + = e κh 1, and the result follows. 163

175 4.3. PATH DECOMPOSITION OF A SPECTRALLY NEGATIVE LÉVY PROCESS We can now get the asymptotic behavior of m h. Proposition e κh m h L h + Eq, where Eq is the exponential distribution of parameter q := NF 1,+ e κ E[e κv 1τV 1,],] ]/ + ζxndx. The parameter q here is the one appearing in Theorem If V = W κ, the κ-drifted brownian motion, the calculations can be made more explicit. Using 2.7 of [37], we can prove that the parameter q in the above proposition which is the same q as in Theorem equals κ 2 /2. Démonstration. of Proposition Thanks to Lemma 4.3.9, we are reduced to study S h, T h. We have S h, T h T h = ST h T h Sh,+ T h T h According to Lemma , PT h > M converges to 1 when h goes to infinity, for any M >. According to Lemma 4.3.1, S1 has finite expectation so we can use the law of large number for Lévy processes see for example Theorem 36.5 in [59] and deduce that St/t converges almost surely and therefore in distribution to E[S1]. We deduce that ST h /T h L E[S1] h + We now deal with the second term of the right hand side of From the independence between S h,+ and T h we have E [ ] + S h,+ T h /T h = E [ S h,+ u/u ] L Th du = E [ S h,+ 1 ]. Then, ζnf h,+., the Lévy measure of S h,+, converges to when h goes to infinity. SinceS h,+ is pure jump, we deduce thats h,+ 1 converges toin probability when h goes to infinity. Then, S h,+ 1 S1 and S1 has finite expectation, so we deduce by dominated converges that E[S h,+ 1] converges to, we thus get S h,+ T h /T h Combining , and we get S h, T h /T h L h + L E[S1] h + 164

176 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 Recall also that E[S1] = E[L 1 1] = + x ζndx. Then, e κh S h, T h = e κh /NF h,+ NF h,+ T h S h, T h /T h, so combining Lemma , the definition of T h, and Slutsky Lemma, we get that e κh S h, T h converges to an exponential distribution with parameter q := NF 1,+ e κ E[e κv 1τV 1,],] ]/ + ζxndx. Thanks to Lemma we get the result. We can now prove Theorem Démonstration. of Theorem As we said in the proof of Proposition 4.3.2, Lemma 1 of [21] applies here, so the random variables m i+1 m i i 1 are iid, and m 1 is also independent from this sequence. To prove Theorem 4.1.4, we thus only need to prove that the random variables e κh m 1 and e κh m 2 m 1 converge in distribution to an exponential distribution with parameter q. Then, note that according to Lemma applied with n := 2, we only need to prove this convergence for e κh m 1 and e κh m 2 m 1. Now, according to subsection 4.3.3, m 2 m 1 = τ 1 h m 1 + L 1 τ 1 h+ L 2 L 1 + m 2 L 2, and according to Proposition for the first term and the Markov property at times τ 1 h, L1 and L 2 for the other terms, we get that the terms on the right hand side have respectively the same law as τv,h, τv,], h/2], τv,], e 1 δκh ], and m h. Now, using Lemma for the first term, and Lemma for the second and third terms, we get that these terms, renormalized by e κh converge to when h goes to infinity. Proposition gives the convergence for the last term renormalized by e κh. Combining with Slutsky Lemma we get that e κh m 2 m 1 converge in distribution to an exponential distribution with parameter q. For m 1 it is even simpler, we have m 1 = L 1+ m 1 L 1, so we can conclude the same way. 4.4 Supremum of the local time when < κ < 1 We now generalize, in the context of the diffusion in V, the arguments of [4] to prove the convergence of the supremum of the local time when < κ < 1. First, let us recall some definitions of [3] and [4] for the study of the diffusion across the valleys. 165

177 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 The diffusions moves across valleys, and for these to be neither too small or too large with respect to the time scale, we have to make the size of the valleys grow with time t. We are thus interested in h t -valleys where h t := logt φt, for some function φ with loglogt << φt << logt. We also define N t, the indice of the largest h t -minima visited by X until time t, { } N t := max k N, sup Xs m k. s t We need our estimates on the valleys to be true simultaneously for a large deterministic number of valleys that we will prove to be greater than N t with good probability. This is why we define n t := e κ1+δφt. In all this section, we assume that δ is small enough so that 1+3δκ < 1, and that the hypothesis of Theorem are satisfied : V has unbounded variations and there exists p > 1 such that V1 L p, so all the results of Sections 4.3 and 4.5 apply here Proof of Theorem As in [4], the idea is to approximate L X t/t by a functional of a sequence e i Si,e t i SiR t i t i 1 that is defined later. Our approximation is similar to Proposition 5.1 of [4] and can be stated as follows : Proposition Let us define the repartition functions P 1 ± α := P 1 1 N t1 η e e i S t t iri t Nt1 η SN t t1 η α e Nt1 η SN t t1 η RN t t ±, t1 η i=1 max 1 i N t1 η 1 where α t ± := α1±loglogt 1/2. Let also v be a positive function such that lim η lim t + vη,t =. For all t large enough we then have P1 α vη,t P supl X t,x/t α P 1 + α+vη,t. x R The next step is to identify the objects in P ± 1 α as continus functionals of something that converges to Y 1,Y 2 defined in the Introduction when t goes to infinity. Let D[,+ [,R 2,J 1 be the space of càd-làg functions taking values in R 2, equipped with the J 1 -Skorokhod topology. If, as in [4], we define Y 1,Y 2 t by e i S t i t α ± t, s, Y 1,Y 2 t s := 1 t se κφt j=1 e j S t j,e j S t jr t j, then we have 166

178 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 Proposition Y 1,Y 2 t converges in distribution to Y 1,Y 2 in D[,+ [,R 2,J 1. The objects in P 1 ± α can be written in term of functionals of Y 1,Y 2 t : the functionals J I,a, K I,a, K I,a, KI,a and K I,a with a = 1 η defined in Subsection 4.3 of [4]. Thanks to Lemma 4.5 there, we see that the κ-stable subordinator Y 1,Y 2 is almost surely a point of continuity for these functionals so, by continus mapping theorem and Proposition 4.4.2, we get, when t goes to infinity, the convergence of P 1 ± α to { P max Y1Y η, 1 Y 2 Y2 1 1 η Y 1Y2 1 1 η Y 1 Y2 1 } 1 η Y 2 Y2 1 1 η Y 2 Y2 1 1 η α. Then, we have almost surelyy2 1 1 η = Y2 1 1 for allη small enough since almost surely Y 2 Y2 1 1 < 1. As a consequence, when η goes to, the above expression converges to the repartition function of the random variable I 1 I 2 defined in the Introduction at α. As a consequence, Theorem follows if we prove Propositions and For Proposition 4.4.1, we first need to show that the time spent by the diffusion and the local time are negligible outside the bottoms of the standard valleys, this is the object of the next two facts that are taken from [3] and [4] however, the extension of these results to our context requires some precautions, this is why we give some details in the end of Section 4.5. Fact There exists a positive constant C > such that for t large enough, P A 1 t := n t j=1 { j 1 H m j i=1 The lower bound converges to 1 since 1+2δκ < 1. } H L i H m i 2t 1 Ce φt n t logh t. logh t Before stating the next fact, which proves that the supremum of the local time is negligible outside the "deep bottoms" of the standard valleys, we need to define what we mean exactly by "deep bottom". We define D j := [ τ j φt2, τ + j φt2 ]. Note that the definition we give is different from the one in [4]. Indeed, the presence of negative jumps for the environment in our context implies to take some precautions and to adapt some definitions. 167

179 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 Fact Recall the definitions of τ h and m h in Subsection There are positive constants C 1,C 2,C 3 such that for t large enough, P P nt 1 j= nt j=1 { { sup R P sup L X Hτ h t,x > te κ1+3δ 1φt C 1 x [,m h t] n t e κδφt, } L X H m j+1,x L X H L j,x te κ1+3δ 1φt 1 C 2 e κδφt, sup x [ L j 1, L j ] D c j } L X H L j,x L X H m j,x te 2φt, 1 C 3n t e 2φt The next step in the proof of Proposition is to show that the main contributions to the local time and to the time spent by the diffusion in the bottoms of the standard valleys can be approximated by the sequence e i S t i,e i S t ir t i i 1 : Proposition The random variables e j,s t j,r t j, j 1 are mutually independent and for any ǫ ],max1/8,1 1+δκ/2[ there exists a positive constant c such that for t large enough, P P A 3 t := nt j=1 { } A 2 t := nt j=1 1 e ǫht/7 e j Sj t L X H L j, m j 1+e ǫht/7 e j Sj t 1 e cht, { } 1 e ǫht/7 e j SjR t j t H L j H m j 1+e ǫht/7 e j SjR t j t 1 e cht. This proposition is proved in the following subsection. For the end of the proof of Proposition 4.4.1, the idea is simply to use the previous steps to translate "L X t/t α" in term of events only involving the sequence e i S t i,e i S t ir t i i 1. We do this in Subsection 4.4.3, following the arguments of the proof of Proposition 5.1 of [4]. Finally, Proposition is roughly speaking a Donsker Theorem for heavy tailed random variables, its proof relies on the study of the right tail of the random variables e 1 S t 1 and e 1 S t 1R t 1 which is done in subsection Proof of Proposition and consequences We now prove that L X H L j, m j,h L j H m j j 1 can be approximated by an iid sequence. This generalizes Proposition 3.5 of [4] to our setting. First, note that the proof of the first point of Lemma 3.6 in [4] can be repeated here. We only replace W κ by V, consider the standard valleys to be defined in our sens, replace the random times L j of [4] by L j 1, apply Lemma for the first 168

180 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 n t indices instead of Lemma 3.2 of [3], and use Remark instead of Lemma 2.2 of [4]. We get P L X H L j, m j,h L j H m j = Lj, h j 1 n t e κδht/6, 1 j n t 1 j n t where the sequence L j, h j j 1 is iid and L j, h j is equal to A j L j L B jτb j,1,, Lj L j 1 e Ṽ j u L B j τb j,1,a j u/a j L 1 du, where A j u := u m j eṽ j x dx and B j is a sequence of iid brownian motions starting at, and independent from V. Note that e j := L X H L j, m j /A j L j = L B jτb j,1, follows an exponential distribution with parameter 1/2. We are thus left to give approximations of h j and A j L j. For this, we first prove lemmas to bound exponential functionals of the environment. Lemma Choose ǫ such that < ǫ < max1/8,1 1 + δκ/2. Then, j 1, P τ j ht/2 L j 1 e Ṽ j u du > e ǫht e cht, for t large enough and some positive constant c depending on δ and ǫ. Démonstration. Fix j 1. We have τ j ht/2 L j 1 e Ṽ j u du = τ j ht L j 1 e Ṽ j u du+ The first term of the right hand side is less than τ j h t L j 1 τ j ht/2 sup e Ṽ j. [ L j 1, τ j ht] τ j ht e Ṽ j u du According to Lemma for the first factor, and the definition of L j together with Lemma applied with α = 1,η = ǫ for the second factor, there is a positive constant c 1 depending on δ and ǫ such that for t large enough, P τ j ht L j 1 e Ṽ j u du > e ǫht /2 > e 1+δκ+ǫ 1ht e c 1h t

181 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 According to Propositions and we have, τ j ht/2 τˆv,[h t,+ [ P e Ṽ ju du > e ǫht /2 2P e ˆV u du > e ǫht /2 τ j ht τˆv,[h t/2,+ [ +2e δκht/3, and the integral on the right hand side is less than τ ˆV,[h t,+ [ sup e ˆV. [τˆv,[h t/2,+ [,τˆv,[h t,+ [] Then, according to Lemma applied with y = h t, r = e ht/8 and Lemma applied with a = h t /4,b = h t /2, z =, there is a positive constant c 2 such that for t large enough, τ j ht/2 P e Ṽ ju du > e ǫht /2 > e 1/8 1/4ht e c 2h t τ j ht The combination of 4.4.6, and yields the result. Lemma Choose ǫ such that < ǫ < 1/8. There is a positive constant c such that for t large enough, Lj j 1, P e Ṽ ju du > e ǫht e cht. τ + j ht/2 Démonstration. Fix j 1. We have Lj τ + j ht/2 e Ṽ j u du = τj h t τ + j ht/2 e Ṽ j u du+ Lj τ j h t e Ṽ j u du According to Proposition 4.3.6, the fist term on the right hand side is equal in law to τv,h t τv,h t/2 e V u du which is less than τv,h t sup e V. [τv,h t/2,τv,h t] According, for the first factor, to Lemma applied with y = h t,r = e ht/8 and, for the second factor, to Lemma applied with a = h t /4,b = h t /2, there is a positive constant c 1 such that for t large enough, τj h t P e Ṽ ju du > e ǫht /2 > e 1/8 1/4ht e c 1h t τ + j ht/2 17

182 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 Because of the definition of L j, the second term in the right hand side of is less than e ht/2 L j τ j h t. Because of Proposition 4.3.6, L j τ j h t is equal in law to τv,], h t /2]. Applying Lemma with y = h t /2 and r = e ht/4 we get, for t large enough, Lj P e Ṽ ju du > e ǫht /2 > e 1/4 1/2ht e ht τ j h t The combination of 4.4.9, and yields the result. Lemma Choose ǫ such that < ǫ < 1/4. There is a positive constant c depending on ǫ such that for t large enough, j 1, P sup A j u/a j L j e 1 2ǫht/2 1 e cht. Démonstration. We have u [ τ j ht/2, τ+ j ht/2] A j L j A j τ + j h t/2 L = τj h t m j eṽ j u du L = τv,h t/2 e V u du τv,h t e V u du and where we used Proposition for the equalities in law. For A j L j we use applied with h = h t, η = ǫ/2 and for A j τ + j h t/2 we use applied with h = h t /2, η = ǫ. We get the existence of a positive constant c 1 such that for t large enough, P A j τ +j h t/2/a j L j e 1 2ǫht/2 1 e c 1h t Then, A j τ j h t/2 = τ j ht/2 m eṽ ju j du and according to Propositions and we have, τˆv,h t/2+ P A j τ j h t/2 e ht1+ǫ/2 2P 2e c 4h t +2e δκht/3, eˆv u du e ht1+ǫ/2 +2e δκht/3 where the last inequality is true for c 4 a positive constant and t large enough. It comes from applied with h = h t /2, η = ǫ. Now combining with the lower bound for A j L j given by applied with h = h t, η = ǫ/2 we get P e ht1 2ǫ/2 A j τ j h t/2/a j L j 1 e c 5h t, for some positive constantc 5, and whentis large enough. The combination of and with the increase of A j. yields the result. 171

183 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 We now approximate h j. Lemma Choose ǫ such that < ǫ < max1/8,1 1+δκ/2. There is a positive constant c depending on δ and ǫ such that for all t large enough we have j 1, P hj A j L j Rje t j e ǫh t/6 A j L j Rje t j 1 e cht, where R t j := τ + j ht/2 τ j ht/2 e Ṽ j u du. Démonstration. Our proof has the same spirit as the one of Lemma 4.7 of [3] but relies on our estimates. We give the details. Here again, h j /A j L j can be cut into three parts : h j /A j L j = = Lj e Ṽ ju L B j L j 1 τ j ht/2 + L j 1 τb j,1,a j u/a j L j du τ + j ht/2 Lj + e Ṽ ju L B j τb j,1,a j u/a j L j du τ j ht/2 τ + j ht/2 =: J j +J j 1 +J j We start by bounding J j 2 and J j : J j 2 supl B j τb j,1,. Lj e Ṽ ju du. [,1] τ + j ht/2 Thanks to estimate 7.12 of [4] in which we take x = e ǫht/2 and Lemma 4.4.7, there exists a positive constant c 1 such that for t is large enough, P J j 2 < e ǫht/2 1 e c 1h t For J j : J j sup L B j ],] τb j,1,. τ j ht/2 L j 1 e Ṽ j u du. Thanks to estimate 7.13 of [4] in which we take x = e ǫht/2 and Lemma 4.4.6, there exists a positive constant c 2 such that for t is large enough, P J j < e ǫht/2 1 e c 2h t For J j 1 we apply Lemma and estimate 7.11 of [4] applied with δ = e 1 2ǫht/2, ǫ = e 1 2ǫht/6, we get τ jh t/2 J j 1 e j e Ṽ ju du τ j ht/2 e 1 2ǫht/6 e j τ + j ht/2 τ j ht/2 e Ṽ j u du,

184 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 with probability greater than 1 e c 6h t, for some positive constant c 6, when t is large enough. Then, τ + j ht/2 τ j ht/2 e Ṽ j u du τ + j ǫht/8 τ + j ǫht/16 e Ṽ j u du L = τv,ǫh t/8 τv,ǫh t/16 e V u du, where we used Proposition for the equality in law. We thus get τ + j ht/2 P e Ṽ ju du e ǫht/8 P τv,ǫh t /8 τv,ǫh t /16 1 e c 7h t, τ j ht/ when t is large enough, according to applied with α = ǫ/8, ω = ǫ/16, and where c 7 is a positive constant. We have Pe j e ǫht/8 t + e ǫht/8 /2, so combining with and we get, for some positive constant c 8 and t large enough, P J j 1 > e ǫht/5 1 e c 8h t Combining with and we get P J j +J j 2 2e ǫht/5 J j 1 1 e c 9 h t, for some positive constant c 9, and when t is large enough. Combining with and we get the result. Now, we approximate A j L j. Lemma S t j := Lj τ + j ht/2eṽ j u du is independent from e j,r t j and such that P Sj t A j L j 1+e ht/7 Sj t for some positive constant c, when t is large enough. 1 e cht, Démonstration. Sj t := Lj ju du so S τ j,r t j t only depends on V and is thus + j ht/2eṽ independent from e j which only depends on B j. The independence between Sj t and Rj t comes from Proposition : the latter gives the independence between P j j 2 and P 3 and allows to apply the Markov property at time τ P j 2,h t /2, since P j L 2 = V x x τv,h t. The random variables e j,sj t and Rj t are therefore mutually independent. Then, from the definitions of Sj t and A j L j we have S t j A j L j = τ + j ht/2 m j eṽ j u du+s t j

185 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 Applying successively Proposition and estimate with h = h t /2, η = 1/3, we get τ + j ht/2 P eṽ ju du e 4ht/6 1 e c 1h t, m j for some positive constant c 1, when t is large enough. Then, S t j τj h t τ + j ht/2 eṽ j u du = τj h t m j eṽ j u du τ + j ht/2 m j eṽ j u du. Applying Proposition for each term, and, for the first term, estimate with h = h t, η = 1/6, for the second term, with h = h t /2, η = 1/3, we get P S t j e 5ht/6 e 4ht/6 1 e c 2h t, for some positive constant c 2, when t is large enough. Putting together and we get that τ + j ht/2 P eṽ ju du/sj t e ht/7 1 e c 3h t, m j for some positive constant c 3, when t is large enough. This, combined with , yields the result. Putting together 4.4.4, the iid character of the sequence L j, h j 1 j nt, Lemma and Lemma 4.4.1, we obtain Proposition Thanks to this proposition, we can comparen t with the overshoots of. i=1 e isir t i. t For any a, let us define { } j N a := min j, e i SiR t i t > a. Lemma Fix ǫ as in Proposition and η ],1[. Assume that t is so large such that 1 e ǫht/7 1 < 1+η and 1+e ǫht/ /logh t 1 η. Recall the events A 1 t and A 3 t introduced in respectively Fact and Proposition Then V nt,h t {N t < n t } A 1 t A 3 t { N 1 ηt N t N 1+ηt }. Even though it is used in the following subsection, an other interest of this Lemma is to prepare the further study of the almost sure behavior of L X t. Indeed, for the almost sure behavior, the contribution of the last valley can be sometimes omitted, sometimes totally included, so we are left to study some behavior of N t 1 i=1 e i SiR t i t or N t i=1 e isir t i, t but the above lemma allows to replace N t by some N a, which is more convenient since it only depends on the sequence e i SiR t i, t i 1. i=1 174

186 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 Démonstration. Assume that the event V nt,h t {N t < n t } A 1 t A 3 t is realized. Then, for any k n t we have and k 1 N t k Hm k t H m k t H L i H m i t k 1 e i SiR t i t t1 e ǫht/7 1 1+ηt N 1+ηt k, i=1 i=1 k 1 k 1 N 1 ηt k e i SiR t i t 1 ηt H L i H m i 1 η1+e ǫht/7 t i=1 i=1 H m k t [ 1 η1+e ǫht/7 +2/logh t ] t Hm k t N t k. We have thus proved thatn 1 ηt N t N 1+ηt is satisfied onv nt,h t {N t < n t } A 1 t A 3 t Proof of Proposition We now use the preceding results of this section to approach the repartition function of L X t/t by a repartition function involving the sequencee is t i,e i S t ir t i, i 1. We first state two facts which come from Lemmas 5.2 and 5.3 of [4]. Fact Fix ǫ as in Proposition Let us define the repartition functions, depending on t, F γ,k x := P F ± γ,k x := P max L XH L j, m j γt, 1 j k 1 max e jsj t γt1±2e ǫht/7, 1 j k 1 k 1 H L j H m j xt, j=1 k 1 e j SjR t j t xt1±2e ǫht/7. Then, there is a positive constant c such that for any 2 k n t, < x 1 and γ > possibly depending on t, for all t large enough. j=1 F γ,k x e cht F γ,k x F + γ,k x+e cht, Démonstration. This is a direct consequence of Proposition

187 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 Before stating the next fact we define X mj := X.+ m j which is, according to the Markov property, a diffusion starting from m j. We also define, for any r R, H X mj r to be the hitting time of r by X mj. Fact Fix ǫ as in Proposition and recall the definition of X m1 in Subsection We define the repartition functions, depending on t, f γ x := P L X m1 t1 x, m 1 γt,h X m1 L 1 > t1 x,h X m1 L 1 < H X m1 L, f γ x := P sup L X m1 t1 x,y γt,h X m1 L 1 > t1 x,h X m1 L 1 < H X m1 L, y D 1 f ± γ x := P 1/R t 1 γ1±2e ǫht/7 /1 x, e 1 S t 1R t 1 > t1 x1 2e ǫht/7. Then, there is a positive constant c such that for any < x 1 and γ > possibly depending on t, for all t large enough. f γ x e cht f γ x f γ x f + γ x+e cht, For the justification of this fact in our context, we give some details in the end of Section 4.5. In order to generalize Lemma 5.4 of [4] to our context, let us recall the definition of the functionals K I,a and K I,a defined there : a >, KI,a f 1,f 2 := f 2 f 1 2 a, K I,a f 1,f 2 := f 2 f 1 2 a. Note that these functionals actually do not involve f 1. According to Lemma 4.5 of [4], Y 1,Y 2 is almost surely a point of continuity for these functionals. Thanks to this we can prove : Lemma lim limsup η t + sη,t := k n t P 1 t k e i SiR t i t > 1 η/2, 1 2η < 1 k 1 e i S t t iri t 1 η =. i=1 i= lim limsup sη,t := 1 Pηt Hm Nt 1 ηt = η t + This result is less precise than Lemma 5.4 of [4] of which it is the analogue, but even in [4], they only need that the limits in t converge to when η goes to. We can thus substitute our Lemma to theirs. Démonstration. The events { 1 k e i S t t iri t > 1 η/2, 1 2η < 1 t i=1 } k 1 e i SiR t i t 1 η i=1 176

188 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 for k {1,...,n t } are clearly disjoint so sη,t equals P 1 t N 1 ηt i=1 e i S t ir t i > 1 η/2, 1 2η < 1 t N 1 ηt 1 i=1 e i SiR t i t 1 η, N 1 ηt n t =P Y2Y t t, η > 1 η/2, Y2Y t t, η > 1 2η, N 1 ηt n t =P KI,1 η Y2 t > 1 η/2, K I,1 η Y 2 t > 1 2η, N 1 ηt n t. Now, according to Proposition to be proved in the following subsection, Y2,Y t 2 t converges to Y 2,Y 2 for the convergence in distribution in DR +,R 2 with the J 1 topology, and we have that for any fixed η ],1/2[, Y 1,Y 2 is almost surely a point of continuity for K I,1 η and K I,1 η. We thus get lim sup t + sη,t P Y 2 Y η > 1 η/2, Y 2 Y η > 1 2η, and we now study the limit when η goes to. Since Y 2 is a κ-stable subordinator, it is known that almost surely E := 1 Y 2 Y2 1 1 > so Y2 1 1 η = Y2 1 1 for all < η < E and Y 2 Y2 1 1 η 1 2η for all < η < E /2. This proves that almost surely, the event in the probability in fails to happen for all η small enough, so by dominated convergence, this probability converges to when η goes to. This proves Let us fix ǫ as in Proposition 4.4.5, put ǫ t := e ǫht/7 and choose t large enough so that 2/logh t η,1 ǫ t 1 2,1 3η 1 2η1+ǫ t 1. Using successively the definitions of A 1 t and A 3 t and Lemma , we get that Pηt Hm Nt 1 ηt is greater than P ηt Hm Nt 1 ηt, V nt,ht, N t n t, A 1 t A 3 t N t 1 P ηt H L i H m i 1 2ηt P i=1 N t n t, A 1 t A 3 t η1 ǫ t 1 t 2ηt N t n t, A 1 t A 3 t P 2ηt N 1 ηt 1 i=1 2Y t, 1 N t 1 i=1 e i S t ir t i 1 η 2 logh t t, V nt,h t, e i S t ir t i 1 3ηt 1 2η1+ǫ t 1 t, V nt,h t, N 1+ηt 1 i=1 2Y t, 1 e i SiR t i t 1 3ηt, V nt,ht, N t n t, A 1 t A 3 t =P 2η Y t 2 1 η Y t 2 1+η 1 3η, V nt,ht, N t n t, A 1 t A 3 t =P 2η K I,1 η Y 2 t K I,1+η Y 2 t 1 3η, V nt,ht, N t n t, A 1 t A 3 t. 177

189 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 According to Lemma 4.3.5, Fact 4.4.3, Proposition and Lemma of the following subsection, we have P V nt,h t {N t n t } A 1 t A 3 t t + 1, and again, from the convergence of Y2,Y t 2 t to Y 2,Y 2 and the continuity of K I,1 η, we get lim sup t + sη,t 1 P 2η Y 2 Y η Y 2 Y η 1 3η Here again, we have almost surely E := min{y 2 Y , 1 Y 2 Y2 1 1 } > so Y η = Y2 1 1 for all η E and Y 2 Y η 1 3η for η E /3. Also, Y2 1 1 η = Y2 1 1 for all η E and Y 2 Y2 1 1 η 2η for η min{e,1 E /2}. This proves that almost surely, the event in the probability in happens for all η small enough, so by dominated convergence, this probability converges to 1 when η goes to. This proves Démonstration. of Proposition The proof of Proposition 5.1 of [4] can be identically repeated here. We only replace W κ by V, consider the standard valleys to be defined in our sens, replace the random times L j of [4] by L j 1, and we use our estimates instead of theirs : Estimates 3.1, 3.2, 3.3, 5.2, 5.3, Lemmas 2.2, 3.2, 3.4, 4.1, 5.2, 5.3, Proposition 3.5 and the proof of Lemma 3.6 of [4] have to be replaced here by respectively Fact 4.4.3, the combination of Lemma and Lemma 4.5.2, Lemma of the following subsection, , , the combination of Lemma and Remark 4.3.4, the second point of Fact 4.4.4, the third point of Fact 4.4.4, Lemma of the following subsection, Fact , Fact , Proposition and the discussion before Lemma The only difference is that, since our definition of D j is different, we do not need an analogue of 5.14 from [4] : it follows from our definition of D j that D j [τ 1 h t /2,τ 1 h t /2], so we can use Lemma to bounda j u/a j L j ond j, which is the key to get the analogue of 5.17 from [4] Proof of Proposition and consequences This proposition relies on : Lemma Fix η ],1/3[ and recall the constant C defined in the Introduction. We have lim sup x κ e κφt P e 1 S1/t t > x C =, t + x [e 1 2ηφt,+ [ lim sup t + y [e 1 3ηφt,+ [ y κ e κφt P e 1 S t 1R t 1/t > y C E[R κ ] =

190 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 For any positive α, e κφt Pe 1 S t 1/t x, e 1 S t 1R t 1/t y converges uniformly when t goes to infinity on [α,+ [ [α,+ [ to C y κ E[R κ 1 R y/x ]+C x κ PR > y/x. Démonstration. We start by proving a convergence analogue to 4.3 of [4] : x κ e κφt P S1/t t > x C = lim t + sup x [e 1 ηφt,+ [ and we will deduce the first statement. First, S t 1 = From Proposition we get and 1 L1 eṽ 1u du = L eht t τ 1 h t t τ1 h t L1 eṽ 1u du+ τ + 1 ht/2 τv,], ht/2] eṽ 1u du τ 1 h t τv,], ht/2] e Vy dy = e φt e Vy dy, τ1 h t 1u du = L 1 τv,h t e Vy dy e φt τv,h t t τ 1 h t/2eṽ t τv,h t/2 By the second assertion of Lemma 4.5.3, + P e Vy dy > u u + Cu κ We define ǫ t := e ηφt/2 and take x [ e 1 ηφt,+ [ this implies xe φt e ηφt which converges to infinity. From , , and , x κ e κφt PS1/t t > x is less than x κ e κφt P + τv,h t > xe φt ǫ t +x κ e κφt P e Vy dy > 1 ǫ t xe φt. From Lemma 4.5.6, the limit of the first term is, and from the limit of the first second is C. We get lim sup t + sup x [e 1 ηφt,+ [ x κ e κφt P S t 1/t > x C From the Markov property applied at time τv,], h t /2] we have + τv,], ht/2] e φt e Vy dy = L e φt e Vy dy +e VτV,], ht/2] φt + eṽy dy, 179

191 4.4. SUPREMUM OF THE LOCAL TIME WHEN < κ < 1 where Ṽ is an independent copy of V. We now put ǫ t := e ht/4 and note that VτV,], h t /2] < h t /2. Then, Pe φt + e Vy dy > x1+ǫ t is less than τv,], ht/2] + P e φt e Vy dy > x +P e ht/2 φt eṽy dy > xǫ t P S1/t t > x + +P e ht/2 φt eṽy dy > xǫ t, where we used and for the first term. Using in the above inequality we get lim inf t + inf x [e 1 ηφt,+ [ xκ e κφt P S1/t t > x C. Combining this with we get Now that we have 4.4.3, the rest of the proof is exactly the same as the proof of Lemma 4.1 in [4] once they have proved 4.3. The argument crucially needs the fact that R t 1 t>1 converges in distribution to R and is bounded in all L p spaces. This is true from Proposition applied with h = h t, so the lemma is proved. In [4], the proof of Proposition 1.4 that is, the convergence of Y 1,Y 2 t toward Y 1,Y 2 relies only on their Lemma 4.1 from which is proved the tightness of the family Y 1,Y 2 t and the identification of the limit distribution. Using Lemma instead of Lemma 4.1 of [4], the same proof can be repeated here and we get Proposition As an other consequence of Lemma , we can prove that no more than n t valleys are visited until instant t. This is fundamental since most of the estimates we have are true not for all but for the first n t valleys. Lemma There is a positive constant c such that for all t large enough, PN t n t e cht. Démonstration. We have {N t n t } = {Hm nt t} and n t 1 i=1 H L i H m i H m nt = Hm nt on V nt,h t. Let us fix ǫ as in Proposition Using the definition of A 3 t there, we get that P{N t n t } V nt,h t A 3 t is less than P nt 1 i=1 H L i H m i t, A 3 t P nt 1 i=1 e i S t ir t i t1 e ǫht/7 1 P sup e i SiR t i t t1 e ǫht/7 1 [ 1 P e 1 S1R t 1/t t > 1 e ǫht/7 1] n t 1 1 i n t 1 exp [ n t 1P e 1 S t 1R t 1/t > 1 e ǫht/7 1], 18

192 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V where the last inequality comes from log1 x x for x [,1[. According to Lemma and the definition of n t, we have that n t 1Pe 1 S t 1R t 1/t > 1 e ǫht/7 1 t + C E[R κ ]e κδφt, so P{N t n t } V nt,ht A 3 t e ht for t large enough. Also, PVn c t,h t n t e δκht/3 for t large enough according to Lemma and PA 3,c t has an upper bound given by Proposition The result follows. 4.5 Some estimates on V, V, ˆV and the diffusion in V In this section, we prove some estimates for the processesv,v and ˆV, especially about the hitting times and the exponential functionals of these processes. We also prove some facts used in Section 4.4. Even though Section 4.4 gives the main ideas, the estimates we prove here actually represent the biggest part of the work for the proof of Theorem Some of them are rather classical but some others, especially those of Subsection 4.5.5, are new and technical. They are used in this paper in place of simpler estimates that are true in the case of a drifted brownian potential, but not for a general spectrally negative Lévy potential Estimates on V Lemma Let a, b be positive numbers and define T := inf{x, Vx / [ a,b]}, then 1 e κa e κb PVT = b e κb. Démonstration. For the upper bound, PVT = b Psup [,+ [ V b = e κb. For the lower bound, note that the process e κv. T is a bounded martingale so, by the convergence theorem for martingales and the dominated convergence theorem we get 1 = E [ e κvt] = PVT ae [ e κvt VT a ] +PVT = be κb which yields the result. e κa +PVT = be κb, We now study how V leaves an intervalle from below. More precisely, we control the moments of V τv,], 1]. This is where the assumption V1 L p for some p > 1 becomes necessary for Theorem

193 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Lemma p 1, V1 L p V τv,], 1] L p. Démonstration. We use V < r as defined in Subsection 4.1.2, where r is chosen such that V V < r drifts to. Let κ r denote the non trivial zero of Ψ V V < r. We fix x r. We have P 1+V τv,], 1] > x Pτ V,], x] τv,], 1] PV τ V,], x] 1 P V V < r τ V,], x] 1. Then, since x > r, we have τ V,], x] = τ V < r,], x] which is independent from V V < r, because V V < r and V < r are independent. We thus get that P 1+VτV,], 1] > x is less than P V V < r τ V < r,], x] 1 [ ] e κr/2 E e κrv V < r τ V < r,], x]/2 [ ] = e κr/2 E e Ψ V V < r κ r/2τ V < r,], x], where we used Markov s inequality. Then, note that Ψ V V < rκ r /2 < thanks to the definition of κ r, and that τ V < r,], x] follows an exponential distribution with parameter ν], x]. We thus get e κr/2 P 1+V τv,], 1] > x 1 Ψ V V < rκ r /2/ν], x] Cν], x], where we put C := e κr/2 /Ψ V V < rκ r /2 >. We now choose p 1 and assume V1 L p. Theorem 25.3 in [59] implies that r x p νdx < +, or equivalently + x p 1 ν], x]dx < +. Using we deduce that r r so VτV,], 1] L p. x p 1 P 1+V τv,], 1] > xdx < +, The next lemma is fundamental. In Section 4.4, it allows us to compute precisely the right tails of the contributions to the local and to the time spent by the diffusion in the bottoms of the valleys. This allows to prove that the sum of these contributions converges to the κ-stable subordinator Y 1,Y 2 from which is constructed the limit distribution in Theorem

194 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Lemma There is a positive constant c such that for x small enough, + P e Vu du x c x. + P e Vu du x x + Cx κ, where as in the Introduction, C is the constant in Corollary 5 of [12] applied to V. Démonstration. For the first assertion, our argument is very close to the one given at the beginning of the proof of Theorem 3 in [12], but to be into their setting we would have to assume the existence of finite exponential moments for V which we do not. We thus give some details. We use V < 1 as defined in Subsection For x >, and η > that will be chosen later, the probability P + e Vu du x equals + + P e Vu du x, V < 1 η = +P e Vu du x, V < 1 η η P e V V < 1 u du x, V < 1 η = +P V < 1 η η P e V V < 1 u du x +1 e ην], 1[, because V < 1 is a compound Poisson process that jumps at rate ν], 1[. We get + η P e Vu du x P e V V < 1 u du x +ην], 1[ V V < 1 is a Lévy process with bounded jumps so the arguments of [12] apply and yield η P e V V < 1 u du x P sup V V < 1 logη/x [,η] 2P V V < 1 η logη/x 2x η E [ e V V < 1 η = 2x η eηψ V V < 1 1, where we used Markov s inequality. The crucial point is that the Laplace exponent at 1 : Ψ V V < 1 1 is defined and finite since V V < 1, having bounded jumps, has ] 183

195 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V a Laplace transform defined on the whole complexe plane Theorem 25.3 in [59]. Now, combining with and choosing η = x, we get, for any x small enough so that e xψ V V < 1 1 2, + P e Vu du x 4+ν], 1[ x, which yields the first assertion. We now prove the second assertion, it is only an application of Corollary 5 of [12] to V. Since V is spectrally negative, we know that V has a Laplace transform which is defined on [,+ [ and the θ in the corollary is the non-trivial zero of the Laplace exponent of V, that is κ. To check the Cramer s condition, we use the decomposition V = V V < 1 +V < 1. We have E[ Vx expκvx] E [ V V < 1 x exp κv V < 1 x+κv < 1 x ] +E [ V < 1 x exp κv V < 1 x+κv < 1 x ] E [ V V < 1 x exp κv V < 1 x ] +ME [ exp κv V < 1 x ]. where we used the fact that V < 1 x for the first term and, for the second term, the fact that, since V < 1 x, V < 1 x expκv < 1 x is deterministically bounded bym, the constant boundingy ye κy onr +. Then, sincev V < 1 is a Lévy process with bounded jumps,v V < 1 x admits finite exponential moments of any positive and negative order see Theorem 25.3 in [59]. As a consequence the above expression is finite which yields the Cramer s condition. Finally, V is indeed not arithmetic because it is spectrally negative and not the opposite of a subordinator. The hypothesis of Corollary 5 of [12] are thus satisfied for V and we get the second assertion. Lemma There are two positive constants c 1,c 2 such that y,r >, PτV,], y] > r e c 1y c 2 r. Démonstration. Let us choose c 1 ],κ[ and define c 2 := Ψ V c 1. c 2 is positive because of the definition of κ. We have PτV,], y] > r PVr > y e c 1y E [ e c 1Vr ] = e c 1y c 2 r, where we used Markov s inequality. Lemma Choose η ],1[, then a,b >, PτV V,a < τv,], b] b/ηa+1e κ1 ηa. 184

196 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Démonstration. We first remark that so {τv V,a < τv,], ηa]} { sup V > 1 ηa [,+ [ PτV V,a < τv,], ηa] e κ1 ηa In order to establish a boundary with b instead of ηa, we define the sequence of stopping times T i i by T := and We have T i+1 := min τv T i V T i,a,τ V T i,], ηa]. b/ηa+1 { τv T i V T i,a τv T i,], ηa] } {τv V,a τv,], b]}, i= and by the Markov property applied at the stopping times T i, the events in the intersection have all the same probability, so taking the complementary PτV V,a < τv,], b] b/ηa+1 PτV V,a < τv,], ηa] b/ηa+1e κ1 ηa. } Estimates on V We define V to be "V conditioned to drift to + ", as in [8], page 193. The Laplace exponent Ψ V of V satisfies Ψ V = Ψ V κ+.. As a consequence Ψ >, V so V drift to infinity because of Corollary VII.2 in [8] and it is also proven that V = V. Therefore, for x >, Vx is only Vx conditioned in the usual sense to remain positive, which is a useful property for our proofs. Also, note that from the definition of the law of V in [8], we have [ ] λ > κ, E e λv 1 = E [ e λ+κv1] < +, so the spectrally negative Lévy process V has its Laplace transform, as well as its Laplace exponent Ψ V, defined on the half-plane {Kz > κ}. Lemma There are two positive constants c 1,c 2 such that y,r >, P τv,y > r e c 1y c 2 r. 185

197 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Démonstration. Fix y and r >. From the first point of Lemma 2.6 of [72] we have P τv,y > r P τv,y > r Then, V is a spectrally negative Lévy process, so, according to Theorem 1 page 189 in [8], the process τv,. is a subordinator which Laplace exponent Φ V is defined for λ by [ Φ V λ := log E e λτv,1 ], and we have Φ V = Ψ 1. V From the discussion before the lemma, we know that V has its Laplace transform, as well as its Laplace exponent Ψ V, defined in a neighborhood of. Then, since Ψ >, the holomorphic local inversion theorem tells us that Ψ 1, that is V V Φ V, extends in a neighborhood of. Therefore, the subordinator τv,. has a Laplace transform defined in a neighborhood of. From Markov inequality we get, for a positive c 2 in this neighborhood, P τv,y > r [ e c2r E e c 2τV,y ] = e yφ V c 2 e c2r, Note that c 1 := Φ V c 2 is positive. Combining with we get the result. Lemma There are two positive constants c 1,c 2 such that, for all 1 < a < b, we have P inf V b < a c 2 e c1b a. [,+ [ Démonstration. V b is only V b P inf V [,+ [ conditioned in the usual sense to stay positive, so b < a = P < inf V b /P < a inf V b > [,+ [ [,+ [ P inf V < a b /P inf V > b. [,+ [ [,+ [ Then, according to [8] page 192, the Laplace transform of inf [,+ [ V is given by [ λ, E e λinf [,+ [V ] = Ψ λ V Ψ V λ As we said in the beginning of this subsection, Ψ V extends analytically on a neighborhood ofand has a non null derivative at, so extends too andinf [,+ [ V admits a Laplace transform on a neighborhood of. Choosing c 1 > such that c 1 is in this neighborhood and c 2 := E[e c 1inf [,+ [ V ]/P inf [,+ [ V > 1, we get the result by Chernoff s inequality. 186

198 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Lemma There are two positive constants c 1 and c 2 such that for any < α < ω, η ],1[ and all h large enough, we have P τv,ωh τv,αh 1 e c1ω αh, τv,h P e V u du e 1 ηh 1 e c2ηh/2, τv,h P e V u du e 1+ηh 1 e h Démonstration. From the Markov property at time τv,αh and the fact that V αh is V αh conditioned in the usual sense to stay positive we have that PτV,ωh τv,αh 1 equals P τv αh,ωh 1 for h large enough, = P = P τv αh,ωh 1, inf V αh > /P [,+ [ τv,ω αh 1, inf V > αh [,+ [ 2P τv,ω αh 1, [ ] 2e E e τv,ω αh = 2e e ψ 1 V 1ω αh, inf V αh > [,+ [ /P inf V > αh [,+ [ where we used Chernoff s inequality, the fact that τv,. is a subordinator with Laplace exponent ψ 1 see Theorem VII.1 in [8] and the fact that ψ 1 1 > V V because ψ V = is an easy consequence of Lemma applied with a = 1 ηh, b = 1 η/2h, and applied with α = 1 η/2, ω = 1. Then, we have obviously, τv,h e V u du e h τv,h, so τv,h P e V u du > e 1+ηh P τv,h > e ηh e h, for h large enough according to Lemma This yields

199 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Estimates on ˆV The aim of this subsection is to get, for ˆV, estimates similar to those that we proved for V in the last subsection. We start with a generalization of Lemma Lemma For all z and < a < b with b > z, we have, P inf [τˆv z,[b,+ [,+ [ ˆV z < a e κb a /1 e κb. Démonstration. Let T := τˆv z,[b,+ [. T is a stopping time so, if U is a random variable having the same law as ˆV z T, then ˆV z T +. is equal in law to ˆV U, that is, the Markov process that conditionally on {U = u} has law ˆV u. We thus have + P ˆV z < a = P ˆV U < a = P ˆV u < a L U du,4.5.1 inf [T,+ [ inf [,+ [ b inf [,+ [ because almost surely, U b. Now, since ˆV u is only ˆV u conditioned in the usual sens to remain positive, we have for any u b, P inf [,+ [ ˆV u < a = P < inf ˆV u < a /P [,+ [ P inf ˆV < a u /P inf [,+ [ P ˆV < a b /P inf [,+ [ inf [,+ [ [,+ [ inf [,+ [ ˆV u > ˆV > u ˆV > b, where we used the fact that a u a b for the numerator and the fact that u b for the denominator. Since ˆV is the dual of V, the above estimate can be re-written P ˆV u < a P /P = e κb a /1 e κb. inf [,+ [ sup V > b a [,+ [ Putting into we get the result. sup V < b [,+ [ The proof of Lemma relies on the fact that the hitting times of V are stochastically smaller than the hitting times of V, for which we have precise estimates. Here, the procedure to link the hitting times of ˆV and ˆV is different. This is what we do now. Let us fix x >. Let m x be the point where the process ˆV x reaches its infimum, m x := sup{s, ˆV x s ˆV xs = inf [,+ [ ˆV x }. Note that from the absence of negative jumps, the infimum is always reached at least at m x so ˆV y m x = inf [,+ [ ˆV x. The next lemma is contained in Theorem 24 of [31]. 188

200 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Lemma Assume V has unbounded variations, then ˆV xm x +s ˆV xm L x, s = ˆV. We can thus obtain ˆV from ˆV x, and ˆV x is only ˆV x conditioned in the usual sens to remain positive. This allows us to link ˆV and ˆV, so we are now able to prove our estimate : Lemma Assume V has unbounded variations. There are three positive constants c,c 1 and c 2 such that y > 1, r >, P τˆv,[y,+ [ > r c e κy + e c 1y c 2 r. Démonstration. Let us fix r >, y > 1 and choose x ], 1[ for example x := 1/2. According to Lemma we have P τˆv,[y,+ [ > r = P τˆv xm x +. ˆV xm x,[y,+ [ > r P τˆv xm x +.,[y +x,+ [ > r, because ˆV xm x x. Now, if ˆV x never reaches [,x] after the instant T := τˆv x,[y +x,+ [, then the minimum m x is reached before T so τˆv xm x +.,[y +x,+ [ = τˆv x,[y +x,+ [ m x = T m x T. We deduce that P τˆv,[y,+ [ > r PT > r+p inf [T,+ [ ˆV x < x We now bound the two terms of the right hand side. For the first term, since ˆV x is only ˆV x conditioned in the usual sens to remain positive, we have PT > r = P τˆv x,[y +x,+ [ > r, inf [,+ [ P τˆv x,[y +x,+ [ > r /P ˆV x > /P ˆV x > inf [,+ [ = PτV,], y] > r/1 e κx. Combining with Lemma we get inf [,+ [ ˆV x > PT > r c e c 1y c 2 r, where c 1 and c 2 are the constants in the lemma and c := 1/1 e κx. We now turn to the second term of According to Lemma applied with z = a = x and b = x+y we get P inf [T,+ [ ˆV x < x e κy /1 e κx+y c e κy Now, combining and with we get the result. 189

201 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V We can now use our estimate on the hitting times to bound an exponential functional of ˆV. Lemma Assume V has unbounded variations. There is a positive constant c such that for all h large enough, we have τˆv,[h,+ [ P eˆv u du e 1+ηh 1 e ch Démonstration. We choose c ],κ[. We have so P τˆv,[h,+ [ τˆv,[h,+ [ eˆv u du e h τˆv,[h,+ [, eˆv u du > e 1+ηh P τˆv,[h,+ [ > e ηh e ch, for any choice of c ],κ[ and h large enough, according to Lemma This yields Note that we do not prove an analogue of for ˆV, even if it would have been needed to repeat readily the arguments of [4] in our context. This is because the existence of possibly large negative jumps for V do not allow such an estimate to hold in general. Because of this, we have to take some precautions, and in particular, to prove some extra technical estimates in Subsection Estimates on the first ascend of h from the minimum In order to bound the local time and the time spent by the diffusion between two valleys, we have to study the expectation of some functionals of V involving the first ascend of h from the minimum. This subsection uses the notations and estimates of Subsection Lemma There is a positive constant C such that for h large enough, E[τ h] Ce κh. Démonstration. We have τ h = m h+τ h m h, and using Lemma we get E[τ h] = E [ S h, T h ] +E [ τv,h ] For the first term, note that E[S h, T h ] = E[S h, 1] E[T h ] since S h, is a subordinator and is independent fromt h. Also, recall thats h, 1 S1 wheres1 19

202 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V has finite expectation according to Lemma Then, T h follows an exponential distribution with parameter NF h,+ ce κh, according to Lemma , where c is the positive constant obtained in the lemma. For h large enough we thus get E [ S h, T h ] 2E[S1]/ce κh For the second term, we use Lemma 2.6 of [72] and the fact that, since V is a spectrally negative Lévy process drifting to +, τv,. is a subordinator having finite expectation see [8], Section VII.1. We obtain E [ τv,h ] E [ τv,h ] = he [ τv,1 ] Combining , and we get the result. Lemma There is a positive constant C such that [ ] τ h h >, E e Vu du Ce 1 κh. Démonstration. Since V is spectrally negative, single points are not essentially polar for V so, according to Theorem V.1 of [8], there is L V, a local time that satisfies the density of occupations formula for V : almost surely, for all mesurable function f and t > we have t fvsds = fxl R Vt,xdx. Therefore, as in the proof of the majoration of β h in Lemma 3.6 of [3] we get : [ ] τ h h E e Vu du e x E[L V +,x]dx, and, according to the strong Markov property, where E[L V +,x] = PτV,x < + E[L V +,], PτV,x < + P We thus get [ ] τ h E e Vu du sup V x [,+ [ = 1 x +e κx 1 x>. 1 1 κ E[L V+,] e 1 κh κ. Then, considering the Poisson point process of excursions away from associated with the local timel V.,, we have thatl V +, is only the time when occurs the infinite excursion of V, and this follows an exponential distribution with parameter η V ξ, ζξ = + >. As a consequence, E[L V +,] < + and the result follows. 191

203 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Estimates on the valleys We now use the previous results to prove some estimates on the standard valleys. Lemma For all h large enough we have mj j 1, P e V ju du e e1 2δκh 1 e κδh/4. L j 1 Démonstration. Thanks to Remark we only need to prove the result for j = 1. Recall the proof of Lemma in which we set a = h, b = e 1 δκh and η = δ/2. For h 1 let us define : ph := PminτV V,h,τ V,], δh/2] 1 = PT 1 1 p1 >. Since, from the Markov property, the sequence T i T i 1 i 1 is iid, the probability that T i T i 1 < 1 for all i e 1 2δh is 1 ph e1 2δκh 1 p1 e1 2δκh e h, for h large enough. Then, recall that we proved that, with probability greater than 1 b/ηa + 1e κ1 ηa which is more than 1 e κδh/3, at least when h is large enough, the index i such that T i = τv V,h is greater than 2e 1 δκh /δh. It means that after the e 1 2δκh th one,v will still make2e 1 δκh /δh e 1 2δκh e 1 2δκh descents before τv V,h and so, before m 1. In [, m 1 ], there will therefore be an interval larger than 1 on which V 1 is greater than e 1 2δκh. We thus have m1 L e V 1u du e e1 2δκh 1 with probability greater than 1 e h e κδh/3, for h large enough, and the result follows. Lemma Fix < η < α < 1. For h large enough we have j 1, P L j < τ j αh, inf [ L j, τ j αh] V j > α ηh 1 e κηh/3. It would seem convenient to use the time-reverse property to prove this lemma. However this is not possible here so we have to show that V cannot get too close to its future minimum before time τ j αh. Démonstration. Thanks to Remark we only need to prove the result for j = 1. Recall that τ 1 h is the first time after L 1 when V V reaches h and m 1 is the associated minimum, so P L 1 τ 1 αh PτV V,h < τv,], h] 3e κh/2,

204 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V where the last inequality comes from Lemma applied with a = b = h, η = 1/2. When L 1 < τ 1 αh, let u 1 be the unique point where V 1 reaches its minimum on [ L 1, τ 1 αh]. On the event { L 1 < τ 1 αh, inf [ L 1, τ 1 αh]v1 α ηh} we have V m 1 Vu 1 V m 1 +α ηh V τ 1 αh ηh It implies that V V reaches ηh between u 1 and τ 1 αh, then V descends lower than its minimum level Vu 1 because V m 1 [Vu 1 α ηh,vu 1 [ and then, V V reaches h before V reaches ],Vu 1 α ηh[. We thus consider the times when V V reaches ηh and separate them by the times when V gets lower than its previous minimum. We introduce the sequences of stopping times S j j and T j j 1 where S := L 1 and T j := inf{t S j 1, V Vt = ηh}, S j := inf{t T j, Vt < VT j }. Note that for all j 1, VT j = VT j ηh and VS j = VS j. Combining with the Markov property, this implies that the sequences of truncated processes Vt + S j 1 VS j 1, t T j S j 1 j 1 and Vt + T j VT j, t S j T j j 1 are both iid, and the two sequences are independent. We see that τ 1 + ηh = T J where we define J to be the first index j 1 for which inf{t T j, V Vt = h} < S j and m 1 is the minimum of V before T J V m 1 = VT J = VT J ηh. Moreover, we just saw that on the event { L 1 < τ 1 αh, inf [ L 1, τ 1 αh]v1 α ηh}, J 2 and u 1 is the minimum of V before T K for some random K < J Vu 1 = VT K = VT K ηh. Using , we get, VT J 1 VT K = Vu 1 V m 1 +α ηh = VT J +α ηh = VT J +α 2ηh. From the definition of S j we have VS J 1 VT J 1 so we get { } L 1 < τ 1 αh, inf V 1 α ηh [ L 1, τ 1 αh] {VS J 1 VT J +α 2ηh} For any k 1, the event {VS k 1 VT k +α 2ηh} only depends on Vt+ S k 1 VS k 1, t T k S k 1 whereas the event {J = k} only depends on the sequence Vt+T j VT j, t S j T j j 1. Partitioning on the possible values for J and using the fact that the sequences Vt+S j 1 VS j 1, t T j S j 1 j 1 and Vt+T j VT j, t S j T j j 1 are independent and both iid, we get PVS J 1 VT J +α 2ηh = PVS VT 1 +α 2ηh = Pτ V V,ηh < τ V,], α ηh[ 1+α/ηe κηh/2 /2, 193

205 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V where the inequality comes from Lemma applied with a = ηh, b = α ηh, η = 1/2. Combining with and we get the result for h large enough. Lemma For all h large enough we have j 1, P τ j h L j 1 > e 1+δκh e δκh/2. Démonstration. Here again, we only need to prove the result for j = 1. We have τ 1 h L τ 1 h L = τ 1 h L 1 + L 1 L For the first term, according to the Markov property at L 1 and the definition of τ 1 h, we see that τ 1 h L 1 has the same law as τ h. We thus have P τ 1 h L 1 > e 1+δκh /2 = Pτ h > e 1+δκh /2 and combining with Markov s inequality and Lemma , we get for h large enough, P τ 1 h L 1 > e 1+δκh /2 2Ce δκh, where C is the constant in Lemma For the second term, according to the definition of L 1 and Lemma applied with y = e 1 δκh and r = e 1+δκh /2 we have, for h large enough : P L 1 L > e1+δκh = P τ V,], e 1 δκh ] > e1+δκh e h The combination of , and yields the result Estimates on the diffusion in potential V We now give an upper bound for the time spent by the diffusion and the local time in the negative half-line. Restricted to the drifted brownian case, our result is a little stronger than Lemma 3.5 of [3]. It is the only estimate that we need for both cases < κ < 1 and κ > 1. Lemma Recall the definition of H in Subsection There is a positive constant C such that for r large enough, PH + > r Cr κ/2+κ and P inf L X+,. > r 3r κ/2+κ. ],] Démonstration. From the definition of the local time and formula 4.2.1, we have H + = = A V + L X +,xdx = e Vx L B τb,a V +,A V xdx e Vx L B τb,1,a V x/a V + dx 194

206 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V where B := BA V + 2./A V +. By scale invariance, we see that conditionally to V, B is a brownian motion so H + = L A V + e Vx L B τb,1,a V x/a V + dx supl B τb,1,y A V + e Vx dx y = supl B τb,1,y A V + y + eṽx dx, where Ṽx := V x. By time reversing, the process Ṽx, x has the same law as Vx, x. As a consequence, the right hand side of features three factors, the last two of them being equal in law to + e Vu du. We thus have + PH + > r P supl B τb,1,y > r +2P κ/2+κ e Vu du > r 1/2+κ, y and combining with inequality 7.13 of [4] for the first term, and the second assertion of Lemma for the second term, we get the result for r large enough. For the assertion about the local time, we only replace the integrals on ],] by a supremum as well as the integral on [,+ [ for Ṽ. Since supṽ follows an exponential distribution with parameter κ, the result follows. The next lemma provides a useful estimate to bound the time spent by the diffusion between two standard valleys. It generalizes a part of Lemma 3.6 of [3]. Lemma Recall the definition of H + in Subsection There exists a constant C > such that for h large enough, E[H + τ h] Ce h Démonstration. The beginning of the proof is similar to the beginning of the proof of Lemma 3.6 in [3]. In fact 3.37 of [3] is still true in our setting, with V instead of the drifted brownian motion : [ ] τ h E[H + τ h] 2E[τ h] E e Vu du. Combining this with Lemmas and , we get the result. A fundamental point to have the renewal structure for the contributions to time and to local time is the fact that the diffusion never goes back to a previous valley. Let us define X Li := XH L i

207 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V which is, according to the Markov property, a diffusion in the environment V starting from L i. We also denote byh X Li r the hitting time ofr byx Li. The following lemma proves that the diffusion does not go back : Lemma There is a positive constant c such that for h large enough, i 1, P H X Li τ i h < H X Li + e ch. Démonstration. Let us fixi 1. At fixed environmentv,p V H X Li L i < H X Li + equals = + L i e Vu du + τ i h evu du = ev Li V τih + e Vu+ L i V L i du + e Vu+ τ ih V τ i h du := ev Li V τih I 1 /I Since L i and τ i h are stopping times for V, both I 1 and I 2 have the same law as + e Vu du even though there are not independent. Then, P I 1 /I 2 > e h/4 P I 1 > e h/8 +P I 2 < e h/8 2Ce κh/8 +ce h/16, where the last inequality and the constants come from the two assertions of Lemma It holds for h large enough. From the definition of Li we have V L i V τ i h h/2 so the last term of is less than e h/4 with probability greater than 1 2Ce κh/8 + ce h/16 and is bounded by 1 otherwise. Integrating with respect to V we thus get P H X Li τ i h < H X Li + e h/4 +2Ce κh/8 +ce h/16, and the result follows. Recall the definitions of X mj := X.+ m j and H X mj r. The next lemma proves that the standard valleys are left from the right. Lemma For h large enough, i 1, P H X mi L i 1 < H X mi L i e κδh/6. This is where appears a difference with the brownian case. In Lemma 3.2 of [3], they have a similar result with, instead of L i 1, an other random time denoted by L i. In the context of a Lévy environment, the existence of jumps may allow one of the m i L i to be quite small with a non negligible probability, which would allow some standard valleys to be left from L i. In fact, we would need an analogue of for ˆV to have the same result as in Lemma 3.2 of [3]. In order to still 196

208 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V have the standard valleys left from the right, we have to change our definition of "leave from the right" replacing L i by L i 1. A consequence of this is that the study of the descending parts of the standard h-valleys is more technical since we have to consider a bigger part than in [3], and we do not know its law precisely and requires Subsection In particular, our proof that the standard valleys are left from the right requires the technical Lemma , but the idea of the result has no qualitative difference with the drifted brownian case. Démonstration. of Lemma Let us fix i 1. At fixed environment V, P V H X mi L i 1 < H X mi L i equals Li m i eṽ i u du Li L i 1 eṽ i u du = 1+ m i eṽ L i u i 1 du/ Li max We first provide an upper bound for Li m i eṽ i u du = 1, 1 eṽ i u m i du Li mi eṽ iu du/ eṽ iu du m i L i 1 τi h Li eṽ iu du+ eṽ iu du. m i τ i h According to proposition 4.3.6, the terms of the right hand side have respectively the same law as τv,h e V u du and e h τv,], h/2] e Vu du e h + e Vu du. We thus have Li τv,h P eṽ iu du > e 1+δh P e V u du > e 1+δh /2 m i + +P e Vu du > e δh / For h large enough, the first term of the right hand side is bounded by e h because of and the second is bounded by 2C 2 κ e κδh because of the second assertion of Lemma As a consequence, for h large enough, Li P eṽ iu du > e 1+δh e κδh/ m i A lower bound for m i eṽ i L u i 1 du is given by Lemma Combining with 4.5.3, we get that for h large enough Li mi P eṽ iu du e h e κδh/5, L i 1 m i eṽ i u du/ so integrating with respect to V we get the result for h large enough. 197

209 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Proof of some facts We now prove Facts 4.4.3, and Even if they are taken from [3] and [4], their proof use estimates that are only true for the drifted brownian potential. This is why we have to adapt them to our context and to use our estimates instead of the original ones. We shall refer to [3] and [4] for the proofs and only precise what are the differences between their proofs and ours. Démonstration. of Fact This is Lemma 3.7 of [3]. For the proof, here are the differences : W κ is, off course, replaced here by V. The standard valleys are to be considered as the ones defined in our sense. Also, L i of [3] has to be replaced here by τ i h t so Lemma 3.3 of [3] can be replaced here by Lemma and τ i h t is like we defined it in Subsection For the step 1, the event E2 3.7 is replaced here by nt i=1 { τ i h t = τ i h t } note that this includes { τ 1h t = τ 1 h t } and the negligibility of the complementary is proved by Lemma the constant in the exponential might be different from the one in [3] but it does not matter. For the step 2, the evente1 3.3 is replaced here by nt i=1 {H τ X Li i h t > H X Li m i+1 } and the negligibility of the complementary is proved by Lemma Then, estimate 3.34 of [3] which is, in the proof of Lemma 3.7 there, referred to as "Lemma 3.6" has to be replaced here by Lemma For H + m 1, it is obviously less than H + τ 1 h which equals H + τ 1h on { τ 1h t = τ 1 h t } so, here, the expectation E[H + m 1 1 E 3.3] is bounded by E[H 1 + τ h], just as the other terms, and in particular, there is no need, here, to bother with the event E3 3.7 nor to specify we are on the event V t where the classical and standard valleys coincide. Finally, to bound H m 1, we see from t/logh t > e ht which is true at least for large t and Lemma , that, for t large enough, PH m 1 > t/logh t P H m 1 > e ht P H + > e ht Ce κht/2+κ. Démonstration. of Fact The first point is Lemma 3.3 of [4]. For the proof, here are the differences : W κ is replaced here by V and we use the notations m h t and τ h t instead of m 1h t and τ 1h t. We change a little the definition of bt, that is, bt := 6Rφte κht /1 e κ where R := E[VτV,], 1]]. Note that R is finite according to Lemma and the hypothesis assumed on V. For PA c where A has the same definition as in the original proof of [4] with, off course, V instead of W κ, we get a similar upper bound for t large enough, thanks to the second assertion of Lemma Because of the negative jumps, we have, before bounding PA c 2, to 198

210 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V give a different definition for the stopping times f i and to define a new event A 2. First, f := and i 1, f i := inf{x f i 1, Vx Vf i 1 1}. Let It := max{i N, f i 1 m h t }, we now define A 2 by A 2 := {It bt/2r}. We define, for i 1, E i := {sup [fi 1,f i ]V V h t }. Since inf [,fi 1 ]V = Vf i 1 we have { } E i = sup V f i 1 V f i 1 h t, [,f i f i 1 ] and because of the Markov property applied at f i 1, the events E i are independent and have the same probability that we denote by p t. Also, It is the smallest i 1 for which E i is realized, so It follows a geometric distribution with parameter p t. Then, p t := PE 1 PV leaves [ 1,h t ] from above 1 e κ e κht, where, for the last inequality, we applied Lemma with a = 1 and b = h t. We deduce PA c 2 = 1 p t bt/2r = e bt/2r log1 pt e bt/2r log1 1 e κ e κh t e 3φt, where the last inequality holds for t large enough. Then, { } bt/2r V f bt/2r < bt Vf i Vf i 1 +R < bt/2. i=1 The random variables Vf i Vf i 1 +R are iid having the same law as VτV,], 1]+R, in particular they have mean and belong to L P because of Lemma We can thus apply successively Markov s and Von Barh-Esseen s inequalities : P V p bt/2r 2 p f bt/2r < bt E Vf i Vf i 1 +R bt p 2 2 bt i=1 bt 2R E[ VτV,], 1]+R p ] e p 1κht, where the last inequality comes from the definition of bt and holds for t large enough. Also, A 2 { V { } } f bt/2r bt inf V bt = A 2, [,τ h t] so combining with and , the upper bound PA c 2 e 2φt follows for large t. 199

211 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Thanks to our definition of f i, we still have e Vx e Vf i+1 for f i x < f i+1 so 3.8 of [4] is true in our context. Our upper bound for E[ τ h e Vu du] denoted β h in [4] is given by Lemma For PA c 4, we use Lemma and get PA c 4 P τv,h t < 1 P τv,h t τv,h t /2 < 1 e cht/2, where the last inequality holds for some positive constant c and t large enough, according to applied with ω = 1 and α = 1/2. We now justify the second point of Fact We treat separately the case j =. Since H L = H =, the term L X H L j,x can be omitted. Psup R L X H m 1,x > r t is less than P inf ],] L X+,. > r t +P sup L X Hτ h t,. > r t +P m 1 m h= m 1. [,m h t] Putting r t := te κ1+3δ 1φt and applying Lemmas and together with the first point, we get for some constant C 2 and t large enough, P supl X H m 1,x > te κ1+3δ 1φt C 2 R n t eκδφt For j 1, the proof has the same idea as the one of Lemma 3.2 in [4]. Thanks to Lemmas and we have, for some constant c and t large enough, nt 1 { } P HXj m j+1 < H Xj τ j h, τ jh = τ j h, m j = m j 1 n t e cht. j= On this event we have that, for i {1,...,n t 1}, the chain of inequalities 3.11 of [4] is still true when L i, m i and τ i h are defined in our sense in Subsection and L i of [4] is replaced by τ i h. Moreover, the last term of this chain of inequality has the same law as sup x [,m h t]l X Hτ h t,x, so combining and the first point we get P nt 1 j=1 { sup R } L X H m j+1,x L X H L j,x te κ1+3δ 1φt 1 C 2n t 1, n t e κδφt for some constant C 2 and t large enough. The combination of and is the sought result. The third point of Fact is Lemma 3.4 of [4]. For the proof, here are the differences : 2

212 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V W κ is replaced here by V and the standard valleys are to be considered as the ones defined in our sense in Subsection For convenience, we have r t = φt 2 instead of r t = C φt in [4]. In this proof, we systematically replace τ j h+ t of [4] by L j 1 defined in our sens. For A 2, we have that PA c 2 is less than τj h t Lj P eṽ jy dy > e ht+2φt/κ +P eṽ jy dy > e ht+2φt/κ m j τ j h t τv,h t h τv,], t =P e V y dy > e +P ht+2φt/κ 2 ] e ht e Vy dy > e ht+2φt/κ, where we used Proposition for the laws of P τv,h t > e 2φt/κ +P e c 1h t c 2 e 2φt/κ +2Ce 2φt, + P j 2 and P j 3, e Vy dy > e 2φt/κ, for t large enough, according to Lemma and the second assertion of Lemma the constants are the ones defined in there. Fortlarge enough we getpa c 2 ce 2φt, where c is some positive constant. The event A 3 of the original proof is not needed here thanks to our definition of D j, and we redefine { } A 4 := inf Ṽ j φt 2 /2 [ τ + j φt2, τ j h t] According to Proposition and Lemma applied with a = φt 2 /2,b = φt 2, we have PA c 4 e 2φt for t large enough. We also redefine { } { } A 5 := inf Ṽ j φt 2 /2 [ τ j ht, τ j φt2 ] and A 6 :=. inf V j > h t /4 [ L j 1, τ j ht/2] Combining Proposition and Lemma applied withz =,a = φt 2 /2,b = φt 2, we get for t large enough PA c 5 2e δκht/3 +2e κφt2 /2 /1 e κφt2 e 2φt,. where the last inequality holds for t large enough. According to the definition of L j and Lemma applied with α = 1/2,η = 1/4, we have PA c 6 e κht/12 for large t. On 6 i=4a i, we have V j x φt 2 /2, x [ L j 1, L j ] D c j so the conclusion follows as in the proof of Lemma 3.4 in [4] but here we do not need to intersect with V t of [4]. 21

213 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V Démonstration. of Fact This is Lemma 5.3 of [4]. As in there, let σ X a,b := inf{s, L X s,b > a} be the inverse of the local time of X at b. Here are the differences with the proof of Lemma 5.3 of [4] : W κ is replaced here byv and the standard valleys are to be considered as the ones defined in our sense in Subsection In this proof, we systematically replace L 1 of [4] by L =, in particular, the domain of integration in the integral I, which represents the inverse of the local time, is [ L, L 1 ]. We get the analogue of 5.22 of [4] using Proposition Our lower bound for PA 1 τ 1 h t /2 e ht1+ǫ/2 and PA 1 τ 1 h t /2 e ht1+ǫ/2 come from the proof of Lemma Recall also that, since the domain of integration in I is [ L, L 1 ] we have in our case τ I 2 e ǫht/2 1 ht/2 L1 e V 1x dx+ e V 1x dx, L τ 1 h t/2 with probability larger than 1 2e ǫht/2 note that we took ǫ/2 instead of ǫ and h t instead of logt. We bound the two integrals using Lemmas and and get, for t large enough : P I 2 2e ǫht/2 1 e c 2h t, wherec 2 is a positive constant. Our lower bound forr t 1 comes from Combing with we get for some positive constant c 3 and t large enough, P I 2 e ǫht/4 R t 1 1 e c 3 h t so in place of 5.24 of [4] we have P I γtr1 t e ǫht/5 γtr1,h t X m1 L 1 > σγt, m 1 t1 x,h X m1 L 1 < H X m1 L e c 4h t for all t large enough, and where c 4 is a positive constant. From our definition of e 1 in Subsection we still have σγt, m 1 > H L 1 γt > A 1 L 1 e 1 γtr 1 > A 1 L 1 e 1 R of [4] has to be replaced here by Lemma and, as we already mentioned, 5.22 by Proposition In place of 5.26 of [4] we thus have σγt, m 1 > H L 1 γtr 1 > 1 e c 5h t H L 1, for some positive constant c 5, except on a event which probability is less than e c 6h t for some positive constantc 6. The fact that the constant are modified with respect to the ones in the original proof has no importance so we omit to mention it for the rest of the proof. 22

214 4.5. SOME ESTIMATES ON V, V, ˆV AND THE DIFFUSION IN V For f γ x, note that we use our definition of D 1. According to this definition we have D 1 [τ 1 h t /2,τ 1 h t /2] for t large enough, so we can use the lower bounds for PA 1 τ 1 h t /2 e ht1+ǫ/2 and PA 1 τ 1 h t /2 e ht1+ǫ/2 from the proof of Lemma to bound ã on D of [4] has, off course, to be replaced here by Finally, 3.2 of [4] has to be replaced here by Lemma

215

216 Chapitre 5 Almost sure behavior for the local time of a diffusion in a spectrally negative Lévy environment This work has been the object of an article [73] that will be shortly submitted. 5.1 Introduction We study the almost sure asymptotic behavior of the supremum of the local time for a transient diffusion in a spectrally negative Lévy environment. Let V be a twosided spectrally negative Lévy process which is not the opposite of a subordinator, drifts to at +, and such that V =. We denote its Laplace exponent by Ψ V : t,λ, E [ e λvt] = e tψ V λ. It is well-known, for such V, that Ψ V admits a non trivial zero that we denote here by κ, κ := inf{λ >, Ψ V λ = } >. We are here interested in a diffusion in this potentialv.such a diffusionxt, t is defined informally by X = and dxt = dβt 1 2 V Xtdt, where β is a Brownian motion independent from V. Rigorously, X is defined by its conditional generator given V, 1 d 2 evx e Vx d. dx dx 25

217 5.1. INTRODUCTION The fact that V drifts to puts us in the case where the diffusion X is a.s. transient to the right. The asymptotic behavior of this diffusion has been studied by Singh [66], he distinguishes three main possible behaviors depending on < κ < 1, κ = 1 or κ > 1 the case κ > 1 being also divided into three subcases. We denote by L X t,x,t,x R the version of the local time that is continus in time and càd-làg in space, and we define the supremum of the local time until instant t as L Xt = supl X t,x. x R Here, we study the almost sure asymptotic behavior ofl X t. When the environment is a brownian motion with no drift, Shi [61] has studied this behavior and shown that P-a.s. lim sup t + L X t tlogloglogt 1 32, where P is the so-called annealed probability measure which definition is recalled in Subsection In the same case, Andreoletti and Diel [5] have proved more recently the convergence in distribution of L X t/t. Diel [29] has then continued the study by giving a finite upper bound for the limsup in and doing the same study for the liminf : P-a.s. lim sup t + L X t tlogloglogt e2 2 and j2 64 liminf t + where j is the smallest positive root of the Bessel function J. L X t t/logloglogt e2 π 2 4, In the case of a drifted-brownian environment, the almost sure behavior of L X t has been studied by Devulder [28] using annealed methods. He totally characterizes the almost sure behavior when κ > 1. In this case, for any positive non decreasing function a we have and + n=1 1 nan P-a.s. lim inf t + When κ = 1 he obtains { < + = + limsup t + { L X t tat = P-a.s., /κ + L X t t/loglogt 1/κ = 4κ2 κ 1/8 1/κ P-a.s. lim inf t + L X t t/logtloglogt 1/2. When < κ < 1, his method fails and only provides partial results for the almost sure behavior of the local time. More precisely he proves that the renormalisation for the limsup is greater than t : P-a.s. lim sup t + L X t t 26 = +, 5.1.4

218 5.1. INTRODUCTION and that the renormalization for the liminf is at most t/loglogt and greater than t/logt 1/κ loglogt 2/κ+ǫ for any ǫ > : P-a.s. lim inf t + ǫ >, P-a.s. liminf t + where Cκ is a positive non explicit constant. L X t Cκ, t/loglogt L X t = +, t/logt 1/κ loglogt2/κ+ǫ For the discrete transient Random Walk in Random Enrvironment RWRE, the almost sure behavior of the supremum of the local time has been studied by Gantert and Shi [41], they obtain the behavior of the limsup in the two subcases < κ 1 and κ > 1. For the diffusion in the general potential V, the convergence in distribution of L X t/t has been studied by the author in [74] using two different methods : when < κ < 1 a path decomposition of the environment that provides an interesting renewal structure to study the diffusion is used, this method is inspired from Andreoletti and al. [4], [3] which are themselves inspired from the work of Enriquez and al. [34] in the discrete case. When κ > 1 an equality in law between the local time and a generalized Ornstein-Uhlenbeck process that was introduced in [66] is used. The main contribution of this paper is in the case < κ < 1, pushing further the ideas of [74], we make a deep study of the renewal structure of the diffusion in order to establish the almost sure asymptotic behavior of L X t. We characterize this behavior when < κ < 1 and κ > 1. In particular, the restriction of our results to the case of a drifted brownian potential with < κ < 1 improves the results 5.1.4, and of [28] by giving the exact renormalizations and even the exact value of the constant for the limsup, for the liminf we get the exact renormalizations and an explicit upper bound of the constant Main results We start with the case < κ < 1. In that case, the limit distribution of L X t/t given in [74] depends on exponential functionals of V and its dual conditioned to stay positive. They are defined as follow : IV := + e V t dt and IˆV := + e ˆV t dt, where ˆV, the dual of V, is equal in law to V. In Subsection it is precised how V and ˆV are defined rigorously. These functionals are studied by the author in [72] where it is proved in Theorems 1.1 and 1.13 that they are indeed finite and well-defined. Let G 1 and G 2 be two independent random variables with G 1 L = IV and G 2 L = IˆV. We define R := G 1 +G 2. 27

219 5.1. INTRODUCTION To study the limsup of L X t, we link the almost sur asymptotic behavior of L X t with the left tail of IV or of R according to the case. Before stating our results, let us recall what is known about the left tail of IV. In [72], the left tail of IV is linked to the asymptotic behavior of Ψ V. This asymptotic behavior is usually quantified thanks to two real numbers σ and β : { } σ := sup α, lim λ + λ α Ψ V λ =, { } β := inf α, lim λ + λ α Ψ V λ =. If Ψ V has α-regular variation for α [1,2] for example if V is a drifted α-stable Lévy process with no positive jumps, we have σ = β = α. Note that when Q, the brownian component of V, is positive, then Ψ V has 2-regular variation, and when Q =, 1 σ β 2. The asymptotic behavior of PIV x as x goes to is given by the following theorem from [72] : Theorem [Véchambre, [72]] There is a positive constant K depending on V such that for x small enough More precisely we have P IV x e K /x l < 1 β 1, P IV x e 1/xl If σ > 1, l > 1 σ 1, P IV x e 1/xl If there are two positive constants c < C and α ]1,2] such that cλ α Ψ V λ Cλ α for λ large enough, then for any δ > 1 we have, when x is small enough, exp δα α α 1 P IV x exp α 1, cx 1 α 1 δcx 1 α 1 If there is a positive constant C and α ]1,2] such that Ψ V λ λ + Cλ α, then log P IV x Cx 1 x α 1 α is Remark 1.6 of [72], and are a reformulation of Theorem 1.4 of [72], and comes from Theorem 1.2 of [72] is Theorem 1.5 of [72]. 28

220 5.1. INTRODUCTION We can now state our results for the limsup : Theorem Assume that < κ < 1,V has unbounded variation,v1 L p for some p > 1 and V possesses negative jumps. If there is γ > 1 and C > such that for x small enough P IV x exp C, x 1 γ 1 then we have P-a.s. lim sup t + L X t tloglogt γ 1 C1 γ If there is γ > 1 and C > such that for x small enough P IV x exp C, x 1 γ 1 then we have P-a.s. lim sup t + L X t tloglogt γ 1 C1 γ Assume now that Vt = W κ t := Wt κ t with < κ < 1, i.e. V is the 2 κ-drifted brownian motion, then the above implications and are still true with IV replaced by R. In the above theorem we had to distinguish the case where V possesses negative jumps and the case where V is a drifted brownian motion. First, note that this is a true alternative : since V is spectrally negative, the case where V do not possess negative jumps is the case where V do not possess jumps at all and is therefore a drifted brownian motion. In this case we assumed for convenience that the gaussian component ofv is normalized to1. The difference between the two cases in the above proposition comes from the absence or presence of symmetry for the environment. When V possesses negative jumps, ˆV possesses positives jumps that might repulse it very fast from, this is why the left tail of IV is thiner than the left tail of IˆV, as we can see comparing Theorems 1.4 and 1.14 of [72]. As a consequence, only the left tail of IV is relevant in the left tail of R. When V is a drifted brownian motion, a symmetry appears : ˆV and V are equal in law, R is then the sum of two independent random variables having the same law as IV and none of them can be neglected. Remark It has to be noted that the limsup above is P-almost surely equal to a constant belonging to [, + ] and that the inequalities and are inequalities relative to this value that the lim sup equals P-almost surely. The same will be true in all the results below : all the limsup and liminf considered are P-almost surely equal to a constant. This fact is justified in Subsection

221 5.1. INTRODUCTION Putting together Theorem and what is known for the left tail of IV Theorem 5.1.1, we can state precise results for the limsup : Theorem If < κ < 1, V has unbounded variation and V1 L p for some p > 1, then we have and β > β, P-a.s. limsup t + If σ > 1, σ ]1,σ[, P-a.s. limsup t + L X t tloglogt β 1 L X t tloglogt σ 1 =, = If we make further hypothesis on the regularity of the variation of Ψ V we can give the exact order of L X t : Theorem Assume that < κ < 1,V has unbounded variation,v1 L p for some p > 1 and V possesses negative jumps. If there are two positive constants c < C and α ]1,2] such that cλ α Ψ V λ Cλ α for λ large enough, then we have P-a.s. c α α limsup t + L X t tloglogt α 1 C α 1 α 1. If, more precisely, there is a positive constant C and α ]1,2] such that Ψ V λ Cλ α for large λ, then we have P-a.s. lim sup t + L X t tloglogt α 1 = C α 1 α 1. Assume now that V = W κ, the κ-drifted brownian motion, with < κ < 1, then we have L X P-a.s. lim sup t t + tloglogt = 1 8. Remark According to the combination of Theorem and we see that, if < κ < 1, V has unbounded variation and V1 L p for some p > 1, then we always have P-a.s. lim sup t + L X t tloglogt < +. In other words, tloglogt is the maximal possible renormalisation for the lim sup. Remark As it was noticed by Shi [61] and Diel [29] for the recurrent case, we also notice a difference between the renormalization of the local time for the discrete transient RWRE with zero speed given by Gantert and Shi [41] and the renormalization of the local time that we give for the transient diffusion with zero speed. This difference can be explained as in the recurrent case : the valleys can potentially be much steeper in the continus case with a potential having unbounded variation than in the discrete case, so the local maxima of the local time can potentially be higher in the first case. 21

222 5.1. INTRODUCTION We see in the above two theorems that the renormalization of L X t for the limsup depends directly on the asymptotic behavior of Ψ V. In particular, Theorem says that for a driftedα-stable environment with no positive jumps andα > 1, the renormalization of L X t is tloglogtα 1. We see that we have much more possible behaviors with general spectrally negative Lévy environments, than with drifted brownian environments. Even if, for technical reasons, the above theorems do not apply when the environment V has bounded variation, we can conjecture that the behavior of L X t remains linked in the same way to the left tail of IV which is given by Remark 1.7 of [72]. This implies Conjecture When V has bounded variation, we have P-a.s. < limsup t + L X t t < +. If this conjecture is true, we would have, when the environment has bounded variation, the same renormalization as in the discrete transient case given by Theorem 1.1 of [41]. This would not be surprising since the discrete case gives rise to potentials of bounded variation. Moreover, if V has bounded variation then it is known to be the difference of a deterministic positive drift and a subordinator. The valleys can then not be steeper than the deterministic drift so, according to Remark 5.1.7, the expected renormalization of L X t has to be the same as in the discrete case. For theliminf, there is only one possible renormalization. Our result is as follows : Theorem If < κ < 1, V has unbounded variation and V1 L p for some p > 1, then we have P-a.s. < liminf t + L X t t/loglogt 1 κ κe[iv ]+E[IˆV ] Note that the expectations E[IV ] and E[IˆV ] are finite and well defined since IV and IˆV both admit some finite exponential moments according to Theorems 1.1 and 1.13 of [72]. Example : We consider W κ the κ-drifted brownian motion W κ t := Wt κ t, then, the expression of the Laplace transform of 2 IW κ is given by equation 1.9 of [72]. This expression allows to compute the moments of IWκ and gives in particular E[IWκ] = 2/1 + κ. Moreover Wκ and Ŵ κ have the same law so E[IWκ] + E[IŴ κ] = 4/1 + κ. If we choose, as an environment, V = W κ for < κ < 1, then the above upper bound for the liminf becomes 1 κ 2 /4κ. Putting this in relation with the results of [28], we see that the application of Theorem in the special case of a drifted brownian environment improves and completes by proving that this renormalization is exact and by providing an explicit upper bound. 211

223 5.1. INTRODUCTION The fact that we have many possible renormalizations for the lim sup, depending on the environment V, while only one for the liminf, whatever is the environment V, might seem surprising, here is an heuristic explanation : In each valley the contribution to the time equals approximately the contribution to the local time multiplied by an exponential functional of the bottom of the valley which is close to R. The limsup concerns large values of the local time at a fixed time, it is reached when the contribution to the local time of some valley is large while the contribution to the time of the same valley has a fixed value, this happens when the exponential functional of the bottom of the valley has a small value. The link between the limsup and the small values of an exponential functional is made rigorously in Theorem The liminf concerns small values of the local time at a fixed time, it is reached when the contributions to the local time of the valleys are small while the sum of their contributions to the time have a fixed value, this happens when the exponential functionals of the bottoms of some valleys are large. We see that the difference between limsup and liminf comes from the difference between the left and right tails of R. The left tail is mainly the left tail of IV which depends on the asymptotic of Ψ V, according to Theorem 5.1.1, and Ψ V have many possible behaviors. On the other hand, the right tail is always exponential according to Theorems 1.1 and 1.13 and Remark 3.5 of [72]. This explains the difference of behaviors between the limsup and the liminf. We now treat the case where κ > 1. In this case we use a different method and the results are different. Theorem Let f be a positive non-increasing function. When κ > 1, we have { { + ft κ < + dt t = + limsup ftl X t = t + t 1/κ + P-a.s. 1 The above result is the analogue of Theorem 1.2 of [41] for the continus case. In the special case where V = W κ for κ > 1, our result coincides with proved by Devulder in [28]. Indeed, since f is decreasing we have the easy equivalences + 1 ft κ dt < + t + n=1 fn κ n < + + n=1 f2 n κ < The first equivalence shows that our result agrees with and the second equivalence allows to reformulate the integrability condition in a form that is convenient to prove Theorem Comparing Theorems 5.1.4, and we see that the renormalization for thelimsup is larger in the slow transient case than in the fast transient case : this is in accordance with intuition. However, it is surprising to see that the renormalization in the slow transient case is also greater than the renormalization in the recurrent case, given by Theorem 1.1 of [29]. Here is the heuristic explanation : In the recurrent case 212

224 5.1. INTRODUCTION the diffusion is trapped in the bottom of a large valley while in the slow transient case the diffusion gets successively trapped in the bottom of many valleys, these bottom being much more narrow. This explains that the large values of the local time have the tendency to be higher in the second case. For the liminf, we provide an explicite value. Let the constants K and m be defined similarly as in [66] : [ + κ 1 ] K := E e Vt dt and m := 2 Ψ V 1 >. We have : Theorem When κ > 1, we have P-a.s. lim inf t + L X t t/loglogt 1/κ = 2Γκκ2 K/m 1/κ. Example : If we choose V = W κ for κ > 1, then K = 2 κ 1 /Γκ see Example 1.1 in [66] and m = 4/κ 1. The above limit is then 4κ 2 κ 1/8 1/κ. This coincides with proved by Devulder in [28]. For the liminf the behaviors of the different cases are in accordance with intuition : comparing Theorem 1.1 of [29] with Theorems and we see that the renormalization for the lim inf in the fast transient case is smaller than the renormalization in the slow transient case which is in turn smaller than the renormalization in the recurrent case Sketch of proofs and organisation of the paper The rest of the paper is organized as follows. In section 5.2 we study the case < κ < 1. We first recall the decomposition of the environment into valleys from [74] and the behavior of the diffusion with respect to these valleys. In particular, we recall how the renewal structure of the diffusion allows to approximate the supremum of the local time and the time spent by the diffusion in the bottom of the valleys by an iid sequence of R 2 -valued random variables. For thelimsup, we study the asymptotic of the distribution functionpl X t/t x t, wherex t is a suitably chosen quantity that goes to infinity witht. More precisely, in Proposition we compare the asymptotic of this distribution function with the one of PY,t 1 Y 1,t 2 1 x t, the distribution function of a functional of the above mentioned iid sequence, and in Proposition we link the asymptotic behavior of PY,t 1 Y 1,t 2 1 x t with the left tail of IV or R in the case of a drifted brownian potential. The synthesis of Propositions and allows to compare the distribution function PL X t/t x t with the left tail of 213

225 5.1. INTRODUCTION IV or R in the case of a drifted brownian potential in Proposition This proposition entails Theorem by the mean of the Borel-Cantelli Lemma and the technical Lemma 5.2.1, which decomposes the trajectory of the diffusion into large independent parts in order to get the required independence to apply the Borel-Cantelli Lemma. The combination of Theorem with what is known and recalled in Theorem for the left tail of IV easily yields Theorems and which solves the problem for the limsup. For the liminf, we study the quantity PL X t t/x t. In Proposition it is compared with PY,t 1 Y 1,t 2 1 1/x t, the distribution function of an other functional of the R 2 -valued iid sequence. In Lemmas and we study the Laplace transform of a random variable involved in this functional, this allows to give a lower and an upper bound for PY,t 1 Y 1,t 2 1 1/x t in Proposition The synthesis of Propositions and gives the asymptotic of PL X t t/x t in Proposition This Proposition entails Theorem by the mean of the Borel-Cantelli Lemma and Lemma This solves the problem for the liminf. In Section 5.3 we study the case κ > 1. In this case, the local time at t can be approximated by the local time at an hitting time and the latter has the same law as the generalized Ornstein-Uhlenbeck process introduced in [66]. Using what is known for the excursion measure of this process we prove Theorems and In Section 5.4 we justify some facts about V, V and the diffusion in V that are used along the paper Facts and notations For Y a process and S a borelian set, we denote τy,s := inf{t, Yt S}, KY,S := sup{t, Yt S}. We shall only writeτy,x instead ofτy,{x} andτy,x+ instead ofτy,[x,+ [. Since V has no positive jumps we see that each positive level is reached continuously or not reached at all : x >, τv,x+ = τv,x which is possibly infinite. Moreover, the law of the supremum of V is known, it is an exponential distribution with parameter κ see Corollary VII.2 in Bertoin [8]. If Y is Markovian and x R we denote Y x for the process Y starting from x. For Y we shall only write Y. When it exists we denote by L Y t,x,t,x R the version of the local time that is continus in time and càd-làg in space and by σ Y t,x,t,x R the inverse of the local time : σ Y t,x := inf{s, L Y s,x > t}. Let B be a brownian motion starting at and independent from V. A diffusion in potential V can be defined via the formula : Xt := A 1 V BT 1 V t

226 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 where A V x := T V s := x s e Vu du and for e 2VA 1 V Bu du. s τ B, + e Vu du, It is known that the local time of X at x until instant t has the following expression : L X t,x = e Vx L B T 1 V t,a Vx Recall the notation L X t for the supremum of the local time until time t. We also sometimes use the notation L,+ X for the supremum of the local time on the positive half-line : L,+ X t := sup x [,+ [L X t,x. For the hitting times of r R by the diffusion X we shall use the frequent notation Hr instead of τx,r. DR,R is the space of càd-làg functions from R to R. Let P be the probability measure on DR,R inducing the law of V. For v DR,R, the quenched probability measure P v is the probability measure associated with the diffusion X conditionally on {V = v}. P represents the annealed probability measure, it is defined as P. := DR,R Pv.Pdv. X is a Markovian process under P v but not under P. Note that all the almost sure convergences stated in this Introduction are P-almost sure convergences. For objects not related to the diffusion X we also use the natural notation P for a probability. If Z is a random variable, its law is denoted by LZ and if A is an event of positive probability, LZ A denotes the law of Z conditionally to the event A. For Z an increasing càd-làg process and s, we put respectively Zs, Z s and Z 1 s for respectively the left-limit of Z at s, the largest jump of Z before s and the generalized inverse of Z at s : Zs = lim Zr, r < s Z s := sup Zr Zr, Z 1 s := inf{u, Zu > s}. r s For two quantities a and b depending on a parameter, a b means that loga logb when the parameter converges generally to or infinity. 5.2 Almost sure behavior when < κ < 1 In all this section we assume the hypotheses of Theorems and : < κ < 1, V has unbounded variation and there exists p > 1 such that V1 L p. The hypothesis of unbounded variation is necessary to approximate the law of the left part of a valley by the law of ˆV and the hypothesis about moments for V1 allows to neglect the local time outside the bottom of the valleys. For these reasons, many results of [74] that are recalled in the next subsection have been proved under these hypotheses. 215

227 5.2. ALMOST SURE BEHAVIOR WHEN < κ < Traps for the diffusion We now recall some definitions about valleys and describe how the diffusion gets trapped into successive valleys. The facts and lemmas stated in this subsection are more or less classical and there are all proved or justified in Subsection except Fact which is readily Lemma 3.5 of [74]. We first recall the notion of h-extrema. For h >, we say that x R is an h-minimum for V if there exist u < x < v such that Vy Vy Vx Vx for all y [u,v], Vu Vx Vx + h and Vv Vx Vx + h. Moreover, x is an h-maximum for V if x is an h-minimum for V, and x is an h-extremum for V if it is an h-maximum or an h-minimum for V. Since V is not a compound Poisson process, it is known see Proposition VI.4, in [8] that it takes pairwise distinct values in its local extrema. Combining this with the fact that V has almost surely càd-làg paths and drifts to without being the opposite of a subordinator, we can check that the set of h-extrema is discrete, forms a sequence indexed by Z, unbounded from below and above, and that the h-minima and h-maxima alternate. Let V DR,R be the set of the environments v that satisfy the above properties and that are such that vx x +, vx x +, + Note that the path of V belongs to V with probability 1. e vx dx < +. We denote respectively by m i, i Z and M i, i Z the increasing sequences of h-minima and of h-maxima of V, such that m < m 1 and m i < M i < m i+1 for every i Z. An h-valleys is the fragments of the trajectory of V between two h-maxima. The valleys are visited successively by the diffusions. For the size of the valleys to be well adapted with respect to the time scale, we have to make the size of the valleys grow with time t. We are thus interested in h t -valleys where h t := logt φt, with φt := loglogt ω, where ω > 1 will be chosen later in accordance with some other parameters. We also define N t, the indice of the largest h t -minima visited by X until time t, { } N t := max k N, sup Xs m k. s t We need deterministic bounds for the number of visited valleys. We define n t := e κ1+δφt and ñ t := e ρφt, where we fix ρ ],κ/1+κ[ once and for all in all the paper. The following lemma says that with hight probability, ñ t N t n t : 216

228 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 Lemma There is a positive constant c such that for all t large enough, PN t n t e cht e cφt, PN t ñ t e cφt We recall the definition of the standard valleys given in [74]. Their interest is mainly the fact that they are defined via successive stopping time, which make them convenient to use in the calculations, and also the fact that they take in consideration the descending phases between two h t -minima. Let δ >, small enough so that 1+3δκ < 1, be defined once and for all in the paper. Assume t is large enough so that e 1 δκht h t. We define τ h t = L := and recursively for i 1, L i := inf{x > L i 1, Vx V L i 1 e 1 δκht }, τ i h t := inf { x L i, Vx inf [ L i,x]v = h t}, m i := inf { x L i, Vx = inf [ L i, τ ih t] V}, L i := inf{x > τ i h t, Vx h t /2}, τ i a := sup{x < m i, Vx V m i a}, a [,h t ], τ + i a := inf{x > m i, Vx V m i = a}, a [,h t ]. These random variables depend on h t and therefore on t, even if this does not appear in the notations. We also define Ṽ i x := Vx V m i, x R. We call i th standard valley the re-centered truncated potential Ṽ i x, Li 1 x L i. The law of the bottom of these valleys is given in Fact of Section 5.4. Similarly as in [74] we define the deep bottoms of the j th standard valleys to be the interval D j := [ τ j φt2, τ + j φt2 ]. Remark The random times L i, τ ih t, and L i are stopping times. As a consequence, the sequence Ṽ i x+ m i, Li 1 m i x L i m i i 1 is iid. We also have that the sequence m i i 1 of the minima of the standard valleys coincides with the sequence m i i 1 with hight probability for a large number of indices. Let V t := {v V, i {1,...,n t }, m i = m i }. Note from the definition of V that the sequences m i i 1 and m i i 1 are always defined for any v V so in particular the event V t is well defined. We have : Fact Lemma 3.5 of [74] There is a positive constant c such that for all t large enough, P V V t 1 e cht. 217

229 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 We define X mj := X. + m j which is, according to the Markov property, a diffusion in potential V starting from m j. We also define, for any r R, H X mj r to be the hitting time of r by X mj. When we deal with X mj we often need the notation A j x = x m j eṽ j s ds. As in [4] and [74], we approximate the repartition function of the renormalized local time by repartition functions of functionals of the sequence e j S t j,e j S t jr t j j 1 where e j := L X H L j, m j /A j L j, S t j := Lj τ + j ht/2 eṽ j u du, R t j := τ + j ht/2 τ j ht/2 e Ṽ j u du. In [74], it is shown that e j follows an exponential distribution with parameter 1/2 since the distribution of e j does not depend on t we omit the dependence in t for e j in the notations and that the random variables e j,s t j,r t j, j 1 are mutually independent. To simplify notations we define, as in [4] and [74], the process of the renormalized sum of the contributions : s, Y1,Y t 2s t := 1 se κφt e j S t t j,e j SjR t j, t j=1 and the overshoots of. i=1 e is t ir t i : for any a, let us define N a := min { j, } j e i SiR t i t > a. i=1 We have Fact Proposition 4.2 of [74] Y1,Y t 2 t converges in distribution in D[,+ [,R 2,J 1 to a non-trivial bidimentional subordinator Y 1,Y 2. Let us define some events that happen with hight probability. They describe the behavior of the diffusion and provide effective approximations for the time and the local time, in the study of the case < κ < 1. On these events, the diffusion leaves the valleys from the right and never goes back to a previous valley, the local time and the time spent by the diffusion are negligible, compared with t, outside the bottom of the valleys, the supremum of the local time and the time spent by the diffusion in 218

230 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 the bottom of the valleys are approximated by the iid sequence e j S t j,e j S t jr t j j 1 : E 1 t := E 2 t := E 3 t := E 4 t := E 5 t := n t j=1 n t 1 j= n t j=1 n t j=1 n t j=1 n t Et 6 := j=1 n t Et 7 := j=1 { } H X mj L j < H X mj L j 1, H X Lj + < H X Lj τ j h t, { { { sup y R } L X H m j+1,y L X H L j,y te κ1+3δ 1φt, sup y [ L j 1, L j ] D j sup y D j } L X H L j,y L X H m j,y te 2φt, L X H L j,y L X H m j,y 1+e cht L X H L j, m j { j 1 H m j i=1 } H L i H m i 2t, logh t { 1 e cht e j S t j L X m j,h L j 1+e cht e j S t j { } 1 e cht e j SjR t j t H L j H m j 1+e cht e j SjR t j t. where c is a fixed positive constant that has been chosen small enough the constraints for the choice of c will be precised in the proofs of Fact and Fact Note that the above events depend both on the environment V and the brownian motion driving the diffusion. Fact There is a positive constant L such that for all t large enough, P Et 7 e Lht, 7 i=1 }, P Et i e Lφt Fix η ],1[. If t is so large such that 1 e cht 1 < 1+η and 1+e cht 1 1 2/logh t 1 η, then {V V t } {N t < n t } E 5 t E 7 t { N 1 ηt N t N 1+ηt } Our proofs are based on the study of the asymptotic of the quantities PL X t tx t and PL X t t/x t where x t depends on t and goes to infinity with t. More precisely we define x t such that x t Dloglogt µ 1, t + where D > and µ ]1,2]. Precise choices of x t will be made later, they will all satisfy for some D > and µ ]1,2]. }, 219

231 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 Fix ǫ small enough so that Fact of Section 5.4 is satisfied. We define G t to be the set of "good environments" in the following sens : v V t belongs to G t if it satisfies the following conditions : τ + j ht/2 j {1,...,n t }, e ǫht/4 Rj t = e vu v m j du e ht/8, τ j ht/2 j {1,...,n t }, A j τ j h t/2 τ j ht/2 = e vu v m j du e5ht/8, j {1,...,n t }, A j τ + j h t/2 = j {1,...,n t }, m j τ + j ht/2 m j e vu v m j du e 5ht/8, τ j ht/2 L j 1 e vu v m j du e ǫht, j {1,...,n t }, Lj τ + j ht/2 e vu v m j du e ǫht, P v 7 i=1e i t e Lφt/ where P v. is defined in Subsection and L is the constant defined in Fact Note that, since G t V t, we will often use the fact that m j = m j for v G t and j n t. Lemma There is a positive constant c such that for all t large enough, P V G t 1 e cφt. We need that, as in Lemma 5.3 of [4], an inequality for the local time in the bottom of a valley is related to an inequality for the corresponding random variable Rk t. Since we deal with unlikely inequalities for the local time, we have to prove the negligibility of the event where such inequalities happen but not the corresponding inequalities for Rk t. Recall the notations X m j and H X mj. defined above. For any fixed environment v G t and z [,1] we define { Etv,k,z 8 1+e cht := 1 z 1 e cht < x trk, t supl X mk t1 z,. tx t, D } k H X mk L k < H X mk L k 1, { Etv,k,z 9 := Rk/x t t < 1 e cht 1 z, L X mk t1 z, m k t/x t, } H X mk L k t1 z,h X mk L k < H X mk L k 1, where c is the same as in the definitions of E 4 t,e 6 t and E 7 t. Note that the inequalitiy 1 z1+e cht /1 e cht < x t R t k resp. Rt k /x t < 1 e cht 1 z only depends 22

232 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 on the environment v. We consider E 8 tv,k,z resp. E 9 tv,k,z to equal when v is such that this equality is not satisfied. In the rest of the paper, we use the same convention when we consider events, at fixed environment, that partially depend on the environment. Fact There is a positive constant c such that for t large enough and any fixed environment v G t we have P v nt k=1 {N t k, H m k /t 1 4/logh t } E 8 tv,k,h m k /t e cφt, P v nt k=1 E8 tv,k,1 4/logh t e cφt, P v nt k=1 {N t k} E 9 tv,k,h m k /t e cφt Note that in the above fact, v is fixed in G t V t so N t k implies H m k /t 1. The quantities in the above fact are thus well defined. We know from Fact that the contributions to the local time and to the time spent of the successive valleys are approximated by the iid sequencee j S t j,e j S t jr t j j 1. Since we deal with extreme values of the local time, we need to know the right tail of the distributions of e j S t j and of e j S t jr t j. We also need informations about the extreme values of R t j. All these are given by the next fact from [74] : Fact Fix η ],1/3[ and let C be the constant in Lemma 4.15 of [74]. We have lim sup t + x [e 1 2ηφt,+ [ lim sup t + y [e 1 3ηφt,+ [ x κ e κφt P e 1 S t 1/t > x C =, y κ e κφt P e 1 S t 1R t 1/t > y C E[R κ ] = R t 1 t> converges in distribution to R and there exists a positive λ such that [ ] λ < λ, E e λrt 1 E[ e λr], t + where the above quantities are all finite. This entails the convergence of the moments of any positive order of R t 1 to those of R when t goes to infinity. Finally, let us state a general lemma about the diffusion X : 221

233 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 Lemma Let q be the constant in Theorem 1.4 of [74]. There is a positive constant c such that for all r and t large enough, P supx 2t κ e κδloglogtω /q [,t] e cht, P Xt t κ e ρ κloglogtω /2q e cφt, P inf 3r 1, [,+ [ P inf L X +,. > r ],] 3r κ/2+κ Decomposition of the diffusion into independent parts An important point in our proofs is to give a decomposition of the trajectory of X that makes independence appear in order to apply the Borel-Cantelli Lemma. Let us fix a > 1 and define the sequences t n := e na, u n := e κna 2an a 1 /3, v n := e κna an a 1 /3. Let X n := XHv n +., the diffusion shifted by the hitting time of v n and T n := min{t n,τx n,u n,τx n,u n+1 }. Note that from the Markov property for X at time Hv n and the stationarity of the increments of V, X n v n is equal in law to X under the annealed probability P. Let n be large enough so that u n v n u n+1 for all n n. We define the events C n := {T n = t n } and D n := {Hv n < t n /n}. The idea is that the sequence of processes X n t, t T n n n is independent so, intersecting a sequence B n n n of interesting events where each event B n only depends on X n t, t t n with C n will result in B n C n n n, a sequence of independent events. Since X n is X shifted by Hv n, the event D n is useful to neglect this time shift when dealing with the renormalization of the local time. We will need the following lemma : Lemma and P C n < +, n 1 P D n < n 1 Démonstration. Here again, let q be the constant in Theorem 1.4 of [74]. First, notice from the definitions of t n, u n and v n that for all n large enough, 2t κ ne κδloglogtnω /q < u n+1 v n,

234 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 and t n /n κ e ρ κloglogtn/nω /2q > v n From the definition of C n and the Markov property applied to X at time Hv n we have P C n PτX n,u n+1 < t n +PτX n,u n < + PτX,u n+1 v n < t n +PτX,u n v n < + P supx u n+1 v n +P inf X u n v n [,t n] [,+ [ e chtn +3/v n u n where c is the constant in Lemma The last inequality is true for n large enough and comes, for the first term, from the combination of and , and, for the second term, from Recall that e htn e logtn = e na. We thus deduce From the definition of D n, and we have for all n large enough, P D n PXtn /n v n P Xt n /n t n /n κ e ρ κloglogtn/nω /2q e cφtn/n, where c is the constant in Lemma Since e cφtn/n = e cloglogtn/nω e caω logn ω we deduce The limsup We study the asymptotic of the quantity PL X t/t x t. Recall that x t is defined in whered > andµ ]1,2] are fixed constants. In all this subsection the parameter ω in is fixed in ]1,µ[. We have : Proposition There is a positive constant c such that for all a > 1 and t large enough we have, P Y,t 1 Y 1,t 2 1/a a x t e cφt PL Xt tx t P Y,t 1 Y 1,t 2 a x t /a +P R t1 axt +e cφt. Note that the functional of Y1,Y t 2 t involved in this proposition is Y,t 1 Y 1,t 2. which represents the supremum of the local time before and not including the last valley. Even though, as we see in Proposition 4.1 of [74], the repartition function 223

235 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 of L t X t/t involves a complex functional of Y1,Y2 t that represents the last valley together with the functional Y,t 1 Y 1,t 2. that represents the previous valleys, Proposition says that the right tail of this distribution function does not involves the last valley. Before proving this proposition we prove some lemmas. Lemma There is a positive constant C such that for t large enough, n t k=1 PH m k /t 1 Ce κφt. Démonstration. Since for all k 1 we have almost surely H m k k 1 j=1 H L j H m j, we deduce that n t n t k 1 PH m k /t 1 P H L j H m j /t 1 k=1 k=1 n t P k=1 j=1 k 1 1 e cht e j SjR t j/t t 1 j=1 k=1 +n t P Et 7 n t e 1 e ch t 1 E [e k 1 j=1 e jsj trt j ]+n /t t e Lht n t [ ] k 1 = e 1 e ch t 1 E e e 1S1 trt 1 /t +e 1+δφt Lh t, k=1 where we used the definition of E 7 t, Markov s inequality, 5.2.4, the fact that the sequence e j S t jr t j j 1 is iid and the definition of n t. For t large enough we thus get n t k=1 Then, ] 1 E [e e 1S1 trt 1 /t = PH m k /t 1 e = k=1 [ ] k 1 E e e 1S1 trt 1 /t +e Lh t/2 e 2 1 E [ e e 1S t 1 Rt 1 /t] +e Lht/ e u P e 1 S t 1R t 1/t > u du e φt/2 e u P e 1 S t 1R t 1/t > u du = e κφt + e φt/2 u κ e u u κ e κφt P e 1 S t 1R t 1/t > u du. We can now use with η = 1/6 and get that for all t large enough ] 1 E [e e 1S1 trt 1 /t C E[R κ ] + e κφt u κ e u du 2 e φt/2 224 C E[R κ ] + e κφt u κ e u du. t + 2

236 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 Putting into , we get the result for t large enough. We now link the asymptotic of PR t 1 a/x t with the left tail of IV. We have to make a distinction between the case where V possesses negative jumps and the case where V possesses no negative jumps, that is, V is the κ-drifted brownian motion. R t 1 is an exponential functional of the bottom of the first valley. In the first case, due to the jumps, the left side of the bottom of the valley can be neglected, so only the right side counts. In the second case, both sides have the same law, so both have to be taken in consideration in the left tail of R t 1. Lemma Let z t go to infinity with t satisfying logz t 2 << h t. Assume V possesses negative jumps. There is a positive constant c such that for any a > 1 and t large enough, e clog1 1/a/zt2 P IV 1/az t P R t 1 1/z t 2P IV a/z t If V := W κ, the κ-drifted brownian motion then for t large enough, PR 1/z t 2e δκht/3 P R t 1 1/z t 2PR a/zt +2e δκht/ Démonstration. We first assume that V possesses negative jumps. Recall that R1 t = τ + 1 ht/2 τ 1 ht/2 e Ṽ 1u du so, using the equality in law between Ṽ i m i + x, x τ i h m i and V x, x τv,h given by Fact 5.4.7, we get so R t 1 τ + 1 ht/2 m 1 e Ṽ 1 u du L = τv,h t/2 e V u du, P τv,h t/2 R1 t 1/z t P e V u du 1/z t According to Lemma 5.4.6, for some positive constant c and t large enough, τv 1 e cht P,h t/2 e V u du 1/z t is less than τv,h t/2 + P e V u du 1/z t P e V u du e ht/4 τv,h t/2 Combining with we get that for t large enough, P IV 1/z t +e ht/ P R t 1 1/z t 2P IV 1/z t +e ht/4 2P IV a/z t, because e ht/4 a 1/z t for large t thanks to the hypothesis logz t 2 << h t. This is the asserted upper bound in

237 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 On the other hand, using the independence between the left and right parts of the valleys given by Fact 5.4.7, we get that PR1 t 1/z t is more than τ + 1 ht/2 m1 P e Ṽ 1u du 1/az t P e Ṽ 1u du 1 1/a/z t. τ 1 ht/2 m 1 From Fact 5.4.7, we get that the first factor equals τv,h t/2 P e V u du 1/az t P IV 1/az t, while the second factor is more than [ E c ht P c ht 1 e κˆv τˆv,h t+ ; + τˆv,h t/2+ e ˆV u du 1 1/a/z t ] e ˆV u du 1 1/a/z t 2e δκht/3. 2e δκht/ Then, c ht c 1 when h t 1 so, putting in , we get that for t large enough, P R1 t 1/z t P IV 1/az t c 1 P IˆV 1 1/a/z t 2e δκht/ According to Theorem 1.14 of [72], there is a positive constant c such that for t large enough, P IˆV 1 1/a/z t e clog1 1/a/zt2. Thanks to the hypothesis logz t 2 << h t we deduce that, for c decreased a little, the second factor in the right hand side of is more than e clog1 1/a/zt2. This yields the lower bound in We now consider the case where V is the κ-drifted brownian motion W κ. Let Z 1 and Z 2 be two independent versions of the process W κ. Since W κ has no jumps, the density of the process P 2 in Fact is almost surely constant so P 2 is equal in law to Ŵκ x, x τ Ŵ κ,ht + = W κx, x τw κ,h t the last equality comes from the fact that Ŵ κ = W κ and W κ is continus. Combining this with and the equality in law between Ṽ i m i +x, x τ i h m i and V x, x τv,h both are from Fact 5.4.7, we get P τz1,h t/2 R1 t 1/z t P e Z1u du+ P τz1,h t/2 R1 t 1/z t P e Z1u du+ τz2,h t/2 τz2,h t/2 e Z 2u du 1/z t +2e δκht/3, e Z 2u du 1/z t 2e δκht/

238 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 Reasoning as in we get for some positive constant c : τz1,h t/2 τz2,h t/2 1 e cht 2 P e Z1u du+ e Z2u du 1/z t P IZ 1 +IZ 2 1/z t +2e ht/4. Combining with we get that for t large enough PR t 1 1/z t is less than 2P IZ 1 +IZ 2 1/z t +2e ht/4 +2e δκht/3 2PR a/z t +2e δκht/3, because 2e ht/4 a 1/z t for large t and because, form the definitions of R, Z 1 and Z 2 we have R L = IZ 1 + IZ 2. The above inequality is the asserted upper bound in On the other hand, we have trivially τz1,h t/2 τz2,h t/2 P e Z1u du+ e Z2u du 1/z t PIZ 1 +IZ 2 1/z t, and combining with we get that for t large enough, P R t 1 1/z t PIZ1 +IZ 2 1/z t 2e δκht/3 = PR 1/z t 2e δκht/3. This yields the lower bound in The next lemma studies the contribution of the last valley : Lemma There is a positive constant c such that for all u > 1 and t large enough, P supl X t,. L X H m Nt,. tx t P R t1 uxt +e cφt. D Nt Démonstration. We fix v G t, a realization of the environment. Let us define { E t v,k,z := sup L X mk t1 z,y tx t, y D k } H X mk m k+1 t1 z,h X mk L k < H X mk L k 1. We have P v sup D Nt L X t,. L X H m Nt,. tx t, N t < n t,e 1 t n t k=1 1 P v E t v,k,z, H m k /t dz

239 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 The fact that the sum stops at n t comes from N t < n t together with the fact that v G t V t. From the definitions of E t v,k,z, Etv,k,z, 8 Et 5 and Et 7 we have { } 1 z 1+e cht E t v,k,z Rk t 1 e cht x t, 1+e cht e k SkR t k t t1 z 2/logh t E 8 tv,k,z E 5 t E 7 t. When z 1 4/logh t we have, on the big event in the right hand side, t1 z/2 t1 z 2/logh t 1+e cht e k S t kr t k 1+e cht 2 1 e cht 1 1 ze k S t k/x t 21 ze k S t k/x t, for t large enough, and trivially 1 z1 + e cht 1 e cht e cht/2 for t large enough. As a consequence, for z [,1 4/logh t ] and t large enough, E t v,k,z { R t k 1+e cht/2 /x t, e k S t k/t x t /4 } E 8 tv,k,z E 5 t E 7 t. Note also that the sum in corresponds to disjoint events so it is actually the probability of a union of events. We thus get that nt 1 4/loght k=1 P v E t v,k,z, H m k /t dz is less than n t 1 4/loght k=1 P v R t k 1+e cht/2 /x t, e k S t k/t x t /4, H m k /t dz +P E v t 5 Et 7 nt k=1 {N t k, H m k /t 1 4/logh t } Etv,k,H m 8 k /t. Now, recall that Sk t,rt k only depends on v and that, v being fixed, e k belongs to the σ-field σxt,t H m k. In other words, it only depends on the diffusion after time H m k. On the other hand, H m k is measurable with respect to the σ-field σxt, t H m k. From the Markov property applied to X at H m k, we get that H m k is independent from e k,sk t,rt k so the above is less than n t k=1 P v R t k 1+e cht/2 /x t, e k S t k/t x t /4 P v H m k /t 1 +P E v t 5 Et 7 nt k=1 {N t k, H m k /t 1 4/logh t } Etv,k,H m 8 k /t n t P v Rk t 1+e cht/2 /x t, e k Sk/t t x t /4 P H L v k 1 /t 1 +e cφt, k=1 where c is a positive constant and where we used the fact that L k 1 m k for the first term and the fact that v G t together with and for the second term. Now, note that the first factor in the above product only depends on 228

240 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 ṽ k x, Lk 1 x L k that is, only on v shifted at time L k 1 while the second factor depends on v before time L k 1. This time L k 1 is a stopping time for the Lévy process V of which v is a fixed possible path. As a consequence, when we integrate the above inequality with respect to v over DR,R = G t G t equipped with the probability mesure P, we get that the two factors are independent so E[ n t k=1 n t k=1 k=1 1 4/loght P V E t V,k,z, H m k /t dz], is less than P Rk t 1+e cht/2 /x t, e k Sk/t t x t /4 P H L k 1 /t 1 +e cφt +PV / G t n t P Rk t 1+e cht/2 /x t P ek Sk/t t x t /4 P H L k 1 /t 1 +e cφt where we used the independence between Rk t and e ksk t and Lemma 5.2.6, and where the constant c has been suitably decreased. Since the sequence e k,sk t,rt k k 1 is iid, we deduce that the first part of the right-hand-side of , E[ n t 1 4/loght k=1 P V E t V,k,z, H m k /t dz], is less than P R t1 1+e cht/2 P e 1 S t1/t x t /4 nt P H L k 1 /t 1 +e cφt. x t Using and Lemma to bound respectively the second and third factor of the second term, we get the existence of a positive constant C such that for t large enough, E [ nt k=1 1 4/loght k=1 ] P V E t V,k,z, H m k /t dz C P R t1 1+e cht/2 +e cφt. x κ t x t It remains to study E[ n t 1 k=1 1 4/logh PV t E t V,k,z, H m k /t dz]. For v G t andz [1 4/logh t,1] we have, using the definitions ofe t v,k,z andetv,k,z 8 : { } E t v,k,z sup L X mk 4/logh t,y tx t, H X mk L k < H X mk L k 1 y D k { } 4 1+e cht logh t 1 e cht x trk t Etv,k,1 4/logh 8 t { 8/x t logh t R t k} E 8 t v,k,1 4/logh t, where, in the last inclusion, we used the fact that 1 + e cht /1 e cht 2 for large t. Recall that the sum in corresponds to disjoint events, we thus get 229

241 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 that n t 1 k=1 1 4/logh Pv t E t v,k,z, H m k /t dz is less than n t k=1 n t k=1 n t k=1 P v R t k 8/x t logh t, H m k /t dz +P v nt k=1 E8 tv,k,1 4/logh t 1 R t k 8/x tlogh t P v H m k /t 1+P v nt k=1 E8 tv,k,1 4/logh t 1 R t k 8/x tlogh t P H L v k 1 /t 1 +e cφt, where, for the first inequality, we used the fact that Rk t only depends on v, and for the second we used the fact that L k 1 m k for the first term and for the second term. Here again, the first factor in the above product only depends on ṽ k x, Lk 1 x L k that is, only on v shifted at time L k 1 while the second factor depends on v before time L k 1. As a consequence, when we integrate the above inequality with respect to v over DR,R = G t G t equipped with the probability mesure P, we get that the two factors are independent so E[ n t 1 k=1 1 4/logh PV t E t V,k,z, H m k /t dz], is less than n t k=1 P Rk t 8/x t logh t P H L k 1 /t 1 +e cφt +PV / G t P R1 t 8/x t logh t nt P H L k 1 /t 1 +e cφt, k=1 where we used the fact that the sequence R t k k 1 is iid and Lemma 5.2.6, and where the constant c has been suitably decreased. Note that from the definitions of x t and h t we have x t logh t Kloglogt µ. Therefore, using Lemmas with z t = x t logh t /8, a = 2 and to bound respectively the first and second factor we get that for t large enough the above is less than Ce κφt P IV 1/K loglogt µ +Ce κφt δκht/3 +e cφt, for some positive constants C and K. The term Ce κφt δκht/3 appears when V = W κ because of and it is not necessary otherwise, note that this term is ultimately less than e cφt. Combining with we get [ nt ] 1 E P V E t V,k,z, H m k /t dz Ce κφt K loglogt µ +2e cφt, k=1 1 4/logh t where K is a positive constant. Since we have chosen ω ]1,µ[ in we have loglogt µ >> φt so for t large enough the above inequality yields [ nt ] 1 E P V E t V,k,z, H m k /t dz e cφt, k=1 1 4/logh t 23

242 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 where the constant c has been suitably decreased. Now putting and in we get for some constant c and t large enough : P supl X t,. L X H m Nt,. tx t P R t1 uxt +PN t n t D Nt +P Et 1 +e cφt. Using and we get the result for a suitably chosen constant c and t large enough. Démonstration. of Proposition Upper bound PL Xt tx t P L Xt tx t, V V t, N t < n t, Et,E 1 t 2 +PV / V t +PN t n t +P Et 1 +P The event Et 1 ensures that for j n t, L j 1 is no longer reached after H m j and the event Et 2 ensures that the local time does not grow too much between H L j 1 and H m j. The event {V V t, N t < n t } ensures that at time t the diffusion is trapped in one of the first n t standard valleys. The first term of the right hand side is thus less than P sup L X t,. L X H m Nt,. [ L Nt 1, L Nt ] sup sup L X H L j,. L X H m j,. t x 1 t, V V t, N t < n t, 1 j N t 1 [ L j 1, L j ] where x 1 t := x t e κ1+3δ 1φt x t. Then, since x 1 t converge to +, we have x 1 t e 2φt for t large enough. Using the definition of Et 3, we get that for such large t the above is less than P supl X t,. L X H m Nt,. sup D Nt V V t, N t < n t +P Et 3. 1 j N t 1 E 2 t. sup L X H L j,. L X H m j,. t x 1 t, D j Putting all this together, we see that for t large enough, PL X t tx t is less than P sup sup L X H L j,. L X H m j,. t x 1 t, V V t, N t < n t 1 j N t 1 D j +P supl X t,. L X H m Nt,. t x 1 t D Nt +PV / V t +PN t n t +P Et 1 +P Et 2 +P Et

243 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 We now deal with the first term. Using the definition of Et 4 we get that the first term of is less than P sup L X H L j, m j t x 2 t, V V t, N t < n t +P Et 4, 1 j N t 1 where x 2 t := 1+e cht 1 x 1 t x t. Now using the definition of Et 6, the above is less than P sup e j Sj t x t, V V t, N t < n t +P Et 4 +P Et 6, 1 j N t 1 where x t := 1 + e cht 1 x 2 t x t. Using with η = a 1, the above is less than P sup e j Sj t x t +P Et 4 +P Et 6 +P Et 5 +P Et 7, =P 1 j N at 1 Y,t 1 Y 1,t 2 a x t +P Et 4 +P Et 6 +P Et 5 +P Et 7. Combining with and the fact that x 1 t x t, we get that PL X t tx t is less than P Y,t 1 Y 1,t 2 a x t +P sup +PV / V t +PN t n t +P Et 1 +P Et 2 +P L X t,. L X H m Nt,. t x t D Nt Et 3 +P Et 4 +P Et 5 +P Et 6 +P Et 7. Applying Lemma with u = a and x t replaced by x t which does not change anything since x t also satisfies 5.2.6, Fact 5.2.3, and 5.2.4, we deduce the existence of a positive constant c such that for t large enough, PL Xt tx t P Y,t 1 Y 1,t 2 a x t +P R t 1 a +e cφt. x t Since x t x t and a > 1 we get the upper bound when t is large enough. Lower bound PL Xt tx t P sup L X t, m j tx t 1 j N t 1 P sup 1 j N t 1 L X H L j, m j tx t, V V t, N t < n t, Et 1 because, on {V V t } {N t < n t } Et 1, m j for j < N t is no longer reached between times H L j and t, P sup 1 j N t 1 e j Sj t tˆx t, V V t, N t < n t, Et, 1 Et 6 232

244 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 because of the definition of Et 6 and where ˆx t := 1 e cht 1 x t x t, P sup e j Sj t tˆx t, V V t, N t < n t, Et, 1 Et, 6 Et, 5 Et 7 1 j N t/a 1 where we used with η = 1 a 1, P Y,t 1 Y 1,t 2 1/a ˆx t PV / V t +PN t n t +P Et 1 +P Et 5 +P Et 6 +P Et 7. Applying Fact 5.2.3, and we get the lower bound since ˆx t x t and a > 1. Proposition Let y t go to infinity with t. For any b > and u ],1[ there is a positive constant C depending on b and u such that for all t large enough, C P R1 t ub/y t /y κ t P Y,t 1 Y 1,t 2 b y t P R1 t b/y t. Démonstration. For any z >, let k t z be the first index k 1 such that e k Sk t/t z. We have { Y,t 1 Y 1,t 2 b } y t = { } k t y t k t y t < N tb = e i S t iri t tb i=1 { e k t y ts t k t y tr t k t y t tb }, and e k t y tsk t t y t Rt k t y has the same law as e t 1S1R t 1 t conditionally on e 1 S1 t ty t so, using and the independence between e 1 S1 t and R1 t : P Y,t 1 Y 1,t 2 b y t P e 1 S1R t 1 t tb e 1 S1 t ty t P R t 1 b/y t, which proves the upper bound. For the lower bound, we fix η < 1 u. Note that according to 5.2.4, { ek t y tsk t t y trk t t y t tb1 η } k t y t 1 { e i S t iri t < tbη Y,t 1 Y 1,t 2 b } y t, and both events on the left-hand-side are independent so P Y,t 1 i=1 Y 1,t 2 b y t P e 1 S1R t 1 t tb1 η e 1 S1 t ty t P k t y t 1 i=1 e i SiR t i t < tbη

245 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 We first deal with the second factor. Let Z i i 1 be iid random variables such that LZ 1 = Le 1 S1R t 1 e t 1 S1 t ty t andt be a geometric random variable with parameter Pe 1 S1 t ty t, independent from the sequencesz i i 1 ande i SiR t i t i 1. We then have k t y t 1 T 1 T 1 P e i SiR t i t < tbη = P Z i < tbη P e i SiR t i t < tbη, i=1 i=1 because the random variable e i SiR t i t is stochastically greater than the random variable Z i. Then, T 1 P e i SiR t i t < tbη P T e κφt P e i SiR t i t < tbη. i=1 1 i e κφt On the first hand we have P e i SiR t i t < tbη = P Y21 t < bη PY 2 1 < bη >, t + 1 i e κφt where we used Fact and where Y 2 is as in there, the second component of the limit process Y 1,Y 2. On the second hand P T e κφt = 1 1 Pe 1 S t 1 ty t e κφt = 1 e e κφt ln1 Pe 1 S t 1 tyt. Using we get P T e κφt C /yt κ. Putting all this together, we get t + the existence of a positive constant c 1 > such that for t large enough, k t y t 1 P e i SiR t i t < tbη c 1 /yt. κ i=1 We now study the first factor in the right hand side of From the independence of the two factors e 1 S t 1 and R t 1 in e 1 S t 1R t 1 we have P e 1 S t 1R t 1 tb1 η e 1 S t 1 ty t P R t 1 bu/y t P e1 S t 1 ty t 1 η/u e 1 S t 1 ty t and P e 1 S t 1 ty t 1 η/u e 1 S t 1 ty t = Pty t e 1 S t 1 ty t 1 η/u Pe 1 S t 1 ty t = 1 Pe 1S t 1 > ty t 1 η/u Pe 1 S t 1 ty t t + 1 u/1 ηκ >, i=1 234

246 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 where the limit comes from , because y t goes to infinity. We thus get the existence of a positive constant c 2 > such that for t large enough, P e 1 S t 1R t 1 tb1 η e 1 S t 1 ty t c2 P R t 1 bu/y t Putting and in we get the lower bound. Fix θ > 1. We apply Proposition the upper bound with a = θ 1/3 and the lower bound with a = θ 1/4, Proposition the upper bound applied with b = θ 1/3, y t = θ 1/3 x t and the lower bound with b = θ 1/4, y t = θ 1/4 x t, u = θ 1/4 and Lemma the upper bounds of and applied with a = θ 1/3, z t = θ 2/3 x t, and the lower bounds of these same expressions applied with a = θ 1/4, z t = θ 3/4 x t. We get : Proposition Fix θ > 1. If V possesses negative jumps, there are positive constants C and c such that for t large enough, e clogxt2 P IV Cx 1/θx κ t e cφt PL Xt tx t C P IV θ/x t +e cφt. t If V := W κ, the κ-drifted brownian motion, there are positive constants C and c such that for t large enough, 1 PR 1/θx Cx κ t e cφt PL Xt tx t C PR θ/x t +e cφt. t We can now link the asymptotic behavior of the local time with the left tail of IV : Démonstration. of Theorem First, we assume that V possesses negative jumps. Let us assume that is satisfied with some constants γ > 1 and C >. We now prove Let a > 1 and define the events A n := { } L X sup t t [a n,a n+1 ] tloglogt γ 1 a3 C 1 γ. We define x t := C 1 γ a 2 loglogt/a γ 1. Note that such a choice of x t satisfies with µ = γ and D = C 1 γ. From the increase of L X., the upper 235

247 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 bound applied with t = a n+1, θ = a and we have, for n large enough, PA n P L Xa n+1 C 1 γ a n+3 logloga n γ 1 C P IV 1/aC 1 γ logloga n γ 1 +e cφan+1 C exp a 1 γ 1 logloga n +e cφan+1 = C loga a γ 1 1 n a γ 1 1 +e cφan+1. Since e cφan+1 = e cloglogan+1 ω n 2 for n large enough, the above is the general term of a converging series so, using the Borel-Cantelli lemma, we deduce that P- almost surely, lim sup t + and letting a go to 1 we get L X t tloglogt γ 1 a3 C 1 γ, Now, let us assume that is satisfied with some constants γ > 1 and C >. We now prove Let a > 1 and let t n, u n, v n defined from this a and X n be as in Subsection recall that X n v n is equal in law to X under the annealed probability P. We define { B n := L X nt n C1 γ t n loglogt n γ 1 a 3γ 1 We also define C n and D n to be as in Lemma and E n := B n C n. We define x t := C 1 γ loglogt γ 1 /a 3γ 1. Note that such a choice of x t satisfies with µ = γ and D = C 1 γ /a 3γ 1. According to the lower bound applied with t = t n, θ = a γ 1, the definition of t n, and the fact that n is large we have, for some positive constants c a, K a and n large enough, PB n K a e calogloglogtn2 P IV a 2γ 1 /C 1 γ loglogt n γ 1 /loglogt n κγ 1 e cφtn = K a e calogalogn2 P IV a γ 1 /C 1 γ logn γ 1 /alogn κγ 1 e cφtn }. K a e calogalogn2 e logn/a /alogn κγ 1 e cφtn = K a exp c a logalogn 2 logn/a+κγ 1logalogn e cφtn K a exp logn e cφtn = K a n 1 e cφtn. Since e cφtn = e cloglogtnω n 2 for n large enough, we get PB n = n 1 236

248 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 Then, the combination of and yields n n 1PE n n 1PB P C n = n 1 Note that each event E n belongs to the σ-field σvs Vu n,u n s u n+1, Xt,Hv n t Hv n +T n, in other words, it only depends on the diffusion between times Hv n and Hv n +T n and on the environment between positions u n and u n+1. From the Markov property and the independence of the increments of the environment, we get that the events E n n 1 are independent. Combining this independence with and the Borel-Cantelli Lemma we get that P-almost surely, the event E n is realized infinitely many often. For n such that this event is realized we have L X Hv n+t n t n loglogt n L X nt n γ 1 t n loglogt n γ 1 C1 γ a3γ According to ans the Borel-Cantelli Lemma we have P-almost surely Hv n +t n t n, n + so combining with we deduce that P-almost surely, lim sup t + and letting a go to 1 we get L X t C1 γ tloglogt γ 1 a 3γ 1, If V = W κ, the κ-drifted brownian motion with < κ < 1, we proceed the same proof, only replacing IV by R and by We thus get that the same result is stil true for V = W κ, but with R instead of IV. The other theorems for the limsup are now easy to prove. Démonstration. of Theorems and In the case where V possesses negative jumps, Theorem is a direct consequence of the combination of Theorem 5.1.2, and Similarly, the first point of Theorem is obtained from the combination of Theorem and The second point of Theorem is obtained from the combination of Theorem and In the case where V = W κ, the κ-drifted brownian motion with < κ < 1, we only need to prove the last point of Theorem 5.1.5, and this requires to determine exactly the left tail of R. This variable is equal in law to the sum of two independent random variables having the same law as IW k. We thus have log E[e λr ] 2 = log E[e λiw k ] = 2log E[e λiw k ] 4 2λ, λ + 237

249 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 where the equivalent comes from 1.1 of [72]. Using this together with De Bruijn s Theorem see Theorem in [14] we get 8 logpr x x x. The last point of Theorem follows from this combined with Theorem The liminf For the liminf we study of the asymptotic of the quantity PL X t/t 1/x t. Recall that x t is defined in where D > and µ ]1,2] are fixed constants. In all this subsection we take ω := 2 for the parameter in Proposition Recall the λ defined in Fact There is a positive constant c such that for all a > 1 and t large enough we have, P Y,t 1 Y 1,t 2 a 1/ax t e cφt PL Xt t/x t 2P Y,t 1 Y 1,t 2 1/4 2/x t +e λ x t/8 +e cφt. Here, the functional of Y1,Y t 2 t involved is Y,t 1 Y 1,t 2. which represents the supremum of the local time after leaving the last valley. Démonstration. Lower bound From the definition of N t, we have H m Nt+1 t on {V V t, N t < n t } so PL Xt t/x t P L XH m Nt+1 t/x t, V V t, N t < n t, Et,E 1 t 2. The event Et 1 ensures that for j n t, L j 1 is no longer reached after H m j and the event Et 2 ensures that the local time does not grow too much between H L j 1 and H m j. The event {V V t, N t < n t } ensures that at time t the diffusion is trapped in one of the first n t standard valleys. The right hand side is thus more than P sup sup 1 j N t [ L j 1, L j ] L X H L j,. L X H m j,. t/ x 1 t, V V t, N t < n t, Et,E 1 t 2 where x 1 t := 1/1/x t e κ1+3δ 1φt. Then, since x 1 t x t, we have 1/ x 1 t e 2φt for t large enough. Using the definition of Et 3, we get that for such large t the above is more than P sup sup L X H L j,. L X H m j,. 1 j N t D j t/ x 1 t, V V t, N t < n t, E 1 t,e 2 t,e 3 t,. 238

250 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 From the definition of Et 4 the above is more than P sup 1 j N t L X H L j, m j t/ x 2 t, V V t, N t < n t, E 1 t,e 2 t,e 3 t,e 4 t where x 2 t := 1 + e cht x 1 t x t. Now using the definition of Et 6, the above is more than P sup e j Sj t 1/ x t, V V t, N t < n t, Et,E 1 t,e 2 t,e 3 t,e 4 t 6, 1 j N t where x t := 1+e cht x 2 t x t. Let a > 1. Using with η = a 1, the above is more than P sup e j Sj t 1/ x t, V V t, N t < n t, Et,E 1 t,e 2 t,e 3 t,e 4 t,e 5 t,e 6 t 7 1 j N at P Y,t 1 Y 1,t 2 a 1/ x t PV / V t +PN t n t +PEt+PE 1 t+pe 2 t+pe 3 t+pe 4 t+pe 5 t+pe 6 t 7, where we used the definition of Y1,Y t 2. t Applying Fact 5.2.3, and we get the asserted lower bound for a suitably chosen constant c and t large enough, since x t x t and a > 1. Upper bound PL Xt t/x t P sup L X t, m j t/x t, V V t, N t < n t 1 j N t +PV / V t +PN t n t P L X t, m Nt t/x t, sup L X H L j, m j t/x t, 1 j N t 1 V V t, N t < n t, Et 1 +P Et 1 +PV / V t +PN t n t, because, on {V V t, N t < n t } E 1 t, m j for j < N t is no longer reached between times H L j and t. We fix v G t, a realization of the environment. Let us define E t v,k,z := { } L X mk t1 z, m k t/x t, H X mk m k+1 t1 z, H X mk L k < H X mk L k 1, and ν t v,k,z := P v E t v,k,z. The event E t v,k,z belongs to the σ-field σxt,t H m k. In other words, it only depends on the diffusion after time H m k. On the other hand, H m k is measurable with respect to the σ-field σxt, t H m k. From the Markov property applied to X at H m k, we get that H m k is independent from the event E t v,k,z. As a consequence,, 239

251 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 P v L X t, m Nt t/x t, sup 1 j Nt 1L X H L j, m j t/x t, N t < n t, Et 1 is less than n t 1 ν t v,k,z P v sup L X H L j, m j t/x t, H m k /t dz k=1 1 j k 1 The fact that the sum stops at n t comes from N t < n t together with the fact that v G t V t. From the definition of Etv,k,z 9 we get { 1 z E t v,k,z 1 e cht 1 }, H x X mk m k+1 t1 z Etv,k,z 9 t R t k and, v being fixed, the events in the above expression are independent from the σ- field σxt, t H m k, so putting into 5.2.5, using the independence and the fact that the sum in corresponds to disjoint events so it is actually the probability of a union of events, we get that the first term in the right hand side of is less than 1 H mnt /t P 1 e cht 1/x RN t t, sup L X H L j, m j t/x t t 1 j N t 1 +E [ 1 V Gt P V nt k=1 {N t k} EtV,k,H m 9 k /t ] +PV / G t P 1 e cht 1Rt N t + H m N t 1, sup L X H L j, m j t/x t +e cφt, x t t 1 j N t 1 wherecis a positive constant and where we used for the second term, Lemma for the third term and the fact that t is large enough, P 1 e cht 1Rt N 1+e cht N t t 1 + e j S x t t jr t j t 1 2/logh t, j=1 sup e j Sj t 1 e cht 1 t/x t, V V t, N t < n t, Et,E 5 t,e 6 t 7 1 j N t 1 +PV / V t +PN t n t +P Et 5 +P Et 6 +P Et 7 +e cφt, where we used the definitions of E 5 t, E 6 t and E 7 t, N t 1 R t P Nt + 1 e j S x t t jr t j t 1/2, j=1 Et,E 5 t,e 6 t 7 +e cφt sup e j Sj t < 2t/x t, V V t, N t < n t, 1 j<n t 1 where we used the fact that t is large enough, Fact 5.2.3, and 5.2.4, and where the constant c has been suitably decreased, R t P Nt + 1 x t t +P N t 1 j=1 e j S t jr t j 1/2, sup 1 j N t e j S t j < 2t/x t, V V t, N t < n t, E 5 t,e 6 t,e 7 t sup e j Sj t < 2t/x t, e Nt SN t t 2t/x t 1 j N t 1 +e cφt. 24

252 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 On the event in the probability of the first term, we have N t = k t 2/x t. For the second term we use with η = 3/4. The above is thus less than P Rt k t 2/x t x t + 1 t R t k P t 2/x t 1/4 x t k t 2/x t 1 j=1 +P sup e j Sj t < 2t/x t 1 j N t/4 P R1 t x t /4 +2P +P 1 t Y,t 1 e j SjR t j t 1/2 +P k t 2/x t 1 j=1 +e cφt e j SjR t j t 1/4 Y 1,t 2 1/4 2/x t +e cφt sup e j Sj t < 2t/x t 1 j N t/4 +e cφt where, for the last inequality, we used the fact that the sequence Rj t j 1 is iid and independent from the random index k t 2/x t, with b = 1/4, y t = x t /2, and the definition of Y1,Y t 2. t Then, note that according to we have for all t large enough, P R1 t x t /4 e λ x t/8. Bounding the three terms in the right hand side of thanks to the above, Fact 5.2.3, and we get the upper bound for a suitably chosen constant c and t large enough. We now study the functional involved in Proposition For this we need two lemmas. In the remaining part of this subsection we fix η ],1/3[. Let y t go to infinity withtsatisfyinglogy t << φt. Letp t := Pe 1 S t 1 > t/y t andh i i 1 be iid random variables such that H 1 has the same law as e 1 S t 1R t 1 conditionally to {e 1 S t 1 t/y t } : LH 1 = Le 1 S t 1R t 1 e 1 S t 1 t/y t. Since we have logy t << φt, gives p t C e κφt y κ t. We have Lemma [ ] 1 E e λh 1/t t + λ C κe[r] 1 κe κφt yt 1 κ Démonstration. For any λ, E[e λh1/t ] equals ] ] E [e λe 1S1 trt 1 /t e 1 S t1 t/y t = 1 p t 1 E [e λe 1S1 trt 1 /t 1 e1 S t1 t/yt + ] =1 p t 1 E [e λue 1S1 t/t 1 e1 S t1 LR1du t t/yt + 1/yt =1 p t 1 1 e λu/yt p t λ ue λxu Pe 1 S1/t t > xdx LR1du, t 241

253 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 where we used iteration by parts, ] = 1 p t 1 E [e 1 λrt 1 /yt p t λ where we used Fubini s Theorem, 1/yt ] = 1 p t 1 p 1 t +p t 1 E [e λrt 1 /yt λ [ ] E R1e t λxrt 1 Pe 1 S1/t t > xdx, 1/yt [ ] E R1e t λxrt 1 Pe 1 S1/t t > xdx We now study the second and third term in Using the fact that the difference between two points of a continuously differentiable function is the integral of its derivative, the last part of Fact and the equivalent for p t we get p t 1 E [e λrt 1 /yt ] = p t λ y t 1 E [R t1e λurt 1 /yt ] du t + λ p te[r] y t Then, from the last part of Fact again, 1/yt [ ] E R1e t λxrt 1 Pe 1 S1/t t > xdx E[R] t + Recall that η ],1/3[. 1/y t Pe 1 S t 1/t > xdx equals = e 1 2ηφt e 1 2ηφt Pe 1 S t 1/t > xdx+ e κφt 1/yt Pe 1 S t 1/t > xdx+ e κφt 1/yt 1/yt λ C E[R] t + e κφt yt 1 κ Pe 1 S t 1/t > xdx e 1 2ηφt x κ x κ e κφt Pe 1 S t 1/t > xdx e 1 2ηφt x κ x κ e κφt Pe 1 S t 1/t > x C dx 1/yt e 1 2ηφt + C e κφt x κ dx C e κφt x κ dx Now, the absolute values of the first and fourth terms of are respectively less than e 1 2ηφt and C e 2ηκ+1 2ηφt /1 κ. In particular, thanks to 2ηκ+ 1 2η > κ which is trivial and logy t << φt, both are negligible with respect to e κφt /yt 1 κ. From we also have 1/yt e κφt x κ x κ e κφt Pe 1 S1/t t > x C dx = Oe κφt /yt 1 κ, e 1 2ηφt and the third term of equals C e κφt /1 κyt 1 κ. Combining with we get λ 1/yt [ ] E R1e t λxrt 1 Pe 1 S1/t t > xdx t + λ C E[R] 1 κe κφt yt 1 κ Putting together and in we obtain

254 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 Lemma Recall the λ defined in Fact For any λ [,λ [ we have [ ] E e λyth 1/t 1 /p t 1 E [ e λr] +λ t + and this limit is positive for all λ ],λ [. 1 x κ E [ Re λxr] dx, Démonstration. For anyλ [,λ [ we haveλy t e 1 S1/t t [,λ [ on the event{e 1 S1/t t 1/y t }, so E[e λyth1/t ] equals ] ] E [e λyte 1S1 trt 1 /t e 1 S t1 t/y t = 1 p t 1 E [e λyte 1S1 trt 1 /t 1 e1 S t1 t/yt + ] =1 p t 1 E [e λytue 1S1 t/t 1 e1 S t1 LR1du t t/yt + 1/yt =1 p t 1 1 e λytu/yt p t +λy t ue λytxu Pe 1 S1/t t > xdx LR1du t =1 p t e λu p t +λ 1 ue λyu Pe 1 S1/t t > y/y t dy LR1du, t where we used iteration by parts and made the change of variable y = y t x, [ ] 1 [ ] = 1 p t 1 E 1 e λrt 1 p t +λ E R1e t λyrt 1 Pe 1 S1/t t > y/y t dy where we used Fubini s Theorem, [ ] = 1 p t 1 p 1 t +p t 1 E e λrt 1 +λ 1 [ ] E R1e t λyrt 1 Pe 1 S1/t t > y/y t dy According to the second term is equivalent to p t 1 E[e λr ]. We now study the third term in Recall that η ],1/3[. This term equals yte 1 2ηφt [ ] λ E R1e t λyrt 1 Pe 1 S1/t t > y/y t dy 1 [ ] +λyte κ κφt R1e t λyrt 1 y/y t κ e κφt Pe 1 S1/t t > y/y t dy =λ yte 1 2ηφt +λy κ te κφt 1 y κ E e 1 2ηφt [ E R1e t λyrt 1 y te 1 2ηφt y κ E 1 +λc yte κ κφt y κ E ] Pe 1 S t 1/t > y/y t dy [ ] y/yt R1e t λyrt 1 κ e κφt Pe 1 S1/t t > y/y t C dy [ R1e t λyrt 1 ]dy λc y κte yte 1 2ηφt κφt [ ] y κ E R1e t λyrt 1 dy

255 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 Now, thanks to , the absolute values of the first and fourth terms of are ultimately less than 2λE[R]y t e 1 2ηφt and 2λC E[R]y t e 2ηκ+1 2ηφt /1 κ. In particular, thanks to 2ηκ+1 2η > κ which is trivial and logy t << φt, both are negligible with respect to e κφt y κ t. From and we also have λy κ te κφt 1 =Oe κφt y κ t, y te 1 2ηφt y κ E [ ] y/yt R1e t λyrt 1 κ e κφt Pe 1 S1/t t > y/y t C dy and, thanks to , the third term of is equivalent to λc yte κ κφt 1 y κ E[Re λyr ]dy. We thus get λ 1 [ ] 1 E R1e t λyrt 1 Pe 1 S1/t t > y/y t dy λc y κ t + te κφt Putting into we obtain t + λp t 1 y κ E[Re λyr ]dy. y κ E[Re λyr ]dy We justify the positivity of the limit as follows : we see that the right hand side of is equivalent to λκe[r]/1 κ when λ goes to. The limit in is therefore positive for small λ. On the other hand, E[e λyth 1/t ] 1/p t increases with λ so the limit in is non-decreasing on [,λ [. We thus get the positivity of the limit for all λ ],λ [. We can now study the lower and upper bounds given by Proposition : Proposition Let y t be chosen as before that is, y t + and logy t << φt. There is a positive constant L not depending on the choice of y t, such that for any b >, u > 1 and t large enough we have e ub1 κyt/κe[r] P Y,t 1 Y 1,t 2 b 1/y t e Lbyt. Démonstration. Lower bound Let us fix α ]b1 κ/κe[r],ub1 κ/κe[r][. For any z >, k t z still denotes the first index k 1 such that e k Sk t /t z. We have { Y,t 1 Y 1,t 2 b } 1/y t = { } k t 1/y t 1 k t 1/y t > N tb = e i S t iri t > tb Now, recall thath i i 1 are iid random variables such thatlh 1 = Le 1 S t 1R t 1 e 1 S t 1 t/y t and let T be a geometric random variable with parameter p t = Pe 1 S t 1 > t/y t, i=1 244

256 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 independent from the sequence H i i 1. Recall that p t C e κφt y κ t. We have P k t 1/y t 1 i=1 e i SiR t i t > tb = P H i > tb PT > αht+1 P T 1 i=1 αht i=1 H i > tb, where we put ht := y t /p t e κφt yt 1 κ /C. We give a lower bound for the two factors in the left hand side of We first study the Laplace transform of the normalized sum of the second factor to prove its convergence to a constant number. For any λ, we have ] [ E [e λ αht αht i=1 H i /t = E e 1/t] λh = e αht log1+e[e λh 1 /t ] 1. According to Lemma , the exponent is equivalent to and since ht e κφt yt 1 κ /C we get E [e λ αht i=1 H i /t λc κe[r]αht/1 κe κφt y 1 κ t, ] t + e λακe[r]/1 κ, so αht i=1 H i /t converges in probability to ακe[r]/1 κ which yields since α > b1 κ/κe[r]. P αht i=1 H i > tb 1, t + We now study the first factors in the left hand side of Since T is geometric with parameter p t, we have PT > αht+1 = 1 p t αht+1 = e αht+1 log1 pt. Now, since ht e κφt yt 1 κ /C and p t C e κφt yt κ we get logpt > αht+1 t + αy t Now, putting and into , and combining the latter with 5.2.6, we get the result for t large enough since α < ub1 κ/κe[r]. Upper bound 245

257 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 Recall and the definitions of H i i 1, p t, T and ht. Let us fix α > that will be chosen latter. We have k t t/y t 1 T 1 αht P e i SiR t i t > tb = P H i > tb PT > αht+p H i > tb. i=1 i=1 i= Let us choose λ ],λ [ where λ defined in Fact The second term equals αht αht P y t H i /t > by t e λbyt E exp λ y t H i /t i=1 i=1 = e λbyt 1+E [ exp λy t H i /t ] 1 αht = e λbyt+ αht log1+e[expλyth i/t] 1. According to Lemma , the fact that ht e κφt yt 1 κ /C and p t C e κφt yt κ we have αht log 1+E [ exp λy t H i /t ] 1 t + αy t 1 E [ e λr] +λ 1 x κ E [ Re λxr] dx. Thanks to the positivity of the limit in Lemma we can choose α such that < α < bλ/21 E[e λr ]+λ 1 x κ E[Re λxr ]dx. We thus get for t large enough αht P y t H i /t > by t e λbyt/ i=1 Since T is geometric with parameter p t, we have PT > αht = 1 p t αht = e αht log1 pt e αyt e αyt/2, t + where we used the equivalents for ht and p t and where the last inequality holds for t large enough. Now, putting and into , and combining the latter with 5.2.6, we get the result for t large enough. Fix θ > 1. We apply Proposition with a = θ 1/3 and Proposition the lower bound with b = θ 1/3, u = θ 1/3, y t = θ 1/3 x t and the upper bound with b = 1/4, y t = x t /2. We get : Proposition Let L ],min{l/8,λ /8}[ where L is the positive constant defined in Proposition and λ is defined in Fact There is a positive constant c such that for any θ > 1 and t large enough we have e θ1 κxt/κe[r] e cφt PL Xt t/x t e Lx t +e cφt. 246

258 5.2. ALMOST SURE BEHAVIOR WHEN < κ < 1 We can now prove Theorem Démonstration. of Theorem Recall that L is the positive constant defined in Proposition and let x t := 4loglogt/2 L. Note that such a choice of x t satisfies with µ = 2 and D = 4/2 L. We define the events { A n := inf t [2 n,2 n+1 ] L X t } t/loglogt L/4. From the increase of L X. and Proposition the upper bound applied with t = 2 n we have for n large enough, PA n P L X2 n 2 n+1 L/4loglog2 n exp 2loglog2 n +e cφ2n = log2 2 n 2 +e cφ2n. Sincee cφ2n = e cloglog2n 2 n 2 fornlarge enough, the above is the general term of a converging series so, using the Borel-Cantelli lemma we deduce that P-almost surely, so the liminf is positive. lim inf t + L X t t/loglogt L/4, We now prove the upper bound for the liminf. Let a > 1 and let t n, u n, v n defined from this a and X n be as in Subsection recall that X n v n is equal in law to X under the annealed probability P. We define { B n := L X nt n t n /loglogt n a2 1 κ κe[r] }, B n := { L X Hv n t n /loglogt n 1 n We also define C n and D n to be as in Lemma and E n := B n C n. We define x t := κe[r]loglogt/a 2 1 κ. Note that such a choice of x t satisfies with µ = 2 and D = κe[r]/a 2 1 κ. Recall the notation L,+ X defined in Subsection Let us choose η and C as in Lemma of the next section and Q be as defined in the next section. According to and Lemma applied with u = t n /nloglogt n, v = v n we get for all n large enough, P L,+ B n P X inf L X +,. > t n /nloglogt n +P Hv n ],] t n /loglogt n > 1 n 3t n /nloglogt n κ/2+κ +C v n /Q+vn 7/8 nloglogtn κ /t n κ +vn η. From the definition of t n, the fact that n κ v n /t κ n = n κ e κ2ana 1 /3 and loglogt n = alogn, we get P B n < n }.

259 5.3. ALMOST SURE BEHAVIOR WHEN κ > 1 According to Proposition the lower bound applied with t = t n, θ = a and the definition of t n we have PB n e loglogtn/a e cφtn = e alogn/a e cφtn = n 1 e cφtn. Since e cφtn = e cloglogtn2 n 2 for n large enough, we get PB n = n 1 Then, the combination of and yields n n 1PE n n 1PB P C n = n 1 As in the proof of Theorem 5.1.2, we see that each event E n belongs to the σ-field σvs Vu n,u n s u n+1, Xt,Hv n t Hv n +T n so the eventse n n 1 are independent. Combining this independence with and the Borel-Cantelli Lemma we get that P-almost surely, the event E n is realized infinitely many often. Combining with and the Borel-Cantelli Lemma we get that P-almost surely, the event B n B n C n is realized infinitely many often. For n such that this event is realized we have L X Hv n+t n t n /loglogt n L X nt n t n /loglogt n + L X Hv n t n /loglogt n a2 1 κ + 1 κe[r] n Recall that according to and the Borel-Cantelli Lemma we have P-almost surely Hv n +t n t n, n + so combining with we deduce that P-almost surely, lim inf t + L X t t/loglogt a2 1 κ, κe[r] and letting a go to 1 we get the asserted upper bound for the liminf. 5.3 Almost sure behavior when κ > 1 In this section we prove Theorems and Let us first recall some facts and notations from [66] and [74]. Our proof is based on the study of the so-called generalized Ornstein-Uhlenbeck process defined by x Zx := e Vx R e Vy dy, 248

260 5.3. ALMOST SURE BEHAVIOR WHEN κ > 1 where R is a two-dimensional squared Bessel process independent from V. Let L be the local time of Z for the position 1, n the associated excursion measure, and L 1 the right continus inverse of L. We denote by ξ a generic excursion. Let us denote by Q the positive constant denoted by n[ζ] in [66]. Recall also the notations K and m defined in the Introduction. We have : Fact There is η > and r > such that for all r r and h > 1 we have e r/q+r7/8 nsupξ>h r η P sup Zx h x [,r] e r/q r7/8 nsupξ>h +r η, nsupξ > h h + Q2κ Γκκ 2 K/h κ The first point is Lemma 2.3 of [74] while the second point is Proposition 5.1 of [66]. Note that Fact is true for a general positive κ and not only for κ > 1. Let us recall the link between Z and the local time until the hitting times. The local time at point x and within the hitting time Hr is given by : L X Hr,x = e Vx L B τb,a V r,a V x M 1 r and M 2 r denote respectively the supremum of the above expression for x ],[ and x [,+ [. The supremum of the local time until instant Hr can be written L XHr = max{m 1 r,m 2 r}, where M 1 r M 1 + < + and, as in [74], M 2 r = L sup Zx x [,r] We can now study the behavior of the local time at a hitting time. This allows to prove the following useful lemma. Lemma There exist η >, a positive constant C, u > and v > such that u u,v v, P L,+ X Hv > u Cv/Q+v 7/8 u κ +v η, where L,+ X is as defined in Subsection Démonstration. From the definition of M 2 and we have P L,+ X Hv u = PM 2 v > u = P supz > u [,v]

261 5.3. ALMOST SURE BEHAVIOR WHEN κ > 1 Now let us choose η and v such that is true for all r v and h > 1 with this η. We choose C > Q2 κ Γκκ 2 K and u such that nsupξ > u Cu κ for all u u. Such a u exists thanks to For u u and v v we have P supz > u [,v] 1 e v/q+v7/8 nsupξ>u +v η v/q+v 7/8 nsupξ > u+v η Cv/Q+v 7/8 u κ +v η. Putting into we get the result. Remark Neither Lemma nor its proof require the hypothesis that κ > 1. The lemma is thus true whatever is the value of κ. We need to study the supremum of the local time until a deterministic time. The following fact from [74] says that we can replace a deterministic time by a hitting time when κ > 1. We now assume κ > 1 until the end of this section. We have : Fact For any α ] max{3/4, 1/κ}, 1[ their exists η > such that for r large enough we have PHr/m r α r Hr/m+r α 1 r η Let α ]max{3/4,1/κ},1[ be fixed until the end of this section and η > be small enough so that both and are satisfied with this α The liminf In this subsection we prove Theorem Let us definej := 2Γκκ 2 K/m 1/κ, the expected liminf. We begin to prove that lim inf t + L X t J t/loglogt 1/κ Let a > 1 and define the events { A n := inf t [a n,a n+1 ] L X t t/loglogt J }. 1/κ a 3/κ 25

262 5.3. ALMOST SURE BEHAVIOR WHEN κ > 1 From the increase of L X., 5.3.7, 5.3.4, 5.3.5, and 5.3.1, we have PA n P L Xa n Ja n 2 /logloga n 1/κ P L XHa n /m a αn Ja n 2 /logloga n 1/κ +a nη P M 2 a n /m a αn Ja n 2 /logloga n 1/κ +a nη = P sup Z Ja n 2 /logloga n 1/κ +a nη [,a n /m a αn ] e a n /Qm a αn /Q a n /m a αn n 7/8 supξ>ja n 2 /logloga n 1/κ +a nη +a n /m a αn η According to the equivalent given by 5.3.2, the exponent in the above expression is, for n large enough, less than alogloga n, so for such large n, e a n /Qm a αn /Q a n /m a αn n supξ>ja 7/8 n 2 /logloga n 1/κ nloga a. The other two terms in the right hand side of are also general terms of converging series so we obtain, PA n < +. n 1 According to the Borel-Cantelli lemma we get lim inf t + L X t t/loglogt 1/κ J/a3/κ, in which we can let a go to 1 which yields We now prove that lim inf t + L X t J t/loglogt 1/κ Let us fix a >, u n := n 2n, v n := u n /m + u α n = n 2n /m + n 2αn and X n := XH2v n +., the diffusion shifted by the hitting time of 2v n. Note that from the Markov property for X at time H2v n and the stationarity of the increments of V, X n 2v n is equal in law to X under the annealed probability P. We take n so large such that 2v n < v n+1 and define the events { L,+ X B n := H2v n u n+1 /loglogu n+1 C n := } aj 1/κ { L X n τ X n,v n+1 1+aJ u n+1 /loglogu n+1 1/κ D n := {τ X n,v n+1 < τ X n,v n }, E n := C n D n, F n := { L,+ X u n L,+ X Hv n }., }, 251

263 5.3. ALMOST SURE BEHAVIOR WHEN κ > 1 Recall that η > has been fixed so that is satisfied. For this η and for C > Q2 κ Γκκ 2 K, the inequality of Lemma is true for u and v large enough. According to this lemma applied with u = aju n+1 /loglogu n+1 1/κ, v = 2v n we get for all n large enough, P B n C 2vn /Q+2v n 7/8 loglogu n+1 /J κ a κ u n+1 +2v n η. Since, for n large enough, v n /u n+1 1/mn 2 and loglogu n+1 logn we can deduce that P B n < n 1 From the equality in law between X n 2v n and X under P,, 5.3.4, 5.3.5, , PC n = P L X Hv n+1 2v n 1+aJ u n+1 /loglogu n+1 1/κ P M 2 Hv n+1 2v n 1+aJ u n+1 /loglogu n+1 1/κ P inf L X +,. > 1+aJ u n+1 /loglogu n+1 1/κ ],] P sup Z 1+aJ u n+1 /loglogu n+1 1/κ [,v n+1 2v n] 31+aJ κ/2+κ u n+1 /loglogu n+1 1/2+κ e v n+1 2v n/q+v n+1 2v n n 7/8 supξ>1+aju n+1 /loglogu n+1 1/κ v n+1 2v n η 31+aJ κ/2+κ u n+1 /loglogu n+1 1/2+κ According to the equivalent given by and the definitions of u n and v n, the exponent in the above expression is equivalent to 1 + a κ loglogu n+1 1+a κ logn so for n large enough, e v n+1 2v n/q+v n+1 2v n n supξ>1+aju 7/8 n+1 /loglogu n+1 1/κ 1 n, and the remaining terms in the right hand side of are the general terms of converging series. We thus get PC n = n 1 P D n = Pτ X n,v n+1 > τ X n,v n Pτ X n,v n < τ X n,+ Pτ X, v n < τ X,+ = P inf X < v n, [,+ [ 252

264 5.3. ALMOST SURE BEHAVIOR WHEN κ > 1 where we used the equality in law between X n 2v n and X under P. Combining with applied with r = v n we get P D n < n 1 Then, the combination of and yields n 1PE n n 1PC n n 1 P D n = According to the definitions of the sequences u n n 1 and u n n 1, to 5.3.7, and to the increase of L,+ X, we have, for n large enough, PF n u η n, so P F n < n 1 Note that each event E n belongs to the σ-field σvs Vv n,v n s v n+1, Xt,H2v n t minτx n,v n,τx n,v n+1, in other words, it only depends on the diffusion between timesh2v n andminτx n,v n,τx n,v n+1 and on the environment between positions v n and v n+1. From the Markov property and the independence of the increments of the environment, we get that the events E n n 1 are independent. Combining this independence with and the Borel-Cantelli Lemma we get that P-almost surely, the event E n is realized infinitely many often. Combining , and the Borel-Cantelli Lemma we get that P-almost surely the event B n F n is realized for all large n. We deduce that P-almost surely, the event B n C n D n F n+1 is realized infinitely many often. Then, for n such that this event is realized we have, L,+ X u n+1 u n+1 /loglogu n+1 L,+ X Hv n+1 1/κ u n+1 /loglogu n+1 1/κ L,+ X H2v n u n+1 /loglogu n+1 + L X τ n Xn,v n+1 1/κ u n+1 /loglogu n+1 1/κ aj +1+aJ, so lim inf t + L,+ X t 1+2aJ. t/loglogt 1/κ Now, letting a go to and combining with the finiteness of sup ],[ L X + see and we obtain so Theorem is proved. 253

265 5.3. ALMOST SURE BEHAVIOR WHEN κ > The limsup In this subsection we prove Theorem First, let us assume that and prove that lim sup t + ftl X t t According to Remark the condition n=1 Let us fix a > and define the events { A n := ft κ t dt < = ft κ dt < + is equivalent to t f2 n κ < sup t [2 n,2 n+1 ] ftl,+ X t t 1/κ From the increase of L,+ X., 5.3.7, 5.3.4, 5.3.5, and 5.3.1, we have PA n P L,+ X 2n+1 2 n/κ a/f2 n a H2 n+1 /m+2 αn+1 2 n/κ a/f2 n +2 ηn+1 P L,+ X = P M 2 2 n+1 /m+2 αn+1 2 n/κ a/f2 n +2 ηn+1 = P sup Z 2 n/κ a/f2 n [,2 n+1 /m+2 αn+1 ] 1 e 2 n+1 /Qm+2 αn+1 /Q+2 n+1 /m+2 αn+1 7/8 n +2 ηn n+1 /m+2 αn+1 η } +2 ηn+1. supξ>2 n/κ a/f2 n 2 n+1 /Qm+2 αn+1 /Q+ 2 n+1 /m+2 αn+1 7/8 n supξ > 2 n/κ a/f2 n +2 ηn n+1 /m+2 αn+1 η. According to 5.3.2, the first term in the right hand side is equivalent to 2 1+κ Γκκ 2 Kf2 n κ /ma κ which is the general term of a convergent series, according to The two remaining terms are also the general terms of convergent series so we get PA n < +, n 1 and applying the Borel-Cantelli lemma we deduce lim sup t + ftl,+ X t t a. Now, letting a go to and combining with the finiteness of sup ],[ L X + see and we obtain

266 5.3. ALMOST SURE BEHAVIOR WHEN κ > 1 Let us now assume that + 1 lim sup t + ft κ dt = + and prove that t ftl X t t According to Remark the condition n=1 = ft κ dt = + is equivalent to t f2 n κ = Let M >, u n := 2 n /m 2 αn and X n := XH 2u n +., the diffusion shifted by the hitting time of 2u n. Note that from the Markov property for X at time H 2u n and the stationarity of the increments of V, X n 2u n is equal in law to X under the annealed probability P. We take n so large such that 2u n < u n+1 and define the events { f2 n+1 L X C n := τ } n Xn,u n+1 M, 2 n+1/κ D n := {τ X n,u n+1 < τ X n,u n }, E n := C n D n, F n := {L XHu n L X2 n }. From the equality in law between X n 2u n and X under P, 5.3.4, 5.3.5, and 5.3.1, we have PC n = P L X Hu n+1 2u n 2 n+1/κ M/f2 n+1 P L,+ X Hu n+1 2u n 2 n+1/κ M/f2 n+1 = P M 2 u n+1 2u n 2 n+1/κ M/f2 n+1 = P sup Z 2 n+1/κ M/f2 n+1 [,u n+1 2u n] 1 e u n+1 2u n/q u n+1 2u n n 7/8 supξ>2 n+1/κ M/f2 n+1 u n+1 2u n η According to and the definition of u n, the exponent in the right hand side is equivalent to 2 22 κ 1 Γκκ 2 Kf2 n+1 κ /mm κ. Sincef converges toat infinity, the above is also an equivalent for the term1 e... in the right hand side of Then, combining with and the fact that the other term is the general term of a covering series we get PC n = n 1 255

267 5.4. SOME LEMMAS so Reasoning as in the proof of we can prove that P D n < +, n 1 n 1PE n n 1PC n n 1 P D n = According to 5.3.7, and to the increase of L X, we also prove that P F n < n 1 The eventse n are independent since for eachn,e n belongs to theσ-fieldσvs Vu n,u n s u n+1, Xt,H 2u n t H 2u n +minτx n,u n,τx n+1,u n+1. Combining this independence with and the Borel-Cantelli Lemma we get that P-almost surely, the event E n is realized infinitely many often. Combining with and the Borel-Cantelli Lemma we get that P-almost surely, the event C n D n F n+1 is realized infinitely many often. Then, for n such that this event is realized we have, so f2 n+1 L X2 n+1 /2 n+1/κ f2 n+1 L XHu n+1 /2 n+1/κ lim sup t + f2 n+1 L X n τ Xn,u n+1 /2 n+1/κ M, ftl X t t M. Now, letting M go to infinity we get so Theorem is proved. 5.4 Some lemmas In this section, we justify some technical facts and lemmas for V, V and the diffusion in V. Some of them are known or can be easily obtained from results of [74], and we give some details for their justification when it is necessary. Some of these facts are new, like the approximation of the contributions of the valleys to the traveled distance by an iid sequence Properties of V, V and ˆV Lemma Lemma 5.4 of [74] There are two positive constants c 1,c 2 such that y,r >, P τv,], y] > r e c 1y c 2 r. 256

268 5.4. SOME LEMMAS Lemma There are positive constants c 1,c 2 such that, t,a >, P sup V > a [t,+ [ Démonstration. Let us choose γ ],κ[, we have P sup V > a [t,+ [ P Vt > 2a+P P e γvt > e 2γa +P = e 2γa E [ e γvt] +e κa, e c 1a c 2 t +e κa. Vt 2a, sup [,+ [ sup Vt+. Vt > a [,+ [ Vt+. Vt > a where we used Markov s inequality for the first term and the Markov property at timetfor the second term, together with the fact that the supremum ofv on[,+ [ follows an exponential distribution with parameter κ. Since E[e γvt ] = e tψ V γ and Ψ V γ < because < γ < κ, we get the result with c 1 := 2γ and c 2 := Ψ V γ. Lemma Lemma 5.3 of [74] There is a positive constant C such that + P e Vu du x x + Cx κ. We now state some Lemmas aboutv. First, we recall howv and ˆV are defined. V being spectrally negative, the Markov family Vx,x may be defined as in [8], Section VII.3. For any x, the process Vx must be seen as V conditioned to stay positive and starting from x. We denote V for the process V. It is known that Vx converges in the Skorokhod space to V when x goes to. Also, as well as V, V has no positive jumps so it reaches every positive level continuously. Since V is spectrally negative and not the opposite of a subordinator, it is regular for ],+ [ see [8], Theorem VII.1, so ˆV is for ],[. Moreover, ˆV drifts to +. We can thus define the Markov family ˆV x, x as in Doney [31], Chapter 8. It can be seen from there that the processes such defined are Markov and have infinite life-time. If moreover V has unbounded variation then ˆV is regular for ],+ [, and from Theorem 24 of [31], we have that ˆV, that we denote by ˆV, is well defined. Here again, for any x, the process ˆV x must be seen as ˆV conditioned to stay positive and starting from x. Note that, since ˆV converges almost surely to infinity, for x >, ˆV x is only ˆV x conditioned in the usual sense to remain positive. 257

269 5.4. SOME LEMMAS Lemma Lemma 5.7 of [74] There are two positive constants c 1,c 2 such that, for all 1 < a < b, we have P inf V b < a c 2 e c1b a. [,+ [ Lemma There are positive constants c 3,c 4,c 5,c 6 such that, x < y, r >, P τv x,y > r e c 3y c 4 r, z,r >, P KV z,z > r e 2c 3z c 4 r +c 6 e c 5z Démonstration. Let us fix x < y. From the Markov property applied atτv,x, the hitting time of x by V, we have P τv x,y > r = P τv τv,x+.,y > r P τv,y > r, so follows from Lemma 5.6 of [74]. For the second point, we have { τv z,2z r } { } inf V [τvz z > z { KVz,z r },,2z,+ [ so taking the complementary, P KV z,z > r P τv z,2z > r +P e 2c 3z c 4 r +P inf V 2z z [,+ [ inf [τvz,2z,+ [, V z z where, for the first term, we used with x = z, y = 2z and, for the second term, we used the Markov property at time τvz,2z. Combining with Lemma applied with a = z, b = 2z, we get Lemma There is a positive constant c such that for t large enough, Démonstration. KV,h t/2 τv,h t/2 P + e V x dx e ht/4 e cht. τv,h t/2 e V x dx KV,h t /2 τv,h t /2 L = KV h,h t/2 t/2 sup e V h t /2, [,KV h t /2,ht/2] sup e V [τv,h t/2,kv,h t/2] 258

270 5.4. SOME LEMMAS where we used the Markov property at time τv,h t /2 for the equality in law. We thus get KV,h t/2 P e V x dx e ht/4 /2 P KV h,h t/2 t/2 e ht/8 /2 τv,h t/2 +P inf V h 3h [,+ [ t/2 t/8 e c 3h t c 4 e h t /8 /2 +c 6 e c 5h t/2 +c 2 e c 1h t/8, where, for the first term, we applied with z = h t /2, r = e ht/8 /2 and, for the second term, we applied Lemma with a = 3h t /8, b = h t /2. Then, according to Corollary 19.VI of [8] we have + KV,h t/2 + e V x dx = L e ht/2 e V x dx = e ht/2 IV, and, according to Theorem 1.3 of [72], IV admits some finite exponential moments, so in particular it has finite expectation. We thus get + P e V x dx e ht/4 /2 = P IV e ht/4 /2 2e ht/4 E [ IV ]. KV,h t/ The result follows from the combination of and The next fact gives the law of the bottom of the valleys in terms of the laws of V and ˆV. It is a combination of Propositions 3.2 and 3.6 of [74]. Fact Assume V has unbounded variation. For all i 1 let For all i 1 we have P i := V i m i x, x m i τ i h t P i := Ṽ i m i x, x m i τ i h t. d VT Pi,P 2 2e δκht/ where d VT is the total variation distance. Moreover, the law of P 2 is absolutely continus with respect to the law of the process ˆV x, x τˆv,h t + and has density c ht /1 e κˆv τˆv,h t+ with respect to this law, where c ht is a constant increasing with h t and converging to 1 when t and hence h t goes to infinity. 259

271 5.4. SOME LEMMAS For all i 1, the two processes Ṽ i m i x, x m i τ i h t = P i and Ṽ i m i +x, x τ i h t m i are independent and the second is equal in law to V x, x τv,h t. Let us now recall a fact from from [74] : Fact Assume that the hypotheses of Theorems and are satisfied. Fix ǫ small enough. There is a positive constant c depending on ǫ such that for all t large enough τ j ht/2 j 1, P e Ṽ ju du e ǫht 1 e cht, j 1, P j 1, P j 1, P L j 1 Lj τ + j ht/2 e Ṽ j u du e ǫht sup u [ τ j ht/2, τ+ j ht/2] 1 e cht, A j u/a j L j e ht/3 1 e cht, sup A j u e 5h t/8 u [ τ j ht/2, τ+ j ht/2] where A j is defined in Subsection e cht, Démonstration , and are respectively Lemma 4.6, Lemma 4.7 and Lemma 4.8 applied with ǫ = 1/6 from [74]. For 5.4.9, note that P sup A j u e 5h t/8 u [ τ j ht/2, τ+ j ht/2] P A j τ j h t/2 A j τ + j h t/2 e 5ht/8. Then, A j τ j h t/2 and A j τ + j h t/2 can be bounded as in the proof of Lemma 4.8 of [74] which yields Contribution of the valleys to the traveled distance We need to approximate the contribution to the distance traveled by the diffusion in each valley. We have : Proposition On an enlarged probability space, there is an iid sequenced t j j 1, independent from the sequence e j,s t j,r t j j 1 and such that for t large enough, j 1, P Lj L j 1 Dj t e c 1h t Dj t 1 e c 2h t and Dj t L Ee κht q, where q is the constant in Theorem 1.4 of [74] and c 1,c 2 are positive constants. 26

272 5.4. SOME LEMMAS Démonstration. From the definitions of valleys we have L j L j 1 = L j L j 1 + m j L j + τ jh t m j + L j τ j h t Recall the definition of m h t from [74] : τ h t := inf { } { } u, Vu inf V = h t, m h t := inf u, Vu = inf V. [,u] [,τ h t] From the Markov property at the stopping times L j 1, L j and τ jh t, and Fact we get that the terms in the right hand side of are respectively equal in law to τv,], e 1 δκht ], m h t, τv,h t and τv,], h t /2]. For the first and fourth term, applying Lemma with y = e 1 δκht and r = e 1 δ/2κht /2 we get for t large enough, j 1, P L j L j 1 + L j τ j h t > e 1 δ/2κht e ht For the third term, applying with x =, y = h t and r = e 1 δ/2κht we get for t large enough, j 1, P τ j h t m j > e 1 δ/2κht e ht The main term is the second one, its law is given by Lemma 3.9 of [74]. Before studying it, let us recall some of the notations used in [74]. LetF denote the space of excursions, that is, càd-làg functions from[,+ [ tor, starting at zero and killed at the first positive instant when they reach this instant can possibly be infinite. For ξ F, let us denote ζξ := inf{s >, ξs = } for the length of the excursion ξ. For h >, let F h, and F h,+ denote respectively the set of excursions which height is strictly less than h and the set of excursions higher than h : { } { } F h, := ξ F, supξ < h [,ζ], F h,+ := ξ F, supξ h [,ζ] Let N be the measure defined on F as in Subsection 3.5 of [74]. Let S h, and S h,+ be two independent pure jumps subordinators with Lévy measures respectively ζnf h,. and ζnf h,+., the image measures of respectively NF h,. and NF h,+. by ζ. The sum of these measures equals ζn so S := S h, +S h,+ is a pure jumps subordinator with Lévy measure ζn. Let also T h be an exponential random variable with parameter NF h,+, independent from S h, and S h,+. According to Lemma 3.9 of [74] we have j 1, m j L j L = S ht, T ht.. 261

273 5.4. SOME LEMMAS Recall that the sequence m j L j j 1 is iid because of Remark Therefore, on an enlarged probability space, there is an iid sequence S ht, j,t ht,j j 1 such that for all j 1, S ht, j T ht,j, S ht, j L = S ht,, T ht,j L = T ht and S ht, j T ht,j = m j L j Let d := NF ht,+t ht and more generally d j := NF ht,+t ht,j. Then, d j L E1 and the sequence d j j 1 is iid. For large t, it is natural to approximate S ht, T ht by a multiple of d, for this we write S ht, T ht e κht d/q = S ht, T ht ST ht +ST ht E[S1]T ht + E[S1]T ht e κht d/q For the first term, using the expression of S in terms of S ht,+ and S ht,, the independence between S ht,+ and T ht, the definition of S ht,+, Cauchy-Schwarz s inequality and the definition of S, we have E [ S h t, T ht ST ht ] = E [ S ht,+ T ht ] = = = 1 NF ht,+ F ht,+ 1 NF E[ S ht,+ 1 ] ht,+ ζξndξ 1 NFht,+ ζ 2 ξndξ, 1 NFht,+ VarS VarS1 is indeed finite thanks to Lemma 3.1 of [74]. For the second term in the right hand side of , using Cauchy-Schwarz s inequality and the independence between S and T ht, E[ ST ht E[S1]T ht ] E [ ST ht E[S1]T ht 2] = 1 NFht,+ VarS1. For the third term in the right hand side of , using the definition of d, E[S1]T ht e κht d/q = E[S1]/NF ht,+ e κht /qd. Then, recall from Proposition 3.12 and Lemma 3.11 of [74] that and q := NF 1,+ e κ E[e κv 1τV 1,],] ]/E[S1], NF ht,+ = e κht NF 1,+ e κ E [ e κv 1τV 1,],] ] + O t + e 2κht

274 5.4. SOME LEMMAS We thus get a.s. E[S1]T ht e κht d/q = d O t + e κht Let us define D t := e κht d/q and more generally Dj t := e κht d j /q. We then have indeed Dj t L Ee κht q and the fact that the sequence sequence Dj t j 1 is iid becomes a.s. E[S1]Tht D t D t Ce 2κht, where C is a positive constant. Using Markov s inequality, and , and the asymptotic of NF ht,+ given by Lemma 3.11 of [74], we get the existence of a positive constant C such that for t large enough, P S h t, T ht ST ht + STht E[S1]T ht e 6κht/1 Ce κht/1. P D t e 9κht/1 = P d qe κht/1 qe κht/1. As a consequence there is a positive constant C such that for h t large enough, P S h t, T ht ST ht + STht E[S1]T ht e 3κht/1 D t Ce κht/1. Combining with and putting into we get P S h t, T ht D t e 2κh t/1 D t Ce κht/1. Combining with we get j 1, P mj L j Dt j e 2κht/1 Dj t Ce κht/ Then, we have PDj t e 1 δ/2κht = Pd j qe δκht/2 qe δκht/2 so putting into and we get for all j 1 P L j L j 1 + τ j h t m j + L j τ j h t > 2e δκht/2 Dj t 2e ht +qe δκht/ Combining and 5.4.2, and putting into we get the result for t large enough Proof of some facts and lemmas This subsection is devoted to the justification of the facts stated in Subsection 5.2.1, which mainly come from [74]. As these results are included in Section 5.2, we prove them under the hypotheses of Theorems and : < κ < 1, V has unbounded variation and there exists p > 1 such that V1 L p. 263

275 5.4. SOME LEMMAS In the facts and lemmas considered here, the value of the constant c is not important as long as it is positive so we allow it to decrease from line to line and it is implicit that the estimates are true for t large enough. Démonstration. of Lemma is only Lemma 4.16 of [74]. For 5.2.3, note that{n t ñ t } = {Hmñt+1 > t} and H mñt+1 = Hmñt+1 on {V V t } since ñ t < n t. Let t be large enough so that 1 2/logh t 1+e cht 1 1/2. Using the definitions of Et 5 and Et 7 which is possible here since ñ t < n t we get that P{N t ñ t } {V V t } Et 5 Et 7 is less than P ñt H mñt+1 > t, Et, 5 Et 7 P e i SiR t i t t/2 P sup e i SiR t i t t/2ñ t 1 i ñ t =1 [ 1 P e 1 S t 1R t 1/t 1/2ñ t ]ñt. i=1 = 1 [ P e 1 S t 1R t 1/t < 1/2ñ t ]ñt According to applied with some η ],1 ρ/3[ we have that P e 1 S1R t 1/t t 1/2ñ t t + 2κ C E[R κ ]ñ κ te κφt = 2 κ C E[R κ ]e κρ 1φt. Since ρ < 1 the later converges to so [ 1 P e1 S1R t 1/t t ]ñt 1/2ñ t exp 2 κ C E[R κ ]ñ t e κρ 1φt t + = exp 2 κ C E[R κ ]e ρ+κρ 1φt. Let ρ := ρ+κρ 1. ρ is positive thanks to the hypothesis ρ ],κ/1+κ[ so e ρ+κρ 1φt converges to and we deduce that for t large enough, P{N t ñ t } {V V t } E 5 t E 7 t 2C E[R κ ]e ρ φt. Combining with the bounds for PV / V t and PE 5 t+pe 7 t given by respectively Fact and we get Now, let us recall a fact about brownian local time : Fact There is a positive constant c such that for all t large enough P sup L B τb,1,y L B τb,1, > e h t/9 L B τb,1, y e h t /3 P sup L B σ B 1,,y 1 e h t/2 y e h t /8 P supl B σ B 1,,y u y R e cht, e ht, /u,

276 5.4. SOME LEMMAS Démonstration is 7.11 of [4] applied with δ = e ht/3, ǫ = e ht/ comes from the second Ray-Knight Theorem combined with estimate 7.15 of [4] applied with u = e ht/2, v = e ht/ comes from the second Ray-Knight Theorem combined with estimate 7.16 of [4]. Démonstration. of Fact According to the combination of Lemmas 5.2 and 5.21 of [74], there is a positive constant c such that PE 1 t e cht. Since h t is ultimately greater than φt we get PE 1 t e cφt. PE 2 t e cφt and PE 3 t e cφt come from Fact 4.4 of [74] respectively the second and third point. For PE 4 t. Let us fix j 1 and use the Markov property at H m j and the expression of the local time within an hitting time. We get the existence of a Brownian motion Bs, s, independent from Ṽ j, such that y D j, L X H L j,y L X H m j,y = e Ṽ j y L B τb,a j L j,a j y, where A j is defined in Subsection Let B := B j A j L j 2./A j L j. By the scaling property for brownian motion, we have that, conditionally to V, B is a standard Brownian motion. We have y D j, L X H L j,y L X H m j,y = e Ṽ j y A j L j L Bτ B,1,A j y/a j L j. From the definition of D j we have D j [ τ j h t/2, τ + j h t/2] so, thanks to we have A P sup j y/a j L j e ht/3 1 e cht. y D j Combining with estimate we get P sup y D j L Bτ B,1,A j y/a j L j > 1+e ht/9 L Bτ B,1, so with probability greater than 1 e cht we have L X H L j, m j sup L X H L j,y L X H m j,y y D j 1+e ht/9 A j L j L Bτ B,1, = 1+e ht/9 L X H L j, m j. e cht Since c > has been chosen "small enough" in the Introduction, we can assume c 1/9 so PE 4 t e cht e cφt follows from the above and the fact that h t is ultimately greater than φt. 265

277 5.4. SOME LEMMAS PE 5 t e cφt comes from Fact 4.3 of [74]. c > has been fixed "small enough" in the Introduction. We can assume that it was chosen so small such that Proposition 4.5 of [74] apply with c instead the constant ǫ/7 there. PE 6 t e cht e cφt and PE 7 t e cht e cφt thus come from Proposition 4.5 of [74] and the fact that h t is ultimately greater than φt. The second point of the fact is Lemma 4.11 of [74]. Démonstration. of Lemma According to and Chernoff inequality we have for any j 1 We have P R t j e ǫht/4 P P R t j e ht/8 e λ e h t /8 / R tj e h1/3 t 2P IV 2e h1/3 t +2e δκht/3 1/3 2e K e h t /2 +2e δκht/3, where, for the last two inequalities, we used Lemma applied with z t = e h1/3 t, a = 2 and 5.1.7, and where K is the constant in The term 2e δκht/3 is only necessary in the case V = W κ because of Combining and we get that is satisfied with probability at least 1 e cht. According to Fact we have that 5.2.8, 5.2.9, and are satisfied with probability at least 1 e cht. According to we have P 7 i=1et i e Lφt. Then, e Lφt/2 P P V 7i=1E it [ ] > e Lφt/2 E P V 7 i=1et i = P 7 i=1et i e Lφt, We thus deduce that is satisfied with probability at least 1 e Lφt/2. Combing all this with Fact 5.2.3, we get Lemma for a suitably chosen constant c. Démonstration. of Fact The ideas are very similar to the ones used for the proof of Lemma 5.3 of [4]. Since we are in the more general context of a Lévy potential and since we do not prove exactly the same thing here, we work conditionally to the environment, we give the details. Recall that ǫ > has been fixed in the definition of G t. c > has been fixed "small enough" in the Introduction and we already fixed some constraints about how small it must be in the proof of Fact We can assume further that it was chosen so small such that c < min1/2,ǫ/4. Let us fix an environment v G t, z [,1 4/logh t ] andk n t. We put x t := 1+e cht 1 x t,z := 1 z/1 e cht R t k. 266

278 5.4. SOME LEMMAS To study E 8 tv,k,z, we look at its intersections with the events {σ X mk t x t, m k < H X mk L k } and {σ X mk t x t, m k > H X mk L k }. We have Etv,k,z {σ 8 X mk t x t, m k < H X mk L k } { } = tz t x t, supl X mk t1 z,. tx t, σ X mk t x t, m k < H X mk L k < H X mk L k 1. D k On the above event, σ X mk tz, m k is finite and the diffusion stays in [ L k 1, L k ] until this time. σ X mk tz, m k is thus equal to I := Lk L k 1 e ṽk x L B σ B tz,,a k xdx, where B is the brownian motion driving the diffusion X mk and A k is defined in Subsection Let B := BtZ 2./tZ and Ãk. := A k./tz. From the scaling property B is still a brownian motion. We can write I = tz Lk L k 1 e ṽk x L Bσ B1,,Ãk xdx Since v G t it satisfies Recall also that t e ht. For t large enough so that 4e ht/8 /1 e cht logh t e ht/4 we thus have tz e 3ht/4 for any choice of z [,1 4/logh t ]. Combining this with the fact that v satisfies and we get sup [ τ k ht/2, τ+ k ht/2] Combining with we get P v sup x [ τ k ht/2, τ+ k ht/2] Ãk. A k τ k h t/2 A k τ + k h t/2 /tz e ht/8. L Bσ B1,,Ãk x 1 e ht/2 e ht, and we deduce that τ + P v k ht/2 e ṽk x L Bσ B1,,Ãk xdx R t τ k ht/2 k e ht/2 Rk t 1 e ht Since v G t it satisfies and Combining with applied with 267

279 5.4. SOME LEMMAS u = e ǫht/2 and the lower bound for R t k P v Lk τ P v k ht/2 in we get τ + k ht/2 e ṽk x L Bσ B1,,A k xdx e ǫht/2 e ǫht/4 R t k L k 1 e ṽk x L Bσ B1,,A k xdx e ǫht/2 e ǫht/4 R t k 1 e cht, e cht Now putting 5.4.3, and into and combining with the fact that c < min1/2, ǫ/4 we get P v E 1 neg := { I tzr t k > e ch t tzr t k} e ch t. Combining with the definition of Z we see that t1 z I on the complementary of E 2 neg. Combining this with we get that E 8 tv,k,z {σ X mk t x t, m k < H X mk L k } is included into { tz t x t, supl X mk σ X mk tz, m k,. tx t, D k } σ X mk t x t, m k < H X mk L k < H X mk L k 1 Eneg On the main event, σ X mk tz, m k is finite. On this event, L X mk σ X mk tz, m k,y is thus equal to Ly := e ṽk y L B σ B tz,,a k y. Here again, for B := BtZ 2./tZ and Ãk. := A k./tz. We have Ly = tze ṽk y L Bσ B1,,Ãk y. Since D k [ τ k h t/2, τ + k h t/2] we can apply together with c < 1/2 and get P v E 2 neg := { y D k, Ly 1+e cht tz } e cht On the big event in we thus have both 1 + e cht tz > tx t and tz t x t, expect possibly on the event E 2 neg. Since these two inequalities are not compatible we get P E v tv,k,z {σ 8 X mk t x t, m k < H X mk L k } P v Eneg 1 +P v Eneg 2 2e ch t

280 5.4. SOME LEMMAS We now study the case where σ X mk t x t, m k > H X mk L k. First, we have Etv,k,z {σ 8 X mk t x t, m k > H X mk L k } { } = tz t x t, supl X mk t1 z,. tx t, H X mk L k < H X mk L k 1 σ X mk t x t, m k D { k } supl X mk t1 z,. tx t, L X mk H X mk L k, m k < t x t D { k } supl X mk H X mk L k,. tx t, L X mk H X mk L k, m k < t x t Et. 1 D k Indeed,sup Dk L X mk t1 z,. sup Dk L X mk H X mk L k,. on{t1 z H X mk L k } and sup Dk L X mk t1 z,. = sup Dk L X mk H X mk L k,. on {H X mk L k < t1 z} E 1 t. Since we are dealing with X mk, the diffusion shifted at time H m k, we can see that the main event above is included in E 4 t. We thus get E 8 tv,k,z {σ X mk t x t, m k > H X mk L k } E 1 t E 4 t follows easily from the combination of and applied with z = 1 4/logh t. Then, the right hand sides of and do not depend on z which is arbitrary in [,1 4/logh t ] and H m k /t is independent from X mk. We can thus replace z by H m k /t in and at least on {N t k, H m k /t 1 4/logh t }. Using the combination of and to study the union of events in we get P v nt k=1 {N t k, H m k /t 1 4/logh t } E 8 tv,k,h m k /t 2n t e cht +P v E 1 t E 4 t Since v G t it satisfies We thus have P v E 1 t E 4 t e Lφt/2 where L is the constant defined in Fact and from the definition of h t and φt we have easily 2n t e cht e cφt for large t. For t large enough we thus get We now prove Let us fix an environment v G t, z [,1] and k n t. Inversing the local time in the definition ofetv,k,z 9 we get thatetv,k,z 9 coincides with the event { Rk t/x t < 1 e cht 1 z, σ X mk t/x t, m k H X mk L k t1 z, } H X mk L k < H X mk L k We have to distinguish the cases H X mk L k > σ X mk t/x t, m k and σ X mk t/x t, m k > H X mk L k. On Etv,k,z 9 {σ X mk t x t, m k < H X mk L k }, σ X mk t/x t, m k is finite and the diffusion stays in [ L k 1, L k ] until this time. On this event, σ X mk t/x t, m k is thus equal to I := Lk L k 1 e ṽk x L B σ B t/x t,,a k xdx,. 269

281 5.4. SOME LEMMAS where, as in , B is the brownian motion driving the diffusion X mk. Now, let B := Bt/x t 2./t/x t and Ãk. := A k./t/x t. From the scaling property B is still a brownian motion. We can write Lk I = t/x t e ṽk x L Bσ B1,,Ãk xdx. L k 1 From t e ht, the definition of x t and the definition of h t we see that t/x t e 3ht/4 for t large enough. Recall also that v G t implies that v satisfies and We thus get sup Ãk. A k τ k h t/2 A k τ + k h t/2 /t/x t e ht/8, [ τ k ht/2, τ+ k ht/2] We can now proceed as in the proof of to get analogues of 5.4.3, and We deduce that P v Eneg 3 := { I t/x t Rk t > e cht t/x t Rk} t e ch t. Now, note from that E 9 tv,k,z {1 e cht σ X mk t/x t, m k tr t k /x t}. Since σ X mk t/x t, m k = I on E 9 tv,k,z {σ X mk t/x t, m k < H X mk L k } and 1 + e cht 1 e cht < 1 we deduce that E 9 tv,k,z {σ X mk t/x t, m k < H X mk L k } E 3 neg. Then, P E v tv,k,z {σ 9 X mk t/x t, m k < H X mk L k } e cht We now study the case where σ X mk t/x t, m k > H X mk L k, following the ideas of the proof of Lemma 5.3 of [4]. Recall that B is the brownian motion driving the diffusion X mk. We have σ X mk t/x t, m k > H X mk L k σ B t/x t, > τb,a k L k t/x t > L B τb,a k L k, Now, let B := BA k L k 2./A k L k. From the scaling property B is still a brownian motion. L B τb,a k L k, = A k L k L B τ B,1,. Note that from the definition of e k given in Subsection we have L B τ B1, = e k. As a consequence, σ X mk t/x t, m k > H X mk L k t/x t > A k L k e k tr t k/x t > A k L k e k R t k. According to the trivial inequality A k L j = Lj m j e vu v m j du Lj τ + j ht/2 e vu v m j du = S t j 27

282 5.4. SOME LEMMAS and the definition of Et 7 we deduce that { } { } σ X mk t/x t, m k > H X mk L k trk/x t t > 1+e cht 1 H X mk L k Now, note from that E 9 tv,k,z {1 e cht H X mk L k tr t k /x t}. Since 1 + e cht 1 e cht < 1, the inequality tr t k /x t > 1 + e cht 1 H X mk L k is in contradiction with the event E 9 tv,k,z. We thus get E 7 t E 9 tv,k,z {σ X mk t/x t, m k > H X mk L k } E 7 t Since the right hand sides of and do not depend on z which is arbitrary in [,1] and H m k /t is independent from X mk, we can replace z by H m k /t in and at least on {N t k}. We can thus use the combination of and to study the union of events in We get P v nt k=1 {N t k} E 9 tv,k,h m k /t n t e cht +P v E 7 t Since v G t it satisfies We thus have P v E 7 t e Lφt/2 where L is the constant defined in Fact and from the definition of h t and φt we have easily n t e cht e cφt for large t. For t large enough we thus get Démonstration. of Fact and are included into Lemma 4.15 of [74] while comes from Proposition 3.8 of [74] applied with h = h t. Démonstration. of Lemma First, note that from and the definition of n t just after, we have 2t κ e κδloglogtω /q = 2n t e κht /q. From the definition ofn t, we know that on{v V t }, at time t, m Nt+1 has never been reached by the diffusion and neither L Nt+1 because L Nt+1 > m Nt+1, we thus have P supx 2n t e κht /q [,t] P LNt+1 2n t e κht /q +PV / V t P Lnt 2n t e κht /q +PV / V t +PN t n t n t P 1+e c 1h t Dj t 2n t e κht /q +e cht, where c is a positive constant, c 1 has the same meaning as in Proposition and t is large enough. For the last inequality, we used Proposition 5.4.9, Fact and [ ] e 1+e c 1 h t 1 n t E e qe κh tdj t/2 nt +e ch t = e log2 1+e c 1 h t 1 n t +e cht, j=1 271

283 5.4. SOME LEMMAS where we used Markov s inequality, the fact that the sequence qe κht D t j is an iid sequence of exponential random variable with parameter 1 and the expression of the Laplace transform for the exponential distribution. Since log2 < 1 and n t = e κ1+δloglogtω >> h t, we get the first point for t large enough. We now prove the second point. We first note that from and the definition of ñ t just after, we have t κ e ρ κloglogtω /2q = ñ t e κht /2q. On {V V t } {N t < n t } E 1 t we have Xt L Nt 1 so P Xt ñ t e κht /2q P LNt 1 ñ t e κht /2q +PV / V t +PN t n t +P Et 1 P Lñt ñ t e κht /2q +PV / V t +PN t ñ t +PN t n t +P Et 1 ñ t P 1 e c 1h t Dj t ñ t e κht /2q +e cφt j=1 where c is a positive constant, c 1 has the same meaning as in Proposition and t is large enough. For the last inequality we used Proposition 5.4.9, Fact 5.2.3, Lemma 5.2.1, and the fact that e cht e cφt for large t. [ e 1 e c 1 h t 1 ñ t E e 2qe κh tdj]ñt t +e cφt = e log3+1 e c 1 h t 1 ñ t +e cφt, where we used Markov s inequality, the fact that the sequence qe κht D t j j 1 is an iid sequence of exponential random variable with parameter 1 and the expression for the Laplace transform of the exponential distribution. Since log3 > 1 and ñ t = e ρloglogtω >> h t, we get the second point for t large enough. We now prove Note that for a fixed environment v V we have P v inf X r = P v H r < H+ = [,+ [ and note that r/2 r e Vx dx = r r/2 e V x dx L = + e vx dx + e r vx dx r r/2 e Vx dx, + e vx dx r/2 r e vx dx where the equality in law comes from the time-reverse property. Applying Lemma with t = r/2 and a = r we get r P e Vx dx r r/2 2 e r e c 1 r c2 r/2 +e κ r,

284 5.4. SOME LEMMAS where c 1 and c 2 are the constants in the lemma. Then, applying Lemma we get for r large enough, + P e Vx dx e r e κ r/ Putting and into we get that with P-probability greater than 1 e c 1 r c2 r/2 + e κ r + e κ r/2 : P V inf [,+ [ X r 2r 1 and it is bounded by 1 when this estimates fails so integrating on V with respect to P we get P inf [,+ [ X r 2r 1 +e c 1 r c2 r/2 +e κ r +e κ r/ follows for r large enough. Finally, is included in Lemma 5.18 of [74] Almost sure constantness of limsup and liminf We now use a classical argument involving Kolmogorov 1 law to justify the almost sure constantness of the limsup stated in Remark We here treat the case of the limsup with the renormalisation tloglogt. The same argument can be used with the liminf instead of the limsup, or with any of the other renormalisations used in the paper. We first fix v V, a realization of the environment. For any n N, the process X n := XτX,n+. is, according to the Markov property, a diffusion in the environment vn+. and it is independent from Xs, s τx,n. We have lim sup t + L X t tloglogt = limsup t + L X nt τx,n+tloglogτx,n+t = limsup t + L X nt tloglogt, where the first equality comes from the fact that P-almost surely the favorite site F t goes to +. Indeed, the diffusion P-almost surely converges to + and as we can see from the results of Subsection 5.1.1, L X t converges P-almost surely to infinity, these two facts imply the convergence of F t to +. As a consequence F t will become greater than n for t large enough which imply the first equality in The second equality comes from the equivalence when t goes to infinity between τx,n+tloglogτx,n+t and tloglogt. The limsup in the right hand side of belongs to the σ-field σx n t, t, it is thus independent from σxs, s τx,n. Since this is true for any n N we get, according to Kolmogorov 1 law, that the limsup is constant P v -almost surely. In other words, the limsup is only a deterministic function of the environment v, let us denote it by Lv. 273

285 5.4. SOME LEMMAS Let us fix v V and n N. Note that P v infx n n 1 > so the limsup still equals Lv on {infx n n 1}. As a consequence, Lv is only a function of vx+n 1 vn 1, x. If we consider the space V equipped with probability P, this implies that LV is independent from the σ-field σvx, x n 1. Since this is true for all n N we deduce from Kolmogorov 1 law that LV is constant P-almost surely. This proves that the limsup is in fact constant P-almost surely. In other words, λ [,+ ] such that P a.s. limsup t + L X t tloglogt = λ. 274

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293 Grégoire VÉCHAMBRE Fonctionnelles de processus de Lévy et diffusions en milieux aléatoires Résumé : Pour V un processus aléatoire càd-làg, on appelle diffusion dans le milieu aléatoire V la solution formelle de l équation différentielle stochastique dx t = 1 2 V X t dt+db t, où B est un mouvement brownien indépendant de V. Le temps local au temps t et à la position x de la diffusion, noté L X t,x, donne une mesure de la quantité de temps passé par la diffusion au point x, avant l instant t. Dans cette thèse nous considérons le cas où le milieu V est un processus de Lévy spectralement négatif convergeant presque sûrement vers, et nous nous intéressons au comportement asymptotique lorsque t tend vers l infini de L X t := sup RL X t,., le supremum du temps local de la diffusion, ainsi qu à la localisation du point le plus visité par la diffusion. Nous déterminons notamment la convergence en loi et le comportement presque sûr du supremum du temps local. Cette étude révèle que le comportement asymptotique du supremum du temps local est fortement lié aux propriétés des fonctionnelles exponentielles des processus de Lévy conditionnés à rester positifs et cela nous amène à étudier ces dernières. Si V est un processus de Lévy, V désigne le processus V conditionné à rester positif. La fonctionnelle exponentielle de V est la variable aléatoire + e V t dt. Nous étudions en particulier sa finitude, son auto-décomposabilité, l existence de moments exponentiels, sa queue en, l existence et la régularité de sa densité. Mots clés : Processus de diffusion, potentiel aléatoire, processus de Lévy, processus de renouvellement, temps local, processus de Lévy conditionné à rester positif, fonctionnelles exponentielles. Functionals of Lévy processes and diffusions in random media Abstract : For V a random càd-làg process, we call diffusion in the random medium V the formal solution of the stochastic differential equation dx t = 1 2 V X t dt+db t, where B is a brownian motion independent of V. The local time at time t and at the position x of the diffusion, denoted by L X t,x, gives a measure of the amount of time spent by the diffusion at point x, before instant t. In this thesis we consider the case where the medium V is a spectrally negative Lévy process converging almost surely toward, and we are interested in the asymptotic behavior, when t goes to infinity, of L X t := sup RL X t,., the supremum of the local time of the diffusion. We are also interested in the localization of the point most visited by the diffusion. We notably establish the convergence in distribution and the almost sure behavior of the supremum of the local time. This study reveals that the asymptotic behavior of the supremum of the local time is deeply linked to the properties of the exponential functionals of Lévy processes conditioned to stay positive and this brings us to study them. If V is a Lévy process, V denotes the process V conditioned to stay positive. The exponential functional of V is the random variable + e V t dt. For this object, we study in particular finiteness, self-decomposability, existence of finite exponential moments, asymptotic tail at and smoothness of the density. Keywords : Diffusion, random potential, Lévy process, renewal process, local time, Lévy processes conditioned to stay positive, exponential functionals. MAPMO UMR CNRS Rue de Chartres - BP ORLÉANS CEDEX