RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT. 1. Introduction

Transcription

1 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT PIERRE ANDREOLETTI, GRÉGOIRE VÉCHAMBRE, AND ALEXIS DEVULDER Abstract. We study a one-dimensional diffusion X in a drifted Brownian potential W κ, with < κ < 1, and focus on the behavior of the local times Lt, x, x of X before time t >. In particular we characterize the limit law of the supremum of the local time, as well as the position of the favorite sites. These limits can be written explicitly from a two dimensional stable Lévy process. Our analysis is based on the study of an extension of the renewal structure which is deeply involved in the asymptotic behavior of X. 1. Introduction Let Xt, t a diffusion in a random càdlàg potential V x, x R, defined informally by X = and dxt = dβt 1 2 V Xtdt, where β is a Brownian motion independent of V. Rigorously, X is defined by its conditional generator given V, 1 x d V x d ev e. 2 dx dx We put ourself in the case where V is a negative drifted brownian motion : V x = W κ x := W x κ 2 x, x R with < κ < 1 and W a two sided Brownian motion. We explain at the end of Section 1.1, what should be done to extend our results to a more general Lévy process. In our case, the diffusion X is a.s. transient and its asymptotic behavior was first studied by K. Kawazu and H. Tanaka : if Hr is the hitting time of r R by X Hr := inf{s >, Xs = r}, 1.1 Kawazu et al. [25] proved that, for < κ < 1 under the so-called annealed probability P, Hr/r 1/κ converges in law to a κ-stable distribution see also Y. Hu et al. [24], and H. Tanaka [33]. Here we are interested in the local time of X denoted Lt, x, x R, t > until an asymptotic instant t. For Brox s diffusion, when κ = it is proved in [4] that the process local time until the instant t re-centered at the localization coordinate b t see [1] converges in law, this allows the author to derive the law of the supremum of the local time before time t R +, L t := sup Lt, x. x R We recall their result below in order to compare it with what we obtain here : Date: 28/5/215 à 1:8: Mathematics Subject Classification. 6K37,6J55. Key words and phrases. Diffusion in a random potential, local time supremum, favorite sites. 1

2 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 2 Theorem 1.1. [4] If κ =, then with R κ := + L t lim = 1, t + t R κ e W κ x dx + + e W κ x dx. 1.2 W κ x, x and W κ, x are two independent copies of the process W κ x, x Doob-conditioned to remain positive. Extending their approach, and following the results of Shi [3], Diel [15] obtains the non-trivial normalisations for the almost sure behavior of the lim sup and lim inf of L t. Notice that these results have been previously established for the discrete analogue of X, the so called Sinai s random walk in [13] and [22]. One of our aim in this paper is to extend the study of the local time in the case < κ < 1, and deduce from that the weak asymptotic behavior of L t. Before going any further let us recall to the reader what is known for the slow transient cases. When the time and space are discrete see [26], for the seminal paper, a result of Gantert and Shi [23] states the almost sure behavior for the lim sup of the supremum of the local time L S n of these random walks denoted S, before time n : a.s. lim sup n + L S n/n = c >. Contrarily to the recurrent case [22] their method, based on a link with the local time of S and a branching process in random environment, is not able to determine the law of the limit of L S n/n. For the continuous time and space case we are treating here, the only paper dealing with L t is Devulder s work [14], who proves that the lim sup t + L t/t = + almost surely. But once again his method can not be used to characterize the limit law of L t/t. 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 like for Brox s diffusion. Second we make a first step on a specific way to study the local time which could be used in estimation problems with random environment [1], [2], [5], [12], [11], [2], [6]. The method we develop here is an improvement of the one used in [3] about the localization of X t for large t. Let us recall the main result of this paper, for that we introduce some new objects. First the notion of h-extrema, with h >, introduced by Neveu et al. [27] and studied more specifically in our case of drifted Brownian motion by Faggionato [19]. For h >, we say that x R is a h-minimum for a given continuous process V if there exist u < x < v such that V y V x for all y [u, v], V u V x + h and V v V x + h. Moreover, x is an h-maximum for V if x is an h-minimum for V, and x is an h-extrema for V iff it is an h-maximum or an h-minimum for V. As we are interested in the process X until time t, we only focus on h t -extrema of W κ where h t := log t φt, with < φt = olog t, log log t = oφt. It is known see [19] 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. Define N t := max { k N, sup Xs m k }, s t the number of h t -minima visited by X until the instant t, we then have the following result

3 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 3 Theorem 1.2. [3] Assume < κ < 1. There exists a constant C 1 >, such that lim P Xt m Nt C 1 φt = 1. t + This result proves that the process X before the instant t visits a sequence of h t -minima, and then gets stuck in an ultimate one. Notice that this result was proved in the discrete case model by Enriquez-Sabot-Zindy in [17]. This phenomenon is due to two facts : the first one is the appearance of a renewal structure which is composed of the time it takes to the process to move from a h t -minima to the following one. The second is the fact that like in Brox s case, the process is trapped a significant amount of time in the neighborhood of the local minima m Nt. It is the extension of this renewal structure to the sequence of local time at the h t -minima that we study here. We now detail our results Results. Let us introduce a few notations involved in the statement of our results. Denote D[, +, R 2, J 1 the space of càdlàg functions 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 with value in R + R + which is a pure positive jump process with κ-stable Lévy measure ν given by x >, y >, ν [x, + [ [y, + [ = 1 y κ E [R κ κ 1 Rκ y x where R κ is defined in 1.2. For a given function f in D[, +, R, define for any s >, a > : ] + 1 x κ P R κ > y, x f s := sup fr fr, f 1 a := inf{x >, fx > a}, r s f s is the largest jump of f before instant s, f 1 a is the first time f is larger than a. Also define the couple of random variables I 1, I 2 I 1 := Y 1 Y 1 2 1, I 2 := 1 Y 2 Y2 1 1 Y 1Y2 1 1 Y 1Y2 1 1 Y 2 Y2 1 1 Y 2Y We are now ready to state the result, the convergence in law denoted L takes place when t goes to infinity : Theorem 1.3. L t t L I = maxi 1, I 2. There is an intuitive interpretation of this theorem which explains the appearance of the Lévy process Y 1, Y 2. We focus on this interpretation now. First for any s >, Y 1 s is the limit of the sum of the first se κφt normalised by t local times taken specifically at the se κφt first h t -minima. Y 2 plays a similar role but for the exit time of the se κφt first h t -valleys. Where an h t -valley is defined as a large neighborhood of an h t -minima, see Section 2.2 for a rigorous definition as well as Figure 1. So by definition I 1 is the largest jump of the process Y 1 before Y 2 is larger than 1 and can be interpreted as the largest local time re-normalized among the local time at the h t -minima visited by X until time t and from where X escape. That is to say I 1 is the limit of the random variable sup k Nt 1 Lm k, t/t. I 2 is a product of two terms : the first 1 Y 2 Y2 1 1 corresponds to the re-normalized amount of time left to the process X before instant t after it has reached the ultimate visited

4 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 4 h t -minima, m Nt. The second term corresponds to the local time of X at this ultimate h t -minima. Intuitively Y 2 is construct from Y 1 multiplying each of its jumps by an independent copy of the variable R κ. Therefore this second term 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 than R of Theorem 1.1. Indeed as in the case κ =, the process X is prisoner in the neighborhood of the last h t -minima visited before time t. We prove this result by showing first that portions of the trajectory of X re-centered at the local h t -minima, until the instant t, is 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. Second we prove that what we seek for the supremum of the local time is, mainly, a function of the sum of theses independent parts, which converges to a Lévy process. Let us give some details about this : Let Qs, s a canonical process, taking values in R +, with infinitesimal generator given for every x > by 1 d 2 2 dx 2 + ζ ζ d 2 coth 2 x dx. This process Q can be thought of as a ζ/2-drifted Brownian motion W ζ Doob-conditioned to stay positive, with the terminology of [7], which is called Doob conditioned to reach + before in [19] see Section 2.1 in [3] for more details. We call BES3, κ/2, the law of Qs, s. For a < b, Wκs, b s τ W κ b a is a κ/2-drifted Brownian motion starting from b and killed when it first hits a. 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 = +. We now introduce functional of W κ and Q : F ± x := τ Q x exp±qsds, x >, G ± a, b := Also for any δ >, and t >, define n t := e κφt1+δ, which is an upper bound of N t as stated in Lemma 3.1. τ W b κa exp ± W b κs ds, a < b. Then let S j t, R j t, e j t, j n t a sequence of i.i.d. random variables with S j, R j and e j independent with S 1 L = F + h t + G + h t, h t /2, 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 an independent copy of F and F + independent of G +. Define l j := e j S j, H j := l j R j, note that for notational simplicity we do not make appear the dependence in t in the sequel. Typically l j plays the roll of the local time at the jth positive h t -minima if the walk escape from it before the instant t and H j the roll of the time it takes for the walk to escape from the corresponding valley. Define the family of processes Y 1, Y 2 t indexed by t, by 1.3 then s, Y 1, Y 2 t s := 1 t se κφt j=1 l j, H j,

5 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 5 Proposition 1.4. We have Y 1, Y 2 t L S Y1, Y 2. Once this is proved, we check that what we need for the supremum of the local time can be written as a function of Y 1, Y 2 t. We obtain an expression in this sens in Proposition 5.1. Then to obtain the limit, we prove the continuity in J 1 -topology of the involved mapping and apply a continuous mapping Theorem see Section 4.3. It appears that with this method we can obtain other asymptotics, like for the supremum of the local time before the last valley is reach before the instant t and once it left it for good as well as for the position of the favorite site : Theorem 1.5. We have L Hm Nt+1 t L Hm Nt t Let us call F t, the position of the first favorite site, F t F t X t L Y 1 Y 1 2 1, L Y 1 Y L BU[,1] + 1 B := inf{s >, Lt, s = L t}, then where B is a Bernoulli with parameter PI 1 < I 2 independent of the uniform random variable on [, 1] : U [,1]. One question we may ask here is what s happen in the discret case or with a more general Lévy process? For the discrete case, we should have a very similar behavior as the renewal structure which appears in both cases continuous and discrete is very similar see the works of Enriquez-Sabot- Zindy [17]. The main difference comes essentially from the functional R κ which should be replaced by a sum of exponential of a simple random walk conditioned to remain positive see [18], [17]. For a more general Lévy process, we think for example, of a spectraly negative Lévy process studied in the case of diffusion in random environment by Singh [32], more work needs to be done, especially for the environment. First to obtain a specific decomposition of the Lévy s path similar to what is done for the drifted Brownian motion in Faggionato [19] 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 Véchambre [34]. The rest of the paper is organized as follows : In Section 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 3 we study the joint process of the first n t hitting times and local times. We show that parts of the trajectory of X is not important for what we seek this part is technical, makes use of some technical results of the paper [3] and can be omitted in the first instance. We then prove the main result of this section : Proposition 3.5. It proves that the joint process exit time, local time can be approximated in probability by i.i.d random variables, again the proofs make use of some technical parts of [3] though the main ideas are discuss in the present paper. In Section 4 we prove Proposition 1.4, and study the continuity of certain functional of Y 1, Y 2 which appears in the expression of the law we have detailed above. This section is independent of the other, we essentially prove a basic functional limit theorem and prepare to the application of continuous mapping theorem.

6 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 6 Section 5 is where we make appear the renewal structure in the problem we want to solve. In particular we prove how the distribution of the supremum of the local time can be approximated by the distribution of a certain function of the couple Y 1, Y 2 t, the main step being Proposition 5.1. The appendix is a reminder of some estimates on the brownian motion, Bessel processes, and functionals of both of these processes Notations. In this section we introduce typical notations for the study of diffusions in random media, as well as elementary tools for the continuous one-dimension case. 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.P W κ dω. We denote respectively by E Wκ, E, and E the expectancies with regard to P Wκ, P and P. 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. We also denote by U a the process U starting from a, and by P a the law of U a ; with the notation U = U. Now let us introduce the following functional of W κ, Ar := r e Wκx dx, r R, and recall that whenever κ >, A := lim r + Ar < a.s. As in Brox [1], there exists a Brownian motion B independent of W κ, such that Xt = A 1 [BT 1 t], where T r := r exp{ 2W κ [A 1 Bs]}ds, r τ B A. 1.4 The local time of the process X at x until instant t simply denoted Lt, x, can be written as see [3] Lt, x = e Wκx L B T 1 t, Ax. 1.5 With these notations, we recall the following expression of Hr, for all r, Hr = T [τ B Ar] = r e Wκu L B [τ B Ar, Au]du Path decomposition and Valleys 2.1. Path decomposition in the neighborhood of the h t -minima m i. First we recall some results for h-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 x = h}, τ i h := inf{s > m i, V i x = h}, h >. The following result has been proved by Faggionato [19] [for i and ii], and the last fact comes from the strong Markov property.

7 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 7 Fact 2.1. path decomposition of W κ around the h t -minima m i i The truncated trajectories V i m i s, s m i τi h t, V i m i + s, s τ i h t m i, i 1 are independent. ii Let Qs, s be a process with law BES3, κ/2. All the truncated trajectories V i m i s, s m i τi 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 Qs, s τ Q h t. iii For i 1, the truncated trajectory V i s + τ i h t, s is independent of W κ 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 Definition of valleys and 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 notations. These notations are used to define valleys of the potential around some m i, which are shown in Lemma 2.2 to be equal to the m i for 1 i n t with large probability. Let h + t := 1 + κ + 2δh t, and define L + :=, m :=, and recursively for i 1 see Figure 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 [ L i,x] W } κ h t, 2.7 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 >, 2.8 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 r.v. depend on t, which we do not write as a subscript to simplify the notations. Notice also that τ i h t is the same in definitions 2.7 and 2.8 with h = h t. Moreover by continuity of W κ, W κ τ i h t = W κ m i + h t. So, 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. Moreover, L + i 1 < L i m i < τ i h t < L i < L + i, i N, 2.9 L + i 1 L i < m i < τ i h t < M i < L + i, i N. 2.1 Furthermore by induction the r.v. 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, i N, are also stopping times. Also 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, 2.11

8 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 8 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 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 j 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 an overwhelming high probability, the first n t + 1 positive h t -minima m i, 1 i n t + 1, coincide with the r.v. m j, 1 i n t + 1. We introduce the corresponding event V t := nt+1 i=1 {m i = m i }. Lemma 2.2. Assume < δ < 1. For t large enough, P V t C1 n t e κht/2 = e [ κ/2+o1]ht, C 1 >. 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 2.3. 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 r.v. with or without tilde. 3. Contributions for hitting and local times 3.1. Negligible parts for hitting times. In the following Lemma we recall results of [3] which tell that the time spent between valleys is negligible compared to the amount of time spent to escape from the valleys. It also gives an upper bound for the number of visited valleys, and the fact that the process never backtracks in a previous visited valley. For any i n t, define U i := H L i H m i, U =

9 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 9 and for any m 1 the events B 1 m := { } m k 1 H m k U i < ṽ t, k=1 where ṽ t := 2t/ log h t. Finally m B 2 m := {H L j H m j < H + L j H m j, H m j+1 H L j < H + L j H L j }, j=1 with H + x j := inf{k > m j, X k = x j } for any x. Lemma 3.1. For any δ small enough and t large enough i=1 P H m 1 < ṽ t P B 1 n t 1 C 2 v t, 3.12 with v t := n t log h t e φt = o1, C 2 >. Also we have : with C 3 > and C 4 >. P B 2 n t 1 C 3 n t e δκht, 3.13 PN t < n t 1 C 4 e δκφt, 3.14 Proof: The first statement is Lemma 3.7 in [3], the second one is proved in Lemmata 3.2 and 3.3 in [3]. Finally 3.14 is proved in Lemma 5.2 of the same paper Negligible parts for local times. We now provide estimations on the local time, more especially we prove that in the complementary of a small interval in the neighborhood of the first n t h t -minima, the local time at each site is negligible compared to t. We split this section into two, the first one deals with coordinate away from the valleys, the second with coordinates in the valleys excluding the points near the bottom 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 valleys is negligible compared to t. More precisely, define the following events { B3m 1 := B 2 3m := sup LH m 1, x ft x [, m 1 ] m 1 j=1 } m 1 j=1 {sup x Lj LH m j+1, x L H Lj, x ft B 3 m := B 1 3m B 2 3m, ft = te [κ1+3δ 1]φt. In this section we prove { sup LH m j+1, x ft x [ L j, m j+1 ] Lemma 3.2. Assume δ small enough such that κ1 + 3δ < 1. There exists C 5 > such that for any large t with w t := e κδφt. P B 3 n t 1 C 5 w t, }, },

10 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 1 The proof is based on the lemma below, let us introduce a few notation with respect to the environment. τ 1 h := inf{u, W κ u inf [,u] W κ h}, h >, m 1h := inf{y, W κ y = inf [,τ 1 h] W κ }. All along this work C + is a positive constant that may grow from line to line. Lemma 3.3. For large t, P sup x [,m 1 h t] L[Hτ1 h t, x] > te [κ1+3δ 1]φt C n t eκδφt Proof of Lemma 3.3: Thanks to 1.5 and 1.6 there exists a Brownian motion Bs, s, independent of W κ, such that L[Hτ 1 h t, x] = e Wκx L B [τ B Aτ 1 h t, Ax], x R By the first Ray Knight theorem see e.g. Revuz and Yor [29], 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 2 α x for every x [, α]. Consequently, using 3.16 and the independence of B and W κ, there exists a 2-dimensional Bessel process Q 2 such that L[Hτ 1 h t, x] = e Wκx Q 2 2[ Aτ 1 h t Ax ] x τ 1 h 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 Wκx dx kt, A 1 := { u, kt], Q 2 2u 2eu [ at + 4 log log[ekt/u] ]}, A 2 := { inf [,τ 1 h t] W κ bt }. We know that P A y C + y κ for y > since 2/A is a gamma variable of parameter κ, 1 see Dufresne [16], so P A C+ kt κ = C + e 2φt. Moreover, P A 1 C+ exp[ at/2] = C + e 2φt by Lemma 6.5. Also we know that inf [,τ 1 h] W κ, denoted by β in Faggionato [19], eq. 2.2 is exponentially distributed with mean 2κ 1 sinhκh/2e κh/2 [19], eq So, P A 2 = P [ inf[,τ 1 h t] W κ > bt] = exp [ btκ/2 sinhκh t /2e κht/2 ] e 2φt for large t. Now assume we are on A A 1 A 2. Due to 3.17, we have for every x < τ 1 h t, since < Aτ 1 h t Ax A kt, L[Hτ 1 h t, x] e Wκx 2e[Aτ 1 h t Ax] { at + 4 log log [ ekt/[aτ 1 h t Ax] ]} We now introduce f i := inf{x, W κ x i} = τ Wκ i, i N, and let x < τ1 h 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 Wκfi+1. All this leads to e Wκx [Aτ1 h t Ax ] τ = e Wκx 1 h t τ e Wκu 1 h t du e e Wκu Wκfi du x f i

11 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 11 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 τ1 u, h t := inf{y u, W κ y inf [u,y] W κ h t } τ1 h 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 τ by the strong Markov property applied at stopping time f i, where β h := E 1 h e Wκu du. We know see [3] eq. 3.38, in the proof of Lemma 3.6 that β h C + e 1 κh for large h. Hence for large t by Markov inequality, P bt A 3 i= f 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, 3.18 and 3.19 lead to L[Hτ 1 h t, x] 2e 2+1 κht btn t e κδφt{ at + 4 log log [ ekt/[aτ 1 h t Ax] ]}. 3.2 For any x m 1 h t, inf [,τ 1 h t] W κ bt so Aτ 1 h t Ax = τ 1 h t x e Wκu du τ 1 h t m 1 ht e Wκu du e bt [τ 1 h t m 1h t ] e bt on the event 4 i= A i with A 4 := {τ 1 h t m 1 h t 1}. Since m 1 = m 1 h t and τ 1 h t = τ 1 h 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] 2κh t e κht + P τ Q h t τ Q h t /2 < 1 2κh t e κht + C + exp[ c h 2 t ] due to [3], eq. 2.8, coming from Faggionato [19], Fact 2.1 ii and c is a positive constant that may decrease from line to line in the sequel of the paper. Now, we have ekt/[aτ 1 h t Ax] ekte bt on 4 i= A i, and then, on this event, 3.2 leads to L[Hτ 1 h t, x] 2e 2+1 κht btn t e κδφt{ at + 4 log log [ ekte bt]}. C + tφte [κ1+δ 1]φt e κδφt h t, since φt = olog t, h t = log t φt and n t = e κ1+δφt. We notice that for large t, C + φth t e κδφt since log log t = oφt. Hence, for large t, L[Hτ 1 h t, x] te [κ1+3δ 1]φt, on 4 i= A i for every x m 1 h t. This gives for large t, P sup x [,m 1 h t] L[Hτ1 h 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 3.2:

12 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 12 Proof of Lemma 3.2: The method is similar to the proof of Lemma 3.7 of [3]. Recall the definition of L i < L i just above 2.8, 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 i+1 ht] W κ}, 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 By the strong Markov property, X i and Xi are diffusions in the environment W κ, starting respectively from L i and L i. We denote respectively by L X i, L X i, H Xi and H X i the local times and hitting times of X i and Xi. 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 nt 1, sup LH m i+1, x L H Li, x = sup LH m i+1, x L H Li, x x R 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, 3.22 L i x m i+1 since m i+1 = m i+1 τ i+1 h t = τ i+1 h t on A 5. Now, notice that the right hand side of 3.22 is the supremum of the local times of Xi L i, up to its first hitting time of τ i+1 h t L i, over all locations in [, m i+1 L i ]. Since X i L i is a diffusion in the environment 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 3.22 has the same law, under the annealed probability P, as sup x [,m 1 h t] L[Hτ1 h t, x]. Consequently, P nt 1 i=1 { } sup LH m i+1, x LH L i, x > te [κ1+3δ 1]φt x R n t [P sup x [,m 1 h t] L [ Hτ1 h t, x ] > te [κ1+3δ 1]φt + P ] A 5 + P A6 C + e κδφt 3.23 by Lemma 3.3, since P A 5 C+ n t h t e κht by [3], eq. 3.41, P A 6 C+ n t e κht/16 by [3], Lemma 3.3, and since φt = olog t. Notice that, as before, m 1 = m 1 = m 1 h t on V t {M }. Finally, P sup x [, m 1 ] LH m 1, x > te [κ1+3δ 1]φt C + e κδφt + P V t + P < M < m 1 C + e κδφt also by Lemma 3.3, Lemma 2.2, and since P < M < m 1 2κh t e κht due to [3], eq This and 3.23 prove the lemma Local time in the valley [ L j, L j ] but far from m j. Let D j := [ m j r t, m j + r t ], with r t := φt

13 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 13 B 4 m := m j=1 sup x D j [τ j h+ t, L j ] with D j the complementary of D j. LH L j, x LH m j, x < te 2φt, m 1. Lemma 3.4. Assume < δ < 1/8. There exists C 6 > such that P [ B 4 n t ] 1 C 6 n t e 2φt, Proof: We make the proof replacing rt = φt 2 by rt = C φt with C a constant large enough, this is a little more precise than what we need here and may be used for other purposes. Let j [1, n t ]. First 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. By scaling, there exists another Brownian motion B that we still denote B for simplicity, independent of Ṽ j, such that 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 terms L B [ τ B 1,. ] and A j L j in 3.25, 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 2 2e 2φt for large t by Lemma 6.4 eq and Moreover on V t, we have by Remark 2.3 and Fact 2.1 ii and iii, A j Lj [ τj h t m j ] e h t + L j τ j h t ev j s ds L = e ht τ Q h t + G + h t /2, h t, where recall that Q has law BES3, κ/2 and is independent of G + h t /2, h t, which is defined in 1.3, and with L j := inf{s > τ j h t, V j s = h t /2}. Consequently, P A 2 P τ Q 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 eq. 6.86, Lemma 6.3 eq and Lemma 2.2, and since φt = olog t and log log t = oφt. Now, we would like to bound the term e Ṽ jx that appears in To this aim, we define 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] We can prove using Fact 2.1 see [3] Lemma 2.5 for details that P A 3 C+ e [κ2 C φt]/16 2 e 2φt if we choose C large enough. Moreover with Fact 2.1 again see [3], eq applied with h = C φt, α = κ/8, γ = κ/16 and ω = h t /C φt we get P A 4 e κ 2 C φt/16 e 2φt for large t. We notice that inf [ mj +C φt, τ j h t] Ṽ j κc φt/16 on A 3 A 4 V t, thanks to Remark 2.3. We prove similarly that P A 5 C+ e κ2 C φt/ P { } V t 2e 2φt, where A 5 := inf [ τ j ht, m j C φt] Ṽ j κc φt/16. Also by [3], Lemma 2.7, P A 6 { } e κht/8, with A 6 := inf [ τ j h+ t, τ j ht] Ṽ j h t /2. We also know that Ṽ j x h t /2 for all τ j h t x L j by definition of L j. Consequently on 6 i=3 A i V t, for all x D j [ τ j h+ t, L j ], we have e Ṽ j x e κc φt/16.

14 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 14 Hence on 6 i=1 A i V t, we have under P Wκ m j, by 3.25 and 3.26, 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 choose C large enough. So, conditioning by W 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 2.2. This proves the lemma Convergence of the main contributions. In this section we make a link between the families {[U j := H L j H m j, LH L j, m j ], j n t }, and the i.i.d. sequence {H j, l j, j n t } described in the introduction. Let F ± 1 a, F ± 2 a and F ± 3 a independent copies of F ± a also independent of G ± a, b. Proposition 3.5. Let t > large, for δ > small enough recall that δ appears in the definitions of n t and h + t there exists d 1 = d 1 δ, κ >, D 1 d 1 > and a sequence {S j t, R j t, e j t, j n t } of i.i.d. random variables with S j, R j and e j independent and S 1 L = F + 1 h t + G + h t, h t /2, L R 1 = F 2 h t /2 + F3 h L t/2 and e 1 = E1/2 such that P nt j=1 { U j H j t H j, LH L j, m j l j t l j } 1 e D 1h t, with l j := S j e j, H j := R j l j, 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 3.6. Let t > large, i There exists a sequence e j t, j n t of i.i.d. random variables with exponential law of mean 2 and independent of W κ such that there exist constants d >, D = D d > possibly depending on κ and δ such that n t P { U j H j e d h t H j, LH L j, m j = L j } 1 e D h t, 3.27 j=1 Lj where L j := e j m eṽ jx j dx, H j := L j R j, R j := τ + j ht/2 τ j ht/2 e Ṽ jx. Moreover the random variables {L j, H j, j 1} are i.i.d. ii Also there exists a sequence of independent and identically distributed random variables S j, j n t, independent of R j, j n t and e j, j n t such that { n t } Lj P eṽ jx dx S j e d h t S j 1 e D h t L, and S 1 = F + h t j=1 m j

15 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 15 Proof of Lemma 3.6 For 3.27 : by the strong Markov property, formula 1.5 and 1.6 the sequence {U j, LH L j, m j, j n t } is equal to the sequence {H j L j, L j H j L j, m j, j n t }, where H j L j := Lj e Ṽ j u L B j[τ Bj A j L j, A j u]du, L j H j L j, m j := L B j[τ Bj A j L j, ], A j u := u m j eṽ j x dx, 3.29 with B j, 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. Also L B j is the local time associated with B j. Define A j := {max u< L L j B j[τ Bj A j L j, A j u] = }, we can prove with exactly the same method than in the proof of Lemma 3.2 in [3] see the estimation of P Wκ E j that there exists c > possibly depending on κ and δ such that P nt j=1 A j 1 e c h t. So n t } P {H j L j = h j 1 e c κh t, with h j := L j L j are strictly function of B j and Ṽ j x + L + j 1, x < L + j L + i 1 < L j j=1 e Ṽ j x L B j[τ Bj A j L j, A j u]du. We also notice that h j, L j H j L j, m j L + j 1, where by construction so the random variables h j, L j H j L j, m j, j n t are i.i.d by the second fact of Lemma 2.2. By scale invariance of brownian motion B j we have that L B j[τ Bj A j L j, A j u], L j u L j is equal in law to A j L j L Bj[τ B j 1, A j u/a j L j ], L j u L j, where B j is a standard Brownian motion independent of W κ which we still denote by B j in the sequel. In particular h j, L j H j L j, m j L = h j, L j, h j := A j L j Lj L j e Ṽ j x L B j[τ Bj 1, A j u/a j L j ]du, Lj := A j L j L B jτ Bj 1,. 3.3 Then let e j := L B jτ Bj 1, and recall that by the first Ray Knight theorem, e j has for law an exponential variable with parameter 1/2. Note then that L j = L j, so to finish the proof of 3.27 we only have to approximate h j in probability. This is what is done in the proof of Lemma 4.7 of [3] : for any 1 j n t and > τ + P j A h j j ht/2 τ + L j e Ṽ jx e j > e 1 3ht/6 A j j ht/2 L j e Ṽ jx e j C + e c h t. τ j ht/2 τ j ht/2 Recall that C + resp. c is a positive constant that may grow resp. decrease from line to line along the paper. For 3.28, the proof can be found in [3] at the end of the proof of Proposition 4.4. Proof of Proposition 3.5 By the first part i of Lemma 3.6 the sequence e j, H j, L j, j n t is i.i.d, moreover e j is independent of H j, L j. This together with part ii yields P nt j=1 { U j e j S j R j t e j S j R j, LH L j, m j e j S j t e j S j } 1 e D 1h t,

16 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 16 with S j independent of R j. Then as m j, j 1 is a subsequence of m j, j 1, using Fact 2.1 for any j we have R j ; S j = L F 1 + h t + G + h t, h t /2; F2 h t/2 + F3 h t/2. 4. Convergence toward the Lévy process Y 1, Y 2 and Continuity 4.1. Preliminaries. We begin this section by the convergence of certain repartition functions, a key results that are essentially improvement of the second part of Lemma 5.1 in [3]. Lemma 4.1. lim sup t + x [e 1 2φt,+ [ lim sup t + y [e 1 3φt,+ [ x κ e κφt P e 1 S 1 /t > x C 2 =, 4.31 y κ e κφt P e 1 S 1 R 1 /t > y C 2 E [R κ κ ] =, 4.32 with C 2 > a known constant [see below 4.39]. 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, + [ [see Section 1.1]. Proof: Proof of 4.31 We first prove that x κ e κφt P S 1 /t > x converges uniformly in x [ e 1 φt, + [ to a constant c 3, that is, we prove that lim t + lim t + sup x [e 1 φt,+ [ For that, let y = e 1 φt x, we have to prove that y κ e κφt P sup y [1,+ [ x κ e κφt P S 1 /t > x c 3 = S 1 /e ht+φt > y c 3 =, 4.34 but this is equivalent to prove that for any function f : ], + [ [1, + [, lim t + ftκ e κφt P S 1 /e ht+φt > ft = c First by definition see Proposition 3.5, S 1 can be writen as a sum of two independent functional, that we denote, for simplicity, as the sum of two generic functionals S 1 /t = F 1 + h t + G + h t, h t /2 /t = e φt e ht F 1 + h t + e ht G + h t, h t / Since we know an asymptotic for the Laplace transform of F + h t /e ht and G + h t /2, h t, the proof of 4.35 is similar to the proof of a Tauberian theorem. First by 6.82 and 6.83 we have [ ] γ >, ω f,t γ := 1 E e γs 1/fte h t +φt γ 1 c 4γ κ 1 ft κ e κφt, 4.37 t + with c 4 a positive constant. Now as ω f,t is the Laplace transform of the measure U f,t := 1 R+ zp S 1 /fte ht+φt > z dz, from 4.37 we have γ >, ω f,t γ ω f,t 1 t + γκ 1. From this we can follow the same line as in the proof of a classical Tauberian theorem see for example [21] volume 2, section XIII.5, page 442. So as for the proof of Theorem 1 in [21] we

17 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 17 can deduce that z >, U f,t [, z] ω f,t 1 t + z 1 κ Γ2 κ. Then as in the proof of Theorem 4 of the same reference page 446, we deduce from the monotony of the densities of measures U f,t that P S 1 /fte ht+φt > z z >, 1 κ ω f,t 1 t + z κ Γ2 κ. Considering this convergence with z = 1 we get exactly 4.35 for c 3 = 1 κ/c 4 Γ2 κ, so 4.33 follows. Let a t := e φt, for any x > at x κ e κφt P e 1 S 1 /t > x, e 1 < a t = 2 1 x/u κ e κφt P S 1 /t > x/u u κ e u/2 du, because e 1 has law E1/2. Taking x arbitrary in [e 1 2φt, + [, we have x/u [e 1 φt, + [, u ], a t ], so thanks to 4.33 we get + xκ e κφt P e 1 S 1 /t > x, e 1 < a t 2 1 c 3 u κ e u/2 du = lim t + sup x [e 1 2φt,+ [ Now for any x > and t large enough such that y 1, y κ e κφt P S 1 /t > y < 2c 3, we have 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 P S 1 /t > x/u e u/2 du a t + = 2 1 u κ x/u κ e κφt P S 1 /t > x/u 1 x u e u/2 du a t u κ x/u κ e κφt P S 1 /t > x/u 1 x>u e u/2 du a t 2 1 e κφt + a t u κ e u/2 du + c 3 + a t u κ e u/2 du. For the second term in the third equality we used the fact that x/u κ e κφt P S 1 /t > x/u < 2c 3 when x u. Since a t = e φt, the last quantities converges to when t goes to infinity. Combining this with 4.38 we get + xκ e κφt P e 1 S 1 /t > x 2 1 c 3 u κ e u/2 du =, 4.39 lim t + sup x [e 1 2φt,+ [ and this is exactly 4.31 with C 2 := 2 1 c 3 + u κ e u/2 du. Proof of 4.32 Let µ R1 the distribution of R 1, for any y, a > and t > by independence of e 1 S 1 and R 1 y κ e κφt P e 1 S 1 R 1 /t > y, R 1 < a = a y/u κ e κφt P e 1 S 1 /t > y/u u κ µ R1 du.

18 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 18 Taking a = a t = e φt and y arbitrary in [e 1 3φt, + [, we have y/u [e 1 2φt, + [, u ], a t ], so, thanks to 4.39 we get + yκ e κφt P e 1 S 1 R 1 /t > y, R 1 < a t C 2 u κ µ R1 du =. lim t + sup y [e 1 3φt,+ [ Also, as + u κ µ R1 du converges to E [R κ κ ], when t goes to infinity y κ e κφt P e 1 S 1 R 1 /t > y, R 1 < a t C 2 E [R κ κ ] =. lim t + sup y [e 1 3φt,+ [ Finally, as the family R 1 t> is bounded in all L p spaces, we can proceed as before to remove the event R 1 < a t and we thus get y κ e κφt P e 1 S 1 R 1 /t > y C 2 E [R κ κ ] =, 4.4 which is lim t + sup y [e 1 3φt,+ [ We now prove the last assertion. For any x, y, a and t >, we have = = e κφt P e 1 S 1 /t > x, e 1 S 1 R 1 /t > y, R 1 < a a e κφt P e 1 S 1 /t > x, e 1 S 1 /t > y/u µ R1 du, a y/x = 1 y κ a y/x + 1 x κ a a y/x a e κφt P e 1 S 1 /t > y/u µ R1 du + a y/x e κφt y/u κ P e 1 S 1 /t > y/u u κ µ R1 du e κφt x κ P e 1 S 1 /t > x µ R1 du. e κφt P e 1 S 1 /t > x µ R1 du, Taking a = a t = e φt and x, y arbitrary in [α, + [ for some α >, we have y/u [e 1 2φt, + [, u ], a t ] whenever t is large enough, so, thanks to 4.39 we get that e κφt P e 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 x κ PR κ > y/x + y κ ER κ κ 1 R y/x. Then as before we can remove the event {R 1 < a t }, we get the last assertion Proof of Proposition 1.4. We start with the finite dimensional convergence Lemma 4.2. For any k N and s i >, i k, Y 1, Y 2 t s i, i k converge in law as t goes to infinity to Y 1, Y 2 si, i k. Proof: Proof is basic here, however we give some details as we deal with a two dimensional walk which increments depend on t itself. As Y1 t s and Y t 2 s are sums of i.i.d sequence we only have to prove the convergence in law for the couple Y 1, Y 2 t s for any s >. Define Y 1 >b, Y 2 >b obtained from Y 1, Y 2 keeping the increments larger than b, Y 1 >b s := 1 se κφt t j=1 l j 1 lj /t>b, and Y 2 >b s := 1 se κφt t j=1 H j 1 Hj /t>b. Also let Y b i := Y i Y i >b. We first prove that for any s >, lim lim t + P Y 1, Y 2 t s > 1 κ2 κ =, 4.41

19 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 19 where for any a R 2, a := max i 2 a i, and 1 κ2 κ > as κ < 1. We compute the first moment of Y 1 s and Y 2 s. Let η > such that κ 1 3η <, applying Fubini we have [ ] e κφt l1 E t 1 l 1 /t = e κφt e1 S 1 E 1 t e1 S 1 /t = e κφt P e 1 S 1 /t > x dx e 1 2ηφt e κφt P e 1 S 1 /t > x dx + e 1 2ηφt e κφt P e 1 S 1 /t > x dx e κ 1 2ηφt + x κ x κ e κφt P e 1 S 1 /t > x dx. e 1 2ηφt The first term converges to when t goes to infinity because κ 1 2η < and, according to 4.31, for t large enough, we have For such t, the second term is less than so we get that x κ e κφt P e 1 S 1 /t > x 2C 2, x e 1 2ηφt. 2C 2 x κ dx = 2C 2 1 κ 1 κ, t 1, ], 1], e κφt l1 E t 1 l 1 /t e κ 1 2ηφt + C + 1 κ Using the same method and applying this time 4.31, we get that t 1, ], 1], e κφt H1 E t 1 H 1 /t e κ 1 3ηφt + C + 1 κ We thus obtain E Y 1 s E Y 2 s se κ 1 2ηφt + C + s 1 κ, 4.44 se κ 1 3ηφt + C + s 1 κ, 4.45 then a Markov inequality yields The next step is to prove that Y 1 >, Y 2 > t s can be written as the integral of a point process which converge to the desired limit. We have s s Y 1 >, Y 2 > t s = xpt 1 dx, dv, xpt 2 dx, dv x> where the measures P 1 t and P 2 t are defined by P 1 t := + i=1 δ t 1 l i,e κφt i, and the same for P2 t replacing l by H. Recall that P 1 t and P 2 t are dependent and now prove that P 1 t, P 2 t converge to a Poisson point measure. For that just use Lemma 4.1 together with Proposition 3.1 in [28] after discretization, it implies that P 1 t, P 2 t converge weakly to the Poisson random measure denoted P 1, P 2 with intensity measure given by ds ν. Then using that for any >, and T < +, on [, T, +, + dsν is finite, we have that Y 1 >, Y 2 > t s converge weakly to s s Y 1 >, Y> 2 s := xp 1 dx, dv, xp 2 dx, dv. x> x> x>

20 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 2 We are left to prove that Y 1 >, Y> 2 converge to Y 1, Y 2 when. This is a straightforward computation, that we detail for completeness. Let ν 1 [x, + [ := + ν [x, + [ [y, + [ dy, we have E x s xp 1 dx, dv = s xν 1 x = C 2 κ, x Then a Markov inequality, proves that for any s >, the process x xp1 dx, dv converge to zero when goes to zero in probability. The same is true for s x xp2 dx, dv, so we obtain that in probability Y 1 >, Y> 2 s converge to Y 1, Y 2 s when. We now prove tightness of the family measures of the processes Y 1, Y 2 t denoted DY 1, Y 2 t t. Lemma 4.3. 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 [8] using Aldous s tightness criterion see Condition 1, and equation page 176 in [8] also used in [9] page 1. We have to check the two following statements: 1 for any >, there exists a such that for any t large enough Psup s [,T ] Y 1, Y 2 t s a. 2 for any >, and η > there exists δ, < δ < T and t > such that for t > t, PωY 1, Y 2 t, δ, T η, 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 is defined by sup r δ u1 <u<u 2 r+δ T {min Y 1, Y 2 t u 2 Y 1, Y 2 t u, Y 1, Y 2 t u Y 1, Y 2 t u 1 }. Also PvY 1, Y 2 t,, δ, T η, and PvY 1, Y 2 t, T, δ, T η, 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 }. s 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 ] Define Y 1 >b obtained from Y 1 where we remove the increments l j /t smaller than b. That is to say Y 1 >b s := 1 se κφt t j=1 l j 1 lj /t>b. Also let Y b 1 := Y 1 Y 1 >b and N u >b := ue κφt i=1 1 lj /t>b. Let < δ 1 < 1, Markov inequality yields PY1 t T a P Y 1 1 T a + P Y 1 >1 T a [ ] a E Y 1 1 T + 1 a δ E N >1 1 T + P Y >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 1 i a δ 1 a δ 1 P 1 T a 2, N >1 T a δ 1 l i /t a 1 δ 1 /2 l i /t 1 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, 4.48 for all t large enough thanks to 4.31 and δ 1 such that δ 1 κ1 δ 1 <.

21 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 21 Also, as for any positive b, N T >b follows a binomial law with parameter T e κφt, Pl 1 /t > b using 4.31 again, and 4.44 we obtain for t is large enough EN >b T 2C 2T b κ, E [ Y b 1 T Collecting 4.48, 4.49 and 4.47 we get the existence of t 1 > such that ] 2C 2 T b 1 κ lim sup PY 1 T a =. 4.5 a + t t 1 The same arguments holds for Y 2 using 4.32 instead of 4.31 and 4.45 instead of 4.44 so 4.5 also holds for Y 2 instead of Y 1. We conclude the proof of 1 by putting 4.5 and its analogous for Y 2 in We now check 2 First as usual we write {ωy 1, Y 2 t, δ, T η} {ωy b 1, Y b 2 t, δ, T η/2} {ωy >b 1, Y >b 2 t, δ, T η/2}. For Y b. we have 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 moreover by positivity of the increments P ωy b 1, δ, T η/2 P k T/δ {Y b 1 k + 1δ Y b 1 kδ η/2} P Y b 1 k + 1δ Y b 1 kδ η/ k T/δ For any k, Y b 1 k + 1δ Y b 1 kδ is the sum of at most δe 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 + 1δ Y b 1 kδ η/2 P Y b 1 2δ η/2 8C 2 δb 1 κ /η, where the first inequality holds for t large enough so that δe κφt 1 and the second from the second expression in 4.49 replacing T by 2δ. Combining with 4.51 we get for large t P ωy b 1, δ, T η/2 8C 2 T 1 + δb 1 κ /η, 4.52 [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 4.51 is less than /4. A similar estimate can be proved for PωY b 2, δ, T η/2. For Y. >b, again we have 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. Let us decrease b in order to get b < η/2 so that {ωy 1 >b, δ, T > η/2} implies that two jumps larger than b occur in an intervall smaller than 2δ. That is {ωy 1 >b, δ, T > η/2} T eκφt j=1 T eκφt i>j,i j/e κφt 2δ {l j l i /t > b}. Applying 4.31 for t large enough P T eκφt j=1 T eκφt i>j,i j/e κφt 2δ {l j l i /t > b} 8C 2 δt b 2κ,

22 RENEWAL STRUCTURE AND LOCAL TIME FOR DIFFUSIONS IN RANDOM ENVIRONMENT 22 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 δ if needed, this also applies to PvY 1, Y 2 t, T, δ, T η Continuity of certain functionals of Y 1, Y 2, in J 1. In this section, we study the continuity of functionals of the Lévy processes Y 1, Y 2. For our purpose we are interested in the following mappings, first the two we have already mention in the introduction which are the basics J : D R +, R D R +, R I : D R +, R, J 1 D R +, R, U f f f f 1 Then we also need the compositions of these two : let J I and J I J I : D R +, R 2 R f = f 1, f 2 f J I : D R +, R 2 R 1 f f = f 1, f 2 f 1 f J I respectively J I produces the largest jump of f 1, just after respectively before f 2 reach 1. Finally let F define by F : D R +, R 2 R f = f 1, f 2 inf { s [, f2 1 1, f 1s = f 1 f }, we need this variable for the characterization of the favorite sites. Lemma 4.4. J is continuous in the J 1 topology. Proof: This fact is basic, but as we have not found a proof in the literature, we give some details. To prove the continuity on D R +, R, we only have to prove it for every compact subset of R +, see [35] Theorem So let f D R +, R and T > for 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 > and g D [, T ], R such that d T f [,T ], g 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 [, 1] onto itself, e : [, T ] [, T ] such that sup es s s [,T ] 2, and sup g es f [,T ] s s [,T ] 2. So for every s [, T ] we have g es f [,T ] s = g es g es f [,T ] s f [,T ] s g es f [,T ] s + g es f [,T ] s 2 2 =, where hs = hs hs. This implies d T JT f [,T ], JT g. Lemma 4.5. The mapping J I DR +, R 2 such that and J I are continuous for J 1 -topology for every couple f 1, f 2