Hardy s inequality in L 2 ([0, 1]) and principal values of Brownian local times

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1 Fields Insiue Communicions Volume, Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes Giovnni Pecci oroire de Proiliés e Modèles léoires Universié Pris VI & Universié Pris VII Pris, Frnce nd Isiuo di Meodi Quniivi Universià. Bocconi Miln, Ily gpecci@pro.jussieu.fr Mrc Yor oroire de Proiliés e Modèles léoires Universié Pris VI & Universié Pris VII Pris, Frnce Dediced o Miklós Csörgő on he occsion of his 7h irhdy Asrc. We presen in unified frmework wo exmples of rndom funcion φ ω, s on R such h he inegrl φ ω, s g s ds is well defined nd finie les, s limi in proiliy for every deerminisic nd squre inegrle funcion g, nd φ does no elong o [,, ds wih proiliy one. In priculr, he second exmple is reled o he exisence of principl vlues of Brownin locl imes. Our key ools re Hrdy s inequliy, some semimringle represenion resuls for Brownin locl imes due o Ry, Knigh nd Jeulin, nd he reformulion of cerin heorems of Jeulin-Yor 979 nd Cherny in erms of ounded operors. We lso eslish, in he ls prgrph, severl wek convergence resuls. Inroducion he im of his pper is o give unified discussion of he wo following resuls, concerning cerin inegrls ssocied o one dimensionl Brownin moion X, sring from zero, nd defined on proiliy spce Ω, F, P. heorem. i For ny f [, ], ds he limi exiss in P. lim ds s f s X s. Mhemics Sujec Clssificion. Primry 6F5, 6F5; Secondry 6G5, 6G7, 6G48. c Americn Mhemicl Sociey

2 Giovnni Pecci nd Mrc Yor ii Xs ds.s.-p s heorem. i For ny g loc R, d nd for ny, he limi lim ds g X s Xs lim X s d g. exiss in proiliy, where, R, denoes joinly coninuous version of he process of Brownin locl imes. ii d.s.-p Boh prs of heorem. hve een proven in [] nd [3], wheres heorem.-i follows from [7, Corollry 3.3]; pr ii of heorem. seems o e new, lhough no quie unexpeced, given he well known Hölder coninuiy properies of Brownin locl imes when regrded s processes indexed y he spce vrile. In Secion we consruc unified proof of oh heorem.-i nd heorem.-i y mens of he well known Hrdy s inequliy in [, ], d see [], [9], [5] nd [], s well s [8] nd [3] for some proilisic discussions. In priculr, our poin is h, vi he use of sochsic Fuini s heorem nd nk s formul for heorem., he limis in heorem. nd. cn e descried in erms of ounded operors respecively from [, ], ds o he firs Wiener chos on [, ], i.e. he spce of r.v. s wih he form Y f s dx s nd f [, ], ds, nd from [, ], d o he spce of r.v. s wih he represenion Y g X u dx u where g. is deerminisic funcion such h [ E g X u ] du < As consequence, oh. nd. correspond o ounded operors from [, ], d o he firs ime-spce Brownin chos see [6], [7] nd [8] for deiled presenion, i.e. he spce of r.v. s of he form Y g u, X u dx u where g.,. is deerminisic nd such h [ E g u, X u ] du < On he oher hnd, heorem.-ii is proven in Secions 3 nd 4 y mens of some well known decomposiion resuls of Ry, Knigh nd Jeulin: such resuls chrcerize he semimringle represenion of he process, when is given nd possily rndom ime. More o he poin, n elemenry pplicion of roer s heorem see e.g. [4, p. 7] permis o prove h he r.v. Λ α d : α.3 is.s. finie or infinie ccording s α < or α. Noe lso h noher key rgumen in our proof is given y n esy modificion of heorem.-ii, which

3 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 3 cn e regrded s very priculr cse of he so clled Jeulin s emm [, emm 3., p. 44]. For he ske of compleeness, we presen version of such resul which is well fied o he frmework of he presen pper. Jeulin s emm. e Y, R e mesurle nd non negive process such h, for fixed, he lw of Y does no depend on, E Y < nd P Y > nd consider deerminisic, posiive nd σ-finie mesure µ d on R. hen, he even Y µ d < R hs proiliy zero or one ccording s µ R, or µ R <. We coninue he pper y proving some wek convergence resul for he rndom funcions convenienly normlized α d nd α d,.4 wih α, s ends o zero: our min resul will e heorem.3 i For every γ > X, γ 4 γ d, w X, [ d γ B d, ],, where B is sndrd Brownin shee on R R independen of X. ii For every γ > X, γ w d [ ] 4 γ d, X, B, γ,, where B is sndrd Brownin shee on R R independen of X iii Define he process τ s, s o e he righ-coninuous inverse of,. hen, he process log/ [ d τ ] 4 log, converges o 4X 4, in he sense of finie dimensionl disriuions s ends o zero. In he ls secion we evenully conjecure some exensions nd improvemens of he ove resuls, nd provide severl suggesions for furher reserch. In priculr, some key fcs re provided o sudy he convergence of he process s ends o zero. [ log/ d 4 log ],,

4 4 Giovnni Pecci nd Mrc Yor Bounded operors nd exisence of principl vlues e H e n operor on [, ], d : [, ], such h for every f [, ] Hf x x f w d, x [, ] for cerin weigh funcion w. > such h he ove inegrl is well defined for every x [, ] nd sup, w y dy <.. I is well known h condiion. is necessry nd sufficien see e.g. [5] or [, pp ] for he operor H o e ounded from [, ] o iself. Noe moreover h he rcheypl exmple of such n operor which is lso he mos useful for our discussion is clssiclly given y he Hrdy s operor for which i is rue h Hf x : x f y y dy Hf [,] f [,] for every f [, ] his is n elemenry version of Hrdy s inequliy: see [8] for proilisic proof. he key resul of he secion is Proposiion. For every H s ove, nd for every f [, ] i ii - lim for every. - lim [ f y Proof i By Fuini s heorem dsf s w s [f s X s ] w s ds [ Xs y]dx s ] w y dy dx u u s Hf s dx s [ Xs Hf X s ] dx s dx u s f s w s ds. Now, since [Hf s] ds c f [,] for cerin consn c, he sochsic inegrl [ ] dx u f s w s ds u u

5 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 5 is well defined. More o he poin [ E dx u c [ u f s w s ds ] dx u [ du f s w s ] s ds u f s ds nd he resul is chieved y pssing o he limi. ii Fuini s heorem yields gin f y w y dy [ Xs y]dx s nd since [ ] E [Hf X s ] Xs ds [ ] ] u s f s w s ds dx s Xs X s y f y w y dy E x x [Hf x] dx c [Hf x] dx c f [,] where c : sup x [,] E x nd c : c c, he inegrl ] dx s [ Xs X s is well defined. he following lso holds [ E dx s Xs f y w y dy X s E x x f y w y dy X s y f y w y dy ] [ f y w y ] y dy dx c f x dx x so h he desired conclusion is oined y leing end o zero. Remrk. Of course, he conclusion of poin sill holds if [ Xs y] is susiued y [ y<xs y]: his simple fc will e needed in he proof of Corollry.. Now, we inroduce he following closed suses of P: C, K nd K noe h C nd K re included in K he spce C : is he firs Wiener chos. he spce K : Y P : Y Y P : Y is he firs ime-spce Brownin chos. g s, X s dx s, g s. f s dx s, f [, ] E g s, X s ds <

6 6 Giovnni Pecci nd Mrc Yor c he spce K is he suse of K composed of r.v. s of he ype Y g X s dx s wih g. deerminisic nd such h E g X s ds < see [6] nd [7] for discussion of hese spces, nd furher ime-spce Brownin choses. hen, consequence of Proposiion. is nd Corollry. he pplicions f - lim f - lim [ f y [f s X s ] w s ds [ Xs y]dx s ] w y dy re ounded operors from [, ] respecively o C nd o K. More generlly, oh pplicions re ounded operors from [, ] o K. As specil cse of he ove discussion, y king w s /s in Proposiion. we oin srighforwrdly heorem.-i, nd moreover Corollry. For every, he limi lim d g. exiss in proiliy for ny g loc R, d. Proof As d g is.s. finie, i is sufficien o prove he resul for g. Bu nk s formul yields d d g g [ X X ] d d g <Xs dx s nd he resul is chieved y using Hrdy s inequliy nd Proposiion., long wih he fc h x x when < < x. We hve herefore proven heorem.-i. Remrk. - he limi in. is usully clled he principl vlue of he funcionl ds X s g X s. See [3] for generl discussion, nd [7] for noher proof of heorem.-i. Now, for fixed, nd for g loc R, d, we define λ g : nd λ g : P-lim [ λ g d, >.3 ] g : i is cler h he process λ g is con- inuous on, : he end of he nex secion we give sufficien condiions on g for he.s. coninuiy in priculr zero of such process.

7 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 7 3 Semimringle decomposiions of Brownin locl imes: Ry, Ry-Knigh, nd Jeulin heorems In order o sudy quniies s.4 or.3 we shll nlyze in some deil he spce indexed process R 3. where is suile rndom ime, such h 3. dmis sufficienly regulr semimringle decomposiion. o mke such frmework precise, we presen in his prgrph unified discussion of some well known represenion resuls for Brownin locl imes, due o Ry, Knigh, nd Jeulin see e.g. [4], [, Ch. XI] nd []. Wih he preceding noion, nd given possily rndom ime, we sy h he process dmis regulr semimringle decomposiion under proiliy mesure Q no necessrily equivlen o P if here exiss righ nd/or lef neighorhood of he origin, noed A, filrion E A nd E, Q - Brownin moion β A such h he process A is E, Q - semimringle on A, wih cnonicl decomposiion dβ K d A R dβ where he process K is E -dped nd sisfies i K d <.s. Q, ii A K d O.s. Q, K d A R 3. s converges o zero. As we shll see elow, he regulriy of given decomposiion serves s key ool o sudy he ehvior of he rndom funcion, A nd, in priculr, o sudy he r.v. s Λ α s defined in.3. However, efore discussing his poin we shll presen in some deil hree exmples of regulr decomposiions h re essenil o he nlysis of he susequen secions. he firs resul concerns deerminisic imes nd is due o Jeulin [, h. II..], u see lso []. heorem 3. Jeulin [] For every deerminisic ime, he process dmis regulr semimringle decomposiion under P, on he se A R. Noe h, whenever is semimringle in some pproprie filrion wih cnonicl decomposiion M C, hen necessrily he locl mringle M is such h M, M 4 d nd herefore he exisence of he BM β cn e deduced y sndrd rgumens see Corollry VI..3 in [] for deils.

8 8 Giovnni Pecci nd Mrc Yor o del wih some specil fmily of rndom imes, we inroduce furher definiions. We sy h process Z is he squre of n Ornsein-Uhleneck process wih prmeers θ nd δ, iniilized x noed OUSQ θ,δ x if Z is he soluion of he sochsic inegrl equion Z x Zs dβ s δ θ Z s ds where β is sndrd BM. A simple pplicion of Girsnov s heorem shows h, for fixed θ, he lw of Z is equivlen o he lw of squre of Bessel process sred from x nd wih dimension δ BESQ δ x, i.e. he soluion of he equion where β is sndrd BM. Now define, for s, Z x Z s dβ s δ τ s : inf l : l > s A well known resul, known s Ry-Knigh heorem, gives precise descripion of he lw of he process τs,. heorem 3. For every s, he process τ s hs regulr decomposiion under P, on he se A R nd wih K. More precisely, he process τs, is BESQ s. Proof See [, Ch. XI]. Suppose now h our reference spce is convenienly enlrged o suppor rndom ime S >, such h S is independen of X nd is exponenilly disriued wih prmeer θ /, θ >. he following resul, known s Ry s heorem, dels wih he lw of he process S, R. heorem 3.3 e he ove noion previl. hen he rndom vriles S nd X S re independen wih disriuion P S ds θ exp θs ds P X S dx θ exp θ x dx Moreover, under he proiliy P s,x : P S s, X S x, x >, he process S, R is Mrkovin nd inhomogeneous nd such h. S, is n OUSQ θ, s,. S, x is n OUSQ θ, s 3. S, x is n OUSQ θ, x S. As consequence, for every posiive s nd rel x, S dmis regulr decomposiion, under P s,x, on he se A R. Proof See [4]. Remrk 3. More generlly, i is shown in [3] h he generl OUSQ θ,δ process my e oined s he process of suile locl imes. As nnounced, consequence of Ry s heorem is he following

9 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 9 Proposiion 3. Fix, le he process nd suppose moreover h λ g d <. hen λ g is.s. coninuous on [,. g e defined s in.3, Proof We sr y considering, s in he semen of Ry s heorem, n independen exponenil ime S nd he mesure P s,x defined herein, for x >. hen, hnks o Girsnov heorem, o show h g i is sufficien o prove h [ S S] d <.s.-p s,x g E Z s d < where Z is BESQ s. Bu he ringulr nd Hölder inequliies yield E Z s 4 E Z d 4 s whence he desired conclusion. One cn lso show h g [ S S] d <.s.-p y slighly modifying he ove rgumens, nd herefore we hve P λ S g is coninuous on [, i.e. P λ g is coninuous on [,.e. d, hence les for one, u in fc for every due o he scling propery: λ w c g g c c d c c for every posiive c. Proposiion 3. implies evenully c Corollry 3. For fixed, he limi lim ds X s g X s Xs 3.3 exiss.s., nd is finie, for every g loc R, d such h g d <. 3.4

10 Giovnni Pecci nd Mrc Yor Remrk 3. Oserve h in [3] i is shown h condiion 3.4 is necessry nd sufficien for he process g s dx s o e n F X σ X - semimringle on [, ], where F X is he nurl filrion of X. he ove corollry is o e conrsed wih [7, heorem 3.], where necessry nd sufficien condiions re given for he.s. exisence of 3.3 when g loc R, d. he nex resul requires h he he process, for given rndom ime, dmis no only regulr decomposiion, u lso n Hölder coninuous modificion such h he Hölder consn is in : his holds, in priculr, when is n inegrle sopping ime wih respec o he nurl filrion of X see e.g. [, formul II γ, ]. Oserve h, o simplify noion, we suppose h he se A, on which is semimringle, coincides wih [, ]; moreover, hroughou he sequel, when deling wih given semimringle decomposiion of, nd when no furher specificion is needed, we will noe β he Brownin moion involved in such decomposiion hus cily implying is dependence on. Proposiion 3. Suppose h for fixed nd sricly posiive rndom ime such h, for some η,, [ ] k, η : E sup, [,] η <, 3.5 he process dmis regulr decomposiion on [, ] wih respec o P: hen, [ d ] 4 dβ converges.s.-p o finie limi s ends o zero. Proof We wrie d d d d 4 : A B C dβ dβ dβ K d nd we evlue ech memer seprely. For B we hve.s. d K d K d d d d K d K d c ω c ω < From now on, when wriing sopping ime we will implicily men sopping ime in he nurl filrion of X.

11 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes wheres for C d d dβ K d K d c ω,.s. - P s, due o he Hölder properies of, d Finlly we wrie, for A d dβ d d 4 < [ dβ d 8 dβ 4 [ d K d [ d.s.-p. dβ ] dβ ] dβ ] dβ. Now, for cerin η,, [ d ] E d dβ E d k, η 4k, η d d η, η d s one deduces from 3.5. Moreover, Hölder inequliy yields E dβ dβ E d nd, since >.s.-p, his implies h lim exiss nd is finie.s.-p. [A B C ] 4 In n nlogous wy one cn lso prove 8k η 8k η d dβ

12 Giovnni Pecci nd Mrc Yor Proposiion 3.3 Suppose h, for rndom ime s in Proposiion 3., he process dmis regulr decomposiion on [, ], wih respec o P: hen, d [ 4 β] converges.s.-p o finie limi s ends o zero. A firs pplicion of Proposiion 3. nd Proposiion 3.3 is given in he nex prgrph, where we presen proof of heorem.-ii. 4 Sudy of he Λ α s Define, for given, he rndom vrile Λ α s in.3: we pply he resuls of he previous secion o chrcerize he proiliy of Λ α eing finie. Proposiion 4. Suppose h, for rndom ime s in Proposiion 3., he process dmis regulr semimringle decomposiion, wih respec o P, on he inervl [, ]: hen P Λ α < equls one or zero, ccording s α, or α [,. Proof Noe h,.s. - P, Λ α is finie if, nd only if d α < - Cse α <. From roer s heorem, here exiss posiive consn K ω such h, for ny η, K ω η for every,.s.-p. So h, for η α / δ, where δ < α/,.s.-p d α K ω η α d <. - Cse α. I is clerly sufficien o prove he clim for α, nd in his cse he conclusion is consequence of Proposiion 3. nd Jeulin s emm s reclled in he Inroducion. As mer of fc, >.s.-p nd, ccording o Proposiion 3., here exiss n.s. finie rndom vrile Y ω such h,.s. - P, lim d d Y 4 lim dβ Finlly, from Jeulin s emm wih Y : dβ / nd µ d : d/, d lim dβ.s-p Anlogously, we hve lso

13 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 3 Proposiion 4. If, for rndom ime s in he Proposiion 3., he process dmis regulr decomposiion on [, ] wih respec o P, hen P lim α d < equls one or zero, ccording s α, or α [,. Since ccording o heorem 3. for deerminisic ime > he process dmis regulr semimringle decomposiion, Proposiion 4. implies srighforwrdly heorem.-ii, whose proof is herefore concluded. Oher pplicions of Proposiion 3. nd Proposiion 3.3 re given in he following prgrph. 5 Wek convergence he im of his prgrph is o sudy he speed which quniies s α d nd α d 5. for α or Xs s ds 5. diverge o infiniy when ends o zero. o do his, we will eslish wek convergence resuls of he ype of heorem.3, y relying oh on he sme line of resoning h led o he proof of Proposiion 3. nd 3.3, nd on limi heorems in disriuion such s he ones discussed e.g. in [, Ch. XIII]. As, for α >, he sudy of such quniies s 5. does essenilly no depend on he res of he prgrph, we will firs del wih he proof of he firs nd second pr of heorem.3 he reder is referred o [6] for severl reled resuls. Proof of heorem.3-i nd.3-ii. I is known h consequence of he Ppnicolou, Sroock nd Vrdhn heorem [, h. XIII..3] is h he lw of he process, indexed y, R, X, converges wekly s ends o zero o h of X, B d, γ 4 where B is sndrd Brownin shee on R R independen of X. hus, o ge he resul we wrie, due o he chnge of vrile /, / d γ d γ γ / d γ 4 γ

14 4 Giovnni Pecci nd Mrc Yor which implies he desired conclusion, s one cn esily check y firs oserving h for ny fixed N, he process N converges o N d γ d γ [ B d, ], hnks o he Coninuous Mpping heorem see e.g. [5], nd hen y considering suile ipschiz coninuous nd ounded funcionls. Pr ii is oined nlogously. Now we concenre on 5.: he following resul is prly consequence of sochsic Fuini s heorem see [9] for severl generlizions. Proposiion 5. e X e sndrd Brownin Moion: hen i For every fixed w dx u X u N, log/ u where N, is normlly disriued, wih zero men nd uni vrince. ii For every fixed [ ] dy X s, s ; log/ y X y log w X s, s ; N, where N, indices sndrd Gussin r. v. independen of X. Proof o ge he firs poin, jus pply he Dmis-Duins-Schwrz heorem see e.g. [, Ch. V] nd he scling properies of BM: dx u X u β log/ u log/ dux u u where β nd β : log β log dux u u o β log/ dux u u re wo sndrd BM s: now wrie duxu Xu du u log Xu du u where he o. is o e inerpreed in he.s. sense, nd finlly, y Fuini s heorem, Xu du log/ log/ u log/ dy dx u X u log/ u y. log/ log/ log/ dx u X u u Esy compuions show h he second erm ends o zero in he sense s ends o zero, nd he resul is compleely oined once he sympoic independence is proved. o see his, noe h Jeulin s emm nd he lw of he iered

15 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 5 logrihm imply s log/ X u du.s. for every s u s ends o zero, so h one cn pply n sympoic version of Knigh s heorem such s he one discussed for insnce in [, h. XIII..3]. he second sserion follows once gin from Fuini s heorem, since dy y X y log dx u X u u. We will lso need he following emm 5. For every p - lim p Xy dy y p. log/ Proof Jus pply Iô s formul, long wih Fuini s heorem o ge p p Xy dy p dy y dy y y log/ log/ y X dx log/ p p X p dx log/ nd one cn esily prove h he second erm converges o zero in, s ends o zero. Proposiion 5. provides n imporn ool o sudy he sympoic ehvior of quniies such s 5., s poined ou in he following Proposiion 5. e e rndom ime s in Proposiion 3., nd such h, under P, dmis regulr semimringle decomposiion on [, ]. hen, s ends o zero, [ d log/ 4 log ] nd 4 [ d log/ β log ] oh converge in lw o 8 N,, where N, indices sndrd Gussin r.v., independen of. Proof According o Proposiion 3.3, [ d lim log/ 4 log ] 4 [ d lim log/ β log ]

16 6 Giovnni Pecci nd Mrc Yor [ ] nd, from Proposiion 5., i is known h log / d β log converges in lw o N,. Evenully, he independence follows from sndrd rgumens. One cn lso prove s consequence of Proposiion 3. Proposiion 5.3 e e rndom ime s in Proposiion 5., such h dmis regulr decomposiion under P, on he se [, ]. Suppose moreover h here exiss funcion φ such h φ, nd fmily of r.v. s C ω such h 4 d φ dβ C converges in disriuion s ends o zero. hen, he rndom vrile φ d C converges in disriuion o he sme limi s ends o zero. Noe h, due o heorem 3. nd n pplicion e.g. of [, formul I γ, ], condiion 3.5, nd hus he ssumpions of Proposiion 5., re sisfied whenever τ s inf l : l > s. his implies h for every s [ d log/ τs s ] 4s log w 4X 4s We cn lso prove more complee resul, corresponding o heorem.3-iii Proof of heorem.3-iii. Since he generl cse cn e sudied y excly he sme echniques plus sndrd recurrence rgumen, we will prove such resul in deil only for he finie dimensionl disriuions of order. We sr y defining for rionl s,, such h s <, nd for Z u τ u, u, s 5.3 Z u Z u u, u, s Z s τ τ s s Z Z s 5.4 nd we noice h, due o he srong Mrkov propery of Brownin moion, he processes Z s nd Z s re independen, nd in priculr, ccording o heorem 3., heir lw is chrcerized y he exisence of wo independen BM β nd β such h Z s Z s Z s dβ Z s sdβ

17 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 7 Now we cn wrie τ Z nd se evenully Z s I, u : I, s : Z Z s Z s d d Z u Z s Z s u, s 5.5 so h he desired resul is compleely proven once i is shown h he lw of he pir I, s 4s log log/, I, I, s 4 s log converges o 4 N, 4s, N, 4 s where he N i s re independen Gussin r.v. s, wih he indiced men nd vrince. As I, I, s I, s I, s d d Z s d Z s dz s : I, s A,s A,s Z s 5.6 Z s dz s we need only show h I, s 4s log log/, I, s 4 s log, A,s, A,s converges in lw o 4 N, 4s, N, 4 s, N 3, 4s s, N 4, 4s s where he N i s re Gussin nd muully independen, nd he noionl convenions re s ove. As he. erms re sympoiclly independen y consrucion of he. erms i is sufficien o show h, [ I, s 4 s log ] 5.7 log/ is sympoiclly independen of log/ / A,s nd h oh converge o he prescried limis, nd h [ I, s 4s log ] 5.8 log/ is sympoiclly independen of log/ / A,s nd h hese wo quniies converge o he correc limis; we will del exclusively wih he cse of 5.7,

18 8 Giovnni Pecci nd Mrc Yor wheres 5.8 cn e sudied nlogously, nd is herefore lef o he reder. o check his, we inroduce he following noion: we wrie C B, whenever C B in proiliy, s ends o zero. Wih his convenion, one cn prove he sympoic relions log/ [I, s 4 s log/] nd, nlogously, [ 4 s ] d log/ β log/ 8 s log/ 8 s log/ A,s log log dβ β dβ β 8 s s log d o prove he resul, consider he mringles Mx, x dβ β log/ Mx, x dβ β log/ 5.9 Z s dz s 5. dβ β., x, x nd cll σy i,, y, i,, he righ-coninuous inverse of he incresing process M i,, M i,. Since for every x x x M,, M, d β x log/ M,, M, x x d β log/ in P in P ccording o he sympoic Knigh s heorem see [, h. XIII..3], he resul is proven once i is shown h for every y P-lim M,, M, P-lim σ M,, M,. y, σy, We now concenre on σ, he cse of σ, cn e hndled nlogously: s M,, M, d β log/.s.

19 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 9 due o he lw of he iered logrihm see [, Ch. II] nd, ccording o emm 5., P- lim M,, M, for every >, one hs he sympoic relions, for fixed y > nd > y, P σ, y < P M,, M, > y P σ, y P M,, M, y. I follows h, for s ove, he following relions hold for every >, when converges o zero: P M,, M, σ > P σ, y, y < ; σy, P σy, < ; M,, M, σ >, d β β o P σy, < ; log/ σ, y log/ β σ, y σ, y σ, y : o Γ,, where he ls equliy derives from x M,, M, d β β x log/ x d β β log/ β σ, y log/ β σ, y dβ x d β β x, log/ s well s he following consequence of he sochsic Fuini s heorem: x x Now we wrie Γ, P d β β log/ β xβ x x log/ x x log/ d β β log/ log/ sup x [, ] [ x >σy, x : P A B C > β dβ β σy, σ, y β dβ x x β σy, y β dβ > β ] dβ. log/ β dβ > 5.

20 Giovnni Pecci nd Mrc Yor so h he resul is proven once i is shown h A, B nd C converge seprely o zero in proiliy for. o see h A converges o zero in nd herefore in proiliy i is sufficien o ke expecions nd le converge o zero. o del wih B jus oserve h i for ny sequence x such h x nd x, due o he lw of he iered logrihm,.s.-p β x β x x log/ x log log/x x log/, where he symol mens usul sympoic relion; ii for ny sequence x such h x nd x <,.s.-p he lw of iered logrihm implies β x β x x log log x x log/ x log/ log log log/ ; log/ iii for ny sequence x such h x c, C for given < c < C <,.s.-p β x β x, x log/ so h he.s. convergence o zero of B is oined y srighforwrd rgumens. Finlly, o del wih C we recll h, for ny N >, he Burkholder-Dvis- Gundy inequliy [, Ch. IV] yields he exisence of universl consn c such h log/ E [ nd [ E log/ hus implying h sup x [,N] sup x [,N] lim log/ in proiliy for ny >. [ x x sup x [, ] β dβ N ] c log/ d 5. β dβ N ] c log/ d 5.3 x β x dβ ] β dβ Conclusion nd suggesions for furher reserch he nex sep owrds full soluion of he prolems rised in his pper, is clerly o sudy he limiing ehvior in lw of he wo processes Y d, :, nd Y d, :,.

21 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes In priculr, we elieve h he following convergence relion holds when ends o zero X, log [ Y 4 log ], w X, 4ξ 4, 6. where ξ is sndrd Brownin moion independen of X noe h his conclusion would provide, y mens of sndrd ime-chnge, n lernive proof o heorem.3-iii. o see why such clim is rher plusile, we refer o he semen of Proposiion 5. wih s deerminisic ime, nd moreover we use nk s formul o wrie, due o sndrd Hölder coninuiy rgumens, nd lso Bu d log/ 4 log/ d [ d <Xs<dX s X X X X ] <Xs<dX s 4 d O log/ <Xs<dX s. d [ d <Xs<ds <Xs<ds log <Xs<dX s log <Xs<dX s ] log. d <Xs<ds 6. dx x o log/ s he inegrl in he middle converges o finie limi when ends o zero. So, y he Iô formul, he relevn erm in 6. is given y d s dx s dx u <Xu< <Xs< s dx s dx u X s X u. According o he DDS heorem, nd y scling, here exiss sequence of rescled BM s β such h s dx s dx u log/ X s X u β 4 log/ ds[ s dxu Xs Xu ] I follows h 6. will e oined once he following clim hs een proven

22 Giovnni Pecci nd Mrc Yor Conjecure 6. As ends o zero, he sequence β is sympoiclly > independen of X, nd he process J : [ s ] log ds dx u X s X u converges in lw o. We conclude he pper y proving wo pril resuls ou he limi ehvior of he sequence J >. Proposiion 6. When ends o zero, he sequence J [ d ds 4 log/ s ] s X s converges.s. o zero, uniformly on he compc of R. Now, Proof nk s formul gives [ s ] ds dx u X s X u [ d ds X s X s X s ] [ d ds 4 X s s ] s [ d d ds X s X s s ] s X s X s : Q Q Q 3. Q ds <Xs< d X d s ds Xs> X s X s nd i is esily verified h oh erms converge.s. o finie limi s ends o zero. On he oher hnd, n nlogous decomposiion holds for Q 3, nd moreover, due o he Hölder coninuiy properies of Brownin locl imes ds Xs< [ d d X s X s ] c η for cerin η,, so h he ls expression converges o zero. Similr rgumens show lso [ d d ds Xs> X s X s X s X s ] c ds Xs>X s X s X η s for η s efore, so h he righ erm converges.s. o finie limi.

23 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 3 J Proposiion 6. When ends o zero, he sequence log/ [ s u ] ds dx u dx v X s X u X s X v converges.s. o, nd he convergence kes plce uniformly on he compc of R. Proof Wrie firs [ s ] ds dx u X s X u [ ds s dx u ds s : A A ] s X s X u du X s X u du X s X u Now, A ds s du Xu X s> X s X u s ds du Xu X s<. We sr y sudying he firs ddend. Esy compuions show h ds s du Xu X s> X s X u [ Xs ds dx x s X s x ds dx x s X s ] Xs> nd moreover ds dx x s X s x ds dx x s s X s x ds dx s X s x.

24 4 Giovnni Pecci nd Mrc Yor Hölder coninuiy implies h.s. here exiss consn c such h, for cerin η, / ds dx x s s X s x o log c ds X s η o log c o log. dsxs η Xs> c η dx x On he oher hnd ds dx s X s x o log o log o o log log ds s dx x log X s dx x ds s >Xs d s x s s he ls equliy eing jusified y he inegrion y prs formul d s x s s s x nd sndrd Hölder coninuiy rgumens. Moreover [ ] Xs ds dx x s Xs> X s o log o log ds X s log Xs d s s x s s ds X s s Xs> dx x s ds s Xs> X s Xs> s, so h log/ ds s du Xu X s> X s X u converges.s. o.

25 Hrdy s inequliy in [, ] nd principl vlues of Brownin locl imes 5 Now wrie ds s du Xu X s< [ Xs <Xs< c c dx x s <Xs< X s dx x s o oin h log/ A s ends o zero uniformly on he compc of R. o conclude, oserve h sndrd inegrion y prs yields [ s u ] A ds dx u dx v X s X u X s X v ] References [] Appel, J. nd Kufner, A. 995, On he wo-dimensionl Hrdy operor in eesgue spces wih mixed norms, Anlysis 5, [] Brlow, M.. nd Yor, M. 98, Semi-mringle Inequliies vi he Grsi-Rodemich- Rumsey emm, nd Applicions o ocl imes, Journl of Funcionl Anlysis 49, [3] Bine, Ph. nd Yor, M. 987, Vleurs principles ssociées ux emps locux rowniens, Bull. Sci. Mh., 3-. [4] Bine, Ph. nd Yor, M. 988, Sur l loi des emps locux rowniens pris en un emps exponeniel, Séminire de Proiliés XXII, ecure noes in Mhemics 3, Springer, Berlin Heidelerg New York, [5] Billingsley, P. 968, Convergence of Proiliy Mesures, Wiley, New York. [6] Csáki, E., Csörgö, M., Földes, A. nd Révész, P. 989, Brownin locl ime pproximed y Wiener shee, he Annls of Proiliy 7, [7] Cherny, A. S., Principl Vlues of he inegrl funcionls of Brownin Moion: Exisence, Coninuiy nd n Exension of Iô s Formul, Séminire de Proiliés XXXV, Springer, Berlin, [8] Doni-Mrin, C. nd Yor, M. 989, Mouvemen Brownien e inéglié d Hrdy dns, Séminire de Proiliés XXIII, ecure noes in Mhemics 37, Springer, Berlin, [9] Hrdy, G., ilewood,. E. nd Póly, G. 934, Inequliies, Cmridge Universiy Press, Cmridge. [] Jeulin,. 98, Semimringles e Grossissemen d une Filrion, ecure Noes in Mhemics 833, Springer, Berlin. [] Jeulin,. 98, Sur l convergence solue de cerines inégrles, Séminire de Proiliés XVI, ecure noes in Mhemics 9, Springer, Berlin Heidelerg New York, [] Jeulin,. 985, Applicion de l héorie du grossissemen à l éude des emps locux Browniens, in: ecure Noes in Mhemics 8, Springer, Berlin, [3] Jeulin,. nd Yor, M. 979, Inéglié de Hrdy, semi-mringles e fux-mis, Séminire de Proiliés XIII, ecure Noes in Mhemics 7, Springer, Berlin Heidelerg New York, [4] Krzs, I. nd Shreve S. E. 988, Brownin Moion nd Sochsic Clculus, Springer, Berlin. [5] Muckenhoup, B. 97, Hrdy s inequliy wih weighs, Sudi Mh. 44, 3-38.

26 6 Giovnni Pecci nd Mrc Yor [6] Pecci, G., A represenion resul for ime-spce Brownin Chos, Annles de l Insiu H. Poincré 37, [7] Pecci, G. 3, Explici formule for ime-spce Brownin chos, Bernoulli 9, [8] Pecci, G., Chos Brownien d espce-emps, décomposiions d Hoeffding e prolèmes de convergence ssociés, Ph. D. hesis, Universié de Pris VI. [9] Pecci, G. nd Yor, M., Four limi heorems for qudric funcionls of Brownin moion nd Brownin ridge, In his volume. [] Perkins, E. 98, ocl ime is semimringle, Z. W. 58, [] Revuz D. nd Yor, M. 994, Coninuous Mringles nd Brownin Moion, Springer, Berlin Heidelerg New York. [] Swyer, E. 985, Weighed inequliies for he wo-dimensionl Hrdy operor, Sudi Mh. 8, -5 [3] Yor, M. 99, Some specs of Brownin moion. Pr I: some specil funcionls. ecures in Mhemics, EH Zürich, Birkhäuser, Bsel.