A characterization of reciprocal processes via an integration by parts formula on the path space

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1 A chaaceizaion of ecipocal pocesses via an inegaion by pas fomula on he pah space Sylvie Rœlly Cene de Mahémaiques Appliquées UMR CNRS 764 École Polyechnique 928 Palaiseau Cédex, Fance oelly@cmapx.polyechnique.f Michèle Thieullen Laboaoie de Pobabiliés e Modèles Aléaoies UMR CNRS 7599 Univesié Pais 6, boie 88 4, place Jussieu Pais Cédex 5, Fance mh@cc.jussieu.f Absac We chaaceize in his pape he class of ecipocal pocesses associaed o a Bownian diffusion heefoe no necessaily Gaussian as he se of Pobabiliy measues unde which a ceain inegaion by pas fomula holds on he pah space C[, ]; R. This funcional equaion can be inepeed as a peubed dualiy equaion beween Malliavin deivaive opeao and sochasic inegaion. An applicaion o peiodic Onsein-Uhlenbeck pocess is pesened. We also deduce fom ou inegaion by pas fomula he exisence of Nelson deivaives fo geneal ecipocal pocesses. AMS Classificaions: 6G5-6G6-6H - 6J6. KEY-WORDS: Recipocal pocess, Inegaion by pas fomula, Sochasic bidge, Sochasic diffeenial equaion wih bounday condiions, Sochasic Newon equaion.

2 Inoducion The pesen pape deals wih ecipocal pocesses which we chaaceize by a simple funcional equaion, an inegaion by pas fomula, on he space of coninuous pahs. Recipocal pocesses ae Makovian fields wih espec o he ime paamee and heefoe a genealizaion of Makov pocesses. The inees in hese pocesses was moivaed by a Confeence of Schödinge [22] abou he mos pobable dynamics fo a Bownian paicle whose laws a wo diffeen imes ae given. Acually, Schödinge was only concened wih Makovian ecipocal pocesses. His inepeaion in ems of lage deviaions fom an expeced behavio was fuhe developed by Föllme, Caiaux and Léonad, Gane. Schödinge pocesses wee also analysed by Zambini and Nagasawa fo hei possible connecions o quanum mechanics. One yea afe Schödinge, Bensein noiced he impoance of non-makovian pocesses wih given condiional dynamics, whee he condiioning is made a wo fixed imes. This was he beginning of he sudy of geneal ecipocal pocesses. Jamison [] poved ha he se of ecipocal pocesses is paiioned ino classes; each subclass is chaaceized by a se of funcions, called Recipocal Chaaceisics [4], [2]. The main esul we obain is ha, fo eal-valued pocesses, each class of ecipocal pocesses wih Recipocal Chaaceisics, F coincides wih he se of soluions of a funcional equaion in which he funcion F plays a simila ole as he Hamilon funcion associaed o a se of Gibbs measues [9]. This funcional equaion is indeed an inegaion by pas fomula on he pah space C[, ]; R and i exhibis a peubed dualiy elaion beween he sochasic inegaion w... a ecipocal pocess and he Malliavin deivaive opeao along a class of es funcions which is smalle han he usual one on he Wiene space. Then, o illusae ou appoach of ecipocal pocesses, we conside some Sochasic Diffeenial Equaions wih ime bounday condiions iniial and final imes. Soluions of such sochasic equaions fom a wide class of non adaped hen anicipaive non Makovian pocesses and we hope ha ou way o idenify hei ecipocal popeies will be a help in he analysis of such pocesses. The seach of a chaaceizaion of ecipocal pocesses as he se of soluions of some second ode equaion was poposed by Kene cf [2]. I was achieved in he Gaussian case by Kene, Fezza and Levy in [4]. Fo he Gaussian saionay case see also [2]. As fa as we know, no such chaaceizaion was available in he non Gaussian case. Ou esul fills his gap in dimension. Concening he geneal non Gaussian case, one of he auhos poved in [23] ha ecipocal pocesses saisfy a sochasic Newon equaion which involves Nelson deivaives and a sochasic vesion of acceleaion. A he end of he pesen pape we sudy he elaionship beween ou esul and he esul of [23]. We also show ha he inegaion by pas fomula which we inoduce povides sufficien condiions fo a ecipocal pocess o be diffeeniable in Nelson s sense. Recipocal pocesses ae ime andom fields defined on a compac ime ineval. When he ime paamee belongs o an ineval wih infinie lengh, he poblemaic is closed o ime Gibbs measue, o quasi-invaian measue on he space of coninuous funcions, as inoduced in he sevenies in he conex of Euclidean uanum Field heoy by Couège and Renouad [5] see also [2]. Sill a lo of poblems in his diecion emain open. 2

3 The pape is devided ino he following secions.. Inoducion. 2. Noaions and famewok. 3. Chaaceizaion of RP, he ecipocal class associaed o he Bownian moion. 4. Chaaceizaion of he ecipocal class associaed o a Bownian diffusion. 5. Applicaion o he peiodic Onsein-Uhlenbeck pocess. 6. Sochasic Newon equaion fo ecipocal pocesses. 2 Noaions and famewok Le Ω = C[, ]; R be he canonical - polish - pah space of coninuous eal-valued funcions on [, ], endowed wih F, he canonical σ-field. Le X [,] denoe he family of canonical pojecions fom Ω ino R. PΩ is he se of pobabiliy measues on Ω. We use equivalenly he noaion f o E f fo he inegal of he funcion f unde a pobabiliy measue. Le P PΩ denoe he Wiene measue on Ω saisfying P X = =. Moe geneally, fo x R, P x is he shifed Wiene measue saisfying P X = x =. We define now he space of smooh cylindical funcionals on Ω : S = {Φ, Φω = ϕω,..., ω n whee ϕ is a bounded C -funcion fom R n in R wih bounded deivaives and... n }. Clealy S L 2 Ω; P. On S we define he deivaion opeao D in he diecion g L 2, by whee D g Φω = = D Φω = n i= n i= ϕ i ω,..., ω n gd x i gd Φωd ϕ x i ω,..., ω n i. I is clea ha D g Φ is also equal o he Gâeaux-deivaive of Φ in he diecion. gd, which is a ypical elemen of he Cameon-Main space. We can now define he space D,2 as he closue of S fo he following nom : Φ 2,2 = E P Φ 2 + E P D Φ 2 d. I is well known see fo example [] ha he opeao D also called Malliavin deivaion is he dual opeao on D,2 of he sochasic inegaion opeao δ defined on Ω by δgω = gdω : g L 2,, Φ D,2, E P D g Φ = E P Φ δg 3

4 The main objec we deal wih in his pape ae he so called ecipocal classes. We conside a given Makov diffusion P PΩ such ha, fo each s <, he map x, y P./X s = x, X = y is coninuous on R 2. The ecipocal class associaed o P is he subse R P of PΩ defined by : R P = { PΩ, s <,./F s ˆF = P./X s, X } 2 whee he fowad esp. backwad filaion F [,] esp. ˆF [,] is given by F = σx s, s, esp. ˆF = σx s, s. Each elemen of R P is called a ecipocal pocess associaed o P. Fom he definiion 2 of a ecipocal class, i is clea ha each ecipocal pocess is a Makovian field in he sense ha, fo s <, F s ˆF and σx ; s ae independen unde condiionnally o σx s, X. Neveheless, a ecipocal pocess is no necessaily a Makov pocess. Jamison gave in [] he following descipion of he subse R M P whose elemens ae he Makovian pocesses of R P : R M P = { R P, ν, ν σ-finie measues on R, X, X dx, dy = p, x,, yν dxν dy} 3 whee ps, x,, y is he pobabiliy ansiion densiy of P which always exiss and is egula in he cases eaed in his pape. Due o hisoical easons ecalled in he inoducion, he elemens of R M P ae called in he lieaue Schödinge pocesses. Le us menion he following equivalen definiion of R P as he class of pocesses having he same bidges as P see [] : R P = { PΩ, m PR 2, = P /X = x, X = ymdx, dy}. 4 R 2 Remak ha fom he above definiion 4 any ecipocal pocess in R P is a mixue of bidges of P. 3 Chaaceizaion of RP, he ecipocal class associaed o he Bownian moion 3. Dualiy unde he Bownian bidge We ecalled in he above equaliy he dualiy beween Malliavin deivaive D and sochasic inegaion δ unde he Wiene measue P. In fac, emains valid if P is eplaced by any ohe Wiene measue P x, x R, and heefoe, by lineaiy of his equaion wih espec o he inegao, equaliy is also ue unde P µ, a µ-mixue of P x, x R: P µ = P x µdx, µ PR. 5 R Wha is moe supising is he fac ha he dualiy beween D and δ holds also unde any desinegaion of he Wiene measue in Bownian bidges, if we esic he class of es funcions g in o a smalle space han L 2,. So le us inoduce he funcion space L 2, = {g L 2,, 4 g d = }.

5 I is he ohogonal subspace in L 2, o he consan funcions. We have Poposiion 3. Le x, y R 2 and P x,y PΩ be he law of he Bownian bidge on [, ] fom x o y. Then g L 2,, Φ S, P x,y D g Φ = P x,y Φ δg. 6 Poof : I is enough o veify 6 fo any Φω = ϕω,..., ω n, < <... < n <. whee ϕ is a bounded C -funcion on R n wih bounded deivaives, and g a sep funcion wih vanishing inegal on [, ], i.e. g = p α i [si,s i+ [ + β {} i= whee s <... s p < s p+ =, α i, β R and p i= α is i+ s i =. We emak ha α p = if he suppo of g is included in [, [. Le τ = maxs p, n. I is clea ha τ < and ha Φ is F τ -measuable. Le also decompose g as : g = g τ + α p [τ,[ + β {} 7 in such a way ha g τ admis a suppo included in [, τ]. Then P x,y Φ δg = P x,y Φ δg τ + α p P x,y y X τ Φ. 8 To manage he fis em of he.h.s. we now use he fac ha P x,y is absoluely coninuous wih espec o P x on C[, τ]; R : fom he semi-maingale decomposiion of he Bownian bidge see fo example [2],IV.4 fo geneal popeies of Bownian bidges, and by Gisanov heoem, he densiy is he following exponenial maingale dp x,y Fτ y X s dp x X = exp s dx s 2 y Xs 2ds s which we denoe by Mτ y X. Remak ha, o ensue he exisence of he densiy M y X, i is necessay o esic he pobabiliy measues P x,y and P x on [, τ] wih τ <, since M y X explodes fo =. We hen anspose he dualiy unde P x Fτ o a peubaed inegaion by pas fomula unde P x,y Fτ : P x,y Φ δg τ = P x Φ δg τ Mτ y By a simple compuaion, D g τ M y τ M y τ = P x D g τ Φ Mτ y = P x,y D g τ Φ X = + P x Φ D g τ M y τ + P x,y Φ D g τ M y τ M y τ g τ D log M y τ Xd 5.

6 whee So and 8 becomes Bu, by 7, D log Mτ y X = y X P x,y D g τ M y τ M y τ = y X τ τ s dx s y X s s ds P x,y a.s.. Φ = P x,y y X τ τ Φ g τ d P x,y Φ δg = P x,y D g τ Φ + P x,y y X τ τ Φ g τ d + α p P x,y y X τ Φ. 9 g τ d = = g τ d gd α p τ = α p τ. Fuhemoe D g τ Φ = D g Φ since Φ is F τ -measuable and g τ [,τ] = g [,τ]. Fom hese las emaks, we can conclude ha 9 implies 6. Remak 3.2 : To pove he dualiy equaion 6 unde P, we could also use he coespondence beween he Gauss space of he Bownian bidge P, and he Wiene space Ω, P, based on he isomophism α beween L 2, and L2, defined by : g L 2,, αg = g + gs ds. In fac, following Gosselin and Wuzbache [9], Poposiion 2.2, if X is a Bownian moion unde P, he image pocess of X unde he ansfomaion dω s Θ : ω Θω = s < is a Bownian bidge wih law P, ; he sochasic inegal δgθx = gdθx is well defined fo g L 2, and moeove : δgθx = δαgx P a.s.. So, o deduce 6 fom i is enough o emak ha, fo g L 2, and Φ S, D g Φ Θ = D αg Φ Θ. Remak 3.3 : As emaked a he beginning of he secion 3., he dualiy P µ Φ δg = P µ D g Φ 6

7 holds fo any g L 2,, Φ S and µ PR. Taking Φω = φ ω φ ω Φω fo φ, φ C R, and Φ S, one obains fom P µ φ X φ X P µ Φδg/X, X = P µ φ X φ X P µ D g Φ/X, X + P µ φ X φ X Φ g d. So, fo g L 2,, he las em vanishes and his implies P X,X Φδg = P µ Φδg/X, X = P X,X D g Φ fo a.s.x, X unde P µ. Anyway, his esul is weake han 6 which holds fo all x, y R 2. Remak 3.4 : One could also pove equaliy 6 fo Φ D, Chaaceizaion of he condiional pobabiliies The naual quesion is now o analyse if he dualiy unde a measue beween D and δ esed on all g, Φ L 2, S chaaceizes he bidges of. The posiive answe is he objec of he following : Poposiion 3.5 Le PΩ such ha sup [,] X < +. If g L 2,, Φ S, D gφ = Φ δg hen./x, X = P X,X a.s.. Poof : Fis, following he same agumen as in Remak 3.3, i is clea ha also holds unde./x, X a.s.. Fo simpliciy, le us denoe by x,y PΩ he law of he bidge of on [, ] beween x and y, x, y R 2. Le g a fixed sep funcion on [, ], and fo λ R, define ψλ = x,y expiλδ g. 2 By eceneing g, we also inoduce he sep funcion g = g Now, emaking ha ψ is diffeeniable on R, we obain ψ λ = i x,y δ g expiλδ g = i x,y δg + y x g d L 2,. 3 g d expiλδ g = ie iλy x R g d x,y δg expiλδg + iy x g d ψλ. Fom, using he fac ha Φ = expiλδg S, we deduce ha fo X, X a.a.x, y, x,y δg expiλδg = x,y D g expiλδg 7

8 which is equivalen o So, ψ λ = x,y δg expiλδg = iλ g 2 d x,y expiλδg iy x g d λ g 2 d g d 2 ψλ. The unique soluion of his diffeenial equaion wih iniial condiion ψ = is ψλ = exp λ2 g 2 d g d 2 + iλy x g d. 4 2 Thus, fo X, X almos all x, y, equaliy 4 holds ue fo all g in he following counable se of sep funcions : { p i= α i [si,s i+ [, = s <... s p < s p+ =, p N, s i, α i }. This se is dense in L 2,, so equaliy 4 holds also ue fo each g L 2,, since is boh sides ae L 2, -coninuous funcionals of g unde he assumpion ha sup [,] X < +. Nex sep is o idenify he pocess wih he above chaaceisic funcional. Le us indicae wo possibiliies : Eihe one veifies ha he following pocess Y = x + B + y B whee B is a Bownian moion, is indeed a Bownian bidge wih law P x,y and admis ψ as chaaceisic funcional cf. fo example Theoem IV.4.3 in [2]. O one emaks ha ψ is associaed o a Gaussian pocess : by aking λ = and g = p α i [i, i [, = < <... < p < p =, i= i is clea ha x,y expiδ g is he exponenial of a bilinea fom in α i. Moeove, aking now g = [s,], we obain he fis wo momens of his Gaussian pocess : implies x,y expiλδ [s,] = e λ2 2 s s2 +iλy x s x,y X = y + x and CovX s, X = s, s. These momens also chaaceize he law of he Bownian bidge. 3.3 The class RP as he se of soluions of a dualiy equaion I is known ha he dualiy holds only when he undelying Pobabiliy measue on Ω belongs o he se {P µ, µ PR} PΩ see [9], Theoem.2. By esicing he class of es funcions g fom L 2, o he smalle se L 2,, i is clea ha he se of Pobabiliy measues unde which he dualiy holds is lage. The following heoem does explici his subse of PΩ. 8

9 Theoem 3.6 Le PΩ such ha sup [,] X < +. The following wo asseions ae equivalen : i g L 2,, Φ S, D g Φ = Φ δg ii RP, i.e. is a ecipocal pocess in he same class as he Bownian moion. Poof : By Poposiion 3.5, i implies he a.s. equaliy beween he bidges of and hose of P. Bu = /X = x, X = y mdx, dy R 2 whee m = X, X.Then using he definiion of RP given in 4 we obain diecly asseion ii. Recipocally, if RP, he desinegaion 4 holds. So is a mixue in x, y of bidges P x,y. Bu, by Poposiion 3., unde each bidge he dualiy beween D and δ holds. This popey emains valid by mixing he undelying measue. So i holds. 4 Chaaceizaion of he ecipocal class associaed o a Bownian diffusion. In his secion we wan o exend he esuls obained in he pevious secion fo ohe classes of ecipocal pocesses han RP. So we ake as efeence pocess no moe a Bownian moion bu a Makovian Bownian semi-maingale, also called Bownian diffusion, and defined as soluion of he sochasic diffeenial equaion : { dx = db + b, X d 5 X = x whee B is a Bownian moion and he dif b saisfies he following egulaiy assumpions : b C,2 [, ] R ; R 6 K >,, x [, ] R, x b, x K + x 2. 7 Since condiion 6 implies ha b is locally lipschiz coninuous unifomly on ime, boh condiions 6 and 7 ensue exisence and uniqueness of a song soluion o equaion 5 see fo example [3] p.234. We denoe by P PΩ he law of his soluion. We inoduce he following supplemenay egulaiy assumpion on he pobabiliy ansiion densiy associaed o P - i will be usefull when we compue he ecipocal chaaceisics of bidges of P - : ps, x,, y = P X dy/x s = x/dy is sicly posiive fo any s, [, ], x, y R and belongs, as funcion of s, xesp., y, o C,3 [, [ R ; R. 8 Le us now inoduce a space-ime funcion F defined on [, ] R and deived fom b by : F, x = b, x + b, x x b, x + b, x. 9 2 x2 Fom [4] and [2] we know ha his funcion ogehe wih he diffusion coefficien due o he fac ha he maingale pa of X is a Bownian moion ae he so-called local ecipocal chaaceisics associaed o P. In fac he funcion F, as funcional of he dif, is invaian on he se R M P and moeove he pai, F chaaceizes compleely he ecipocal class R P see Theoem in [4]. 9 2

10 4. An inegaion by pas fomula Le us now invesigae how he dualiy equaion i in Theoem 3.6 saisfied by evey ecipocal pocess in he Bownian class RP is peubaed when he efeence pocess admis a dif b. Poposiion 4. Le PΩ a ecipocal pocess in he class R P. Suppose moeove ha sup X < + and [,] Then he following inegaion by pas fomula is saisfied unde : g L 2,, Φ S, D g Φ = Φ δg + Φ F, X d < +. 2 g F, X dd. 2 As anounced below, he peubaion em - he second em of he.h.s. - is given by F. In he couse of he poof we will need he following Lemma 4.2 Le P β PC[, τ]; R be he law of a Bownian diffusion wih iniial value x and dif β, fo some < τ. We assume he following : β C,2 [, τ] R ; R and βτ, X τ L P β F β, X L d dp β whee F β = β + β x β + 2 x 2 β. Then, fo g L 2, τ and Φ any F τ -measuable funcion in S, P β D g Φ = P β Φ δg + P β Φ g F β, X dd Poof of Lemma 4.2: Le us denoe by M β, esp. M n,β, he.v. defined by M β τ = exp β, X dx 2 β 2, X d, 2 gd P β Φ βτ, X τ. 22 esp. M n,β = exp χ n log M β whee χ n is a smooh bounded funcion wih bounded deivaive on R saisfying { χn [ n,n+] c = n + ], n [ + n + ]n+,+ [ χ n [ n,n] = Id. [ n,n]. Such a cu-off fo M β appeas in [7]. Remak ha M n,β M β +. Le Pβ n PC[, τ]; R be he pobabiliy measue wih Radon-Nikodym deivaive M n,β wih espec o he Wiene measue P x. By definiion of Pβ n, Pβ n D gφ = P x M n,β D g Φ = P x D g Φ M n,β P x ΦD g M n,β which implies, by inegaion by pas fomula unde he Wiene measue, ha P x M n,β D g Φ = P x Φ M n,β δg P x ΦD g M n,β.

11 By dominaed convegence, he l.h.s. of he above ideniy conveges o P M x β D g Φ = P β D g Φ. The same agumen applies o P Φ x M n,β δg which heefoe conveges o P x Φ M β δg = P β Φδg. By definiion, D g M n,β = M n,β χ nlog M β D g log M β. Moeve, D g log M β = which, by Io s fomula, is equal o g β, X + β, X x β, X d x β, X dx d βτ, X τ gd g F β p, X p dpd. The las em fo which we have o sudy he convegence is heefoe We conclude since P x Φ M n,β χ nlog M β βτ, X τ and, by assumpion, he.v. gd g M n,β χ nlog M β M n,β [ n+,n+] log M β M β βτ, X τ gd M β [ n+,n+] log M β g F β p, X p dpd. F β p, X p dpd. is in L P x since he.v. ino paenhesis is in L P β. Poof of Poposiion 4.: Le us denoe by µ he law of X, X unde. I is sufficien o pove ideniy 2 unde x,y fo µ-a.a. x, y, since i will emain ue by einegaion unde µ. Obviously, assumpion 2 emains ue unde x,y fo µ-a.a. x, y. In he sequel of he pesen poof, we fix such an x, y. Moeove, since R P, x,y coincides wih P x,y and is heefoe he law of a Bownian diffusion on each [, τ], τ < whose dif β saisfies β, z = b, z + log p, z,, y z whee p is he ansiion pobabiliy densiy of P. Le us fis noice ha, when gd =, i is easy o veify ha in he poof of Lemma 4.2 he assumpion βτ, X τ L P β is no moe equied. The emaining assumpions of Lemma 4.2 on β and F β F ae diec consequences of assumpions 8 and 2. Theefoe, fo all Φ S, F τ -measuable and all sep funcions g L 2, τ, one has x,y D g Φ = x,y Φ δg + x,y Φ g F, X dd. 23

12 Le us now fix Φ S, F -measuable, and g a sep funcion in L 2,. These ae he esing objecs which we need in ode o pove 2. Since Φ S, hee exiss a funcion ϕ and a eal numbe τ < such ha ΦX = ϕx, X,, X τ, y, x,y -a.s.. We also fix n lage enough so ha τ < n and g is consan on [ 2 n ; [. Le us se g n = g [, 2 n [ + n gd [ 2 2 n, ]. n n By consucion g n L 2, n since g L2,. Fom Lemma 4.2, we deduce he ideniy x,y D gn Φ = x,y Φ δg n + x,y n Φ g n n F, X dd. I emains o veify ha each em conveges when n ends o infiniy owads he coesponding em in 2 wien unde x,y. We have he following inequaliies : x,y D gn Φ D g Φ DΦ g n g = 2 C n DΦ whee C is he consan value of g on [ 2 n, [. Φ x,y δg n g Φ x,y X X 2 n which conveges o by a.s. coninuiy of pahs and dominaed convegence heoem hanks o assumpion 2. x,y Φ n g n n F, X dd g F, X dd which vanishes hanks o assumpion Chaaceizaion of he ecipocal class R P. We ae now ineesed by he convese saemen of Poposiion 4.. Moe pecisely, ou main esul is o show ha he inegaion by pas fomula 2 chaaceizes he egula elemens of R P. Moe pecisely, ecall ha in he pevious secion, we inoduced he egulaiy assumpions 4.2 and 4.2bis in ode o define he ecipocal chaaceisic F. In he same way in ode o pove a convese saemen o Poposiion 4. we have o conside pobabiliies on ω which a pioi saisfy he following egulaiy condiions denoed by A: Theoem 4.3 Le PΩ saisfying A and such ha sup X 2 < + and [,] If he following inegaion by pas fomula is saisfied unde : F, X 2 d < g sep funcion in L 2,, Φ S, D g Φ = Φ δg + Φ g F, X dd 25 hen is a ecipocal pocess in he class R P. 2

13 Poof : The poof of his heoem divides in hee seps. Sep : We fis pove ha X, [, ] is a -quasi-maingale on [, ]. This amouns o veify ha sup n X i+ X i /F i < + i= whee he supemum is aken ove all he finie paiions = < <... < n = of [, ]. Le us fix such a paiion, and ake g i = [i, i+ ] + i+ i i [i,]. The inegaion by pas fomula 25, applied o g i and any Φ F i -measuable, implies ha, fo i n, X i+ X i /F i We hus have he following inequaliy n i= X X i+ i = i+ i /F i i i + i+ i F, X dd/f i. i X i+ X i /F i n i+ i X X i + 2 i i= i F, X dd/f i F, X d. To pove he boundedness of he.h.s. on all paiions i is sufficien o conol i fo paiions which mesh goes o zeo. Bu hen, we idenify he sum in he.h.s. as a Riemann sum associaed o he inegal X X s s ds. The convegence of his inegal is a diec consequence of he following Lemma 4.4 Le PΩ saisfying he assumpions sup X 2 < + and [,] F, X d 2 < If he inegaion by pas fomula 25 is saisfied unde fo all Φ S, hen i holds also fo he unbounded funcional defined by ΦX = X X s, s <. Moeove, hee exiss a posiive consan C such ha s [, ], X X s 2 C s. Poof of Lemma 4.4 : Le χ n be he cu-off funcion defined in he poof of Lemma 4.2. The inegaion by pas fomula 25 holds ue fo any sep funcion g L 2, and Φ nx = χ n X X s. Due o he assumpions 26, he dominaed convegence heoem applies o each em and hen, 25 holds also fo ΦX = X X s. 3

14 Fo poving he second asseion, le us se g = s [s,] and ΦX = X X s fo s [, ]. Taking = in he fis asseion, one deduces he ideniy : s = We hus conclude ha X X s 2 s X X s 2 s + X X s s X X s X X + 4 sup X [,] s F, X dd sup [,] X 2 2 F, X dd. 2 2 F, X d which is finie by assumpion 26. Remaking ha assumpions 26 ae weake han assumpions 24, his complees he poof of sep. By Rao s heoem cf. [6] Chapie VII, since X, [, ] is a coninuous -quasimaingale, i is hen a coninuous -semi-maingale. Sep 2 : We now idenify he local chaaceisics of he coninuous -semi-maingale X, [, ]. - Le us denoe by A he bounded vaiaion pa of X. We fis pove ha fo any [, ], he andom measue da/f on [, ] is absoluely coninuous wih espec o Lebesgue measue, wih densiy β. saisfying A A β = /F + s F p, X p /F dpds. To his aim, le us ake u > and, as es funcion, a sep funcion g wih suppo in [, u]. We fis show ha u u gda /F = gβ d. Equaion 25 applied o Φ = Φ, F -measuable and o g = g u gda /F = u + u u gd A u A /F u u gd u s u u u u g gd [,u] yields F p, X p /F dpd F p, X p /F dpds. 27 Assumpion 24 implies ha da s < + ; so we can apply Fubini s heoem o he l.h.s. of he above equaliy. Taking u = in 27, we obain ha da/f is absoluely coninuous wih espec o Lebesgue measue on [, ], and is densiy is given by β = A A /F F p, X p /F dp + s F p, X p /F dpds. 28 Fom his expession we obain he coninuiy and even he a.s. deivabiliy of he funcion β fom [,[ o L. Moeove, fo all u >, using he expession given in 27, we also have β = A u A /F u u F p, X p /F dp + u 4 u u s F p, X p /F dpds 29

15 Fo fixed, leing u end o, one obains fom 29 he desied fom fo β : β = A A /F + s F p, X p /F dpds, < <. Fom he expession of A/F, we now wan o deduce he value of A. Fis we pove he following equaliy as pocesses in L d : β. = β../f. 3 Since s β s is coninuous fom [,[ o L, hen β = lim s As A s /F, and we have β /F = lim A s A s s /F /F As A = lim s s /F Bu As A s /F As A = s = s s s A A s /F β s s β s s u F p, X p /F dpdu u F p, X p /F dpdu = β s + β s β s s s F p, X p /F dpdu s s u s F p, X p /F dpdu. When s ends o he fis em of he.h.s ends o β ; he hid em of he.h.s. ends o ; he limis of he second em and he foh ae opposie since, fom 28, fo almos all, β is diffeeniable and β = F, X /F. This complees he poof of 3. Now we conclude obseving ha he pocess A u A u β d u [,] is boh a bounded vaiaion pocess and a coninuous -maingale due o 3. I is hen equal o he consan, which means ha da is indeed absoluely coninuous wih espec o Lebesgue measue d and is densiy is equal o β. So he semi-maingale decomposiion of X, [, ] unde is he following : dx = dm + β, Xd whee M is a -maingale and β, X =: β X is given fo < by β, X = X X /F F p, X p /F dp + s F p, X p /F dpds. 3 Le us show ha he maingale M is in fac a Bownian moion. The assumpion 4.9 and fomula 4.6 imply ha sup [,τ] X L 2 Ω, τ [; [. So, following Meye s eminology, 5

16 M belongs o he class D on [; τ] and, in ode o veify ha M is a bownian moion, i is enough o show ha X+h X 2 lim /F d = τ h h in L cf. [?], Theoems T 28 and T 29 p56. Wih he same agumens as in he poof of Lemma 4.4 we can veify ha 25 holds also fo ΦX = Φ XX +h X, whee [, [, h >, and Φ is F -measuable, and fo g = [,+h] h [,] ; we obain X+h X 2 /F h = h + X +h X X X /F X +h X +h F s, X s dsd/f h + X +h X F s, X s dsd/f. The.h.s. conveges in L o when h ends o hanks o assumpions 24, so is a Bownian semi-maingale. Sep 3: In he las sep, we show ha he coodinae pocess unde is ecipocal, and we idenify is ecipocal class. Since is he mixue of is bidges unde X, X, i is sufficien o pove ha fo X, X -almos all x, y he bidge x,y belongs o he ecipocal class R P. Following he same agumen as in he poof of Poposiion 3.5, fo X, X -almos all x, y, he inegaion by pas fomula 25 holds ue unde x,y. Le us fix such an x, y R 2 and s ], ]. We now show ha x,y is a Makovian semi-maingale. Moe pecisely, we pove ha he law of X, [s, ] is he same unde x,y./f s and x,y./x s. Le us denoe fo simpliciy x,y./f s by x,y F s and x,y./x s by x,y X s. These wo pobabiliies saisfy also equaion 25 fo es funcions g wih suppo in [s, ]. By he same agumens as in Seps and 2, we deduce ha X, [s, ] is a Bownian semi-maingale unde boh pobabiliies whose difs a ime <, compued as in 3, ae especively given by x,y F s U, X/F and x,y X s U, X/F, whee Bu, fo s, U, X = y X F u, X u du + x,y F s./f = x,y X s./f = x,y./f. s F u, X u duds. 32 Then boh difs coincide a.s. which implies ha x,y is Makovian. In paicula is dif pocess is he following funcion β x,y on ime and space : β x,y, z = y z x,y F u, X u /X = z du + s x,y F u, X u /X = z duds. 33 By he same agumens as above, y = /X = y is a makovian semi-maingale. Theefoe, x,y F u, X u /X = z = y F u, X u /X = x, X = z = y F u, X u /X = z 6

17 Ω F u, wq, z, u, w,, ydw. Thanks o hypoheses A,, z β x,y, z belongs o C,2 [, ] R ; R and is ecipocal chaaceisics ae, F x,y, whee F x,y is deived fom β x,y as was F fom b in 9. Le us now pove ha F x,y = F fo all x, y R. Fom 4.8 and assumpions 4.9, he pocess β x,y, X admis a fowad deivaive defined by lim x,y β x,y + h, X +h β x,y, X /F. h h Moeove his deivaive is equal o F, X. Indeed, So, lim x,y β x,y + h, X +h β x,y, X /F h h = lim x,y x,y U + h, X/F +h x,y U, X/F /F h h = lim x,y U + h, X U, X /F h h y = 2 X 2 lim x,y X +h X + F p h h h + x,y = F, X F p, X p dp + 2 since all he ems of he.h.s. vanish excep lim h x,y h +h F p, X pdp/f which ends o he desied expession. Since β x,y, X < and F x, y, X d <, he maingale pa of he semi-maingale β x,y, X is a ue maingale. This popey enables us o idenify he fowad deivaive of β x,y, X wih he finie vaiaion pa of β x,y, X compued by using Io fomula, ha is F, X = F x,y, X. The sic posiiviy of qo, x,, z,, y assumed in A implies F = F x,y. This complees he poof of Theoem 4.3. Remak 4.5 : As coollay of Seps and 2 of he above poof, we obain he fac ha any ecipocal pocess wih ecipocal chaaceisics, F saisfying assumpions 24 is indeed a semimaingale. Sep 3 was devoed o he Makovianiy and idenificaion of x,y. The efeence filaion was he usual canonical one F. Since F = σx is degeneaed unde x,y, we hus concluded o he exisence of a Makovian dif β x,y such ha X x βx,y, X d is a x,y -Bownian moion. We could n use he same agumens diecly on, since a pioi unde, F is no moe degeneaed; we could ecove only he Makovianiy of./x. This emak is made vey clea on he example given in he nex secion : he dif of X a ime in 39 is a funcion of X, X. So i is no Makovian unde he ecipocal pocess, bu clealy, unde x,y = P x,y, i becomes Makovian. +h s F p, X 7

18 5 Applicaion o he peiodic Onsein-Uhlenbeck pocess. Le us denoe by P he law of he eal-valued saionay Onsein-Uhlenbeck pocess, which, fo λ > fixed, is he soluion of he sochasic diffeenial equaion : { dx = db λx d X N ; λ. 34 This is a paicula case of he Bownian diffusion P defined in he las secion, aking b independen of ime and linea wih espec o space. This pocess is Makovian, Gaussian, and admis as ecipocal chaaceisics he funcion F, x = λ 2 x. In he pesen secion we ae ineesed in he soluion of he following s.d.e. wih peiodic bounday condiions : { dx = db λx d 35 X = X. This pocess is called peiodic Onsein-Uhlenbeck pocess, and we denoe is law by P pe. This ype of pocesses has been aleady sudied by seveal auhos wih vaious moivaions. Fis, Kwakenaak [] sudied he momens of such Gaussian pocesses and elaed fileing poblems. Then, he fac ha he soluion of 35 is a ecipocal pocess has been poved fom he analysis of he covaiance kenel in [2]. Neveheless, we popose hee an alenaive poof of he ecipocal popey of he peiodic Onsein-Uhlenbeck pocess based on he inegaion by pas fomula 25. Ou mehod enables us o pove ha he peiodic Onsein-Uhlenbeck pocess is ecipocal, and simulaneously, o idenify is ecipocal class. In his sense, i makes complee, in his vey paicula case, he esul of Ocone and Padoux [7], who sudy he Makov field popey of soluions of geneal linea s.d.e. wih bounday condiions, bu wihou any idenificaion of hei ecipocal classes. We conjecue ha ou mehod, which essenially elies on Gisanov heoem, will exend o moe geneal s.d.e. wih bounday condiions han 35 see [6] fo a descipion of such a geneal class. The mehod of vaiaion of consans yields he following fom fo he unique soluion of 35: X = e λ X + e λ s whee Ψ is he map on Ω defined by : e λ s db s = e λ db e λ+ s s + e λ db s = ΨB 36 Ψω = e λ s e λ dω e λ+ s s + e λ dω s. I is hen saighfowad o veify ha X is also he well known hypebolic cosine pocess, i.e. a zeo mean Gaussian pocess wih covaiance funcion given by cosh λ s 2 CovX s, X = 2λ sinh λ 2 =: R, s 8

19 which implies, in paicula, ha X is saionay. Fom he explici expession of R i is easy o veify ha i solves in a weak sense he second ode paial diffeenial equaion 2 R, s + λ 2 R, s = δ s. Camichael, Masse 2 and Theodoescu chaaceize in [2] he covaiance of saionay gaussian ecipocal pocesses as soluions of such paial diffeenial equaions and in [4], a genealisaion o he non saionay case is poved. Theoem 5. The law P pe of he soluion of 35 is a ecipocal pocess associaed o he saionay Onsein-Uhlenbeck pocess, ha is in he ecipocal class RP. Poof : To pove he heoem we now show ha P pe saisfies he inegaion by pas fomula 25 wih F, x = λ 2 x. Le g L 2, and Φ S. By definiion, P pe D g Φ = P pe. lim ɛ ɛ ΦX + ɛ gsds ΦX = lim ɛ ɛ P pe. ΦX + ɛ gsds ΦX = lim P pe ɛ Φ P pe Φ ɛ ɛ whee P pe ɛ is he image of P pe unde he shif on Ω by he deeminisic pah ɛ. gsds. I is also he law of he soluion of he peiodic s.d.e. { dx = db ɛ λx d 37 X = X whee B ɛ = B +ɛ gsds and gs = gs+λ s gd. By he mehod of vaiaion of consans we deduce ha he soluion of 37 is equal o ΨB ɛ in he same way as he soluion of 35 was equal o ΨB. We hus have P pe D g Φ = lim ɛ ɛ P Eɛ g Φ Ψ whee P is he Wiene measue and Eɛ g denoes he Gisanov densiy : Theefoe Eɛ g = exp ɛ gsdb s ɛ2 2 P pe D g Φ = P gsdb s g 2 sds. Φ Ψ. We can now go back o an expecaion unde P pe fo he igh-hand side using again he fac ha ΨB = X solves P pe -a.s. equaion 35. This yields P pe D g Φ = P pe ΦX gsdx s + gsλx s ds. I emains o subsiue fo gs ino is expession gs + λ s gd and o show ha s gd dx s + gsx s ds vanishes. Fubini s heoem applies o he double inegal since P pe X s ds <. We hus obain ha s gd dx s + gsx s ds = X gd =. 9

20 This complees he poof. The law P pe of he peiodic Onsein-Uhlenbeck pocess being in RP i admis he following decomposiion P pe = P x,y µdx, dy whee µ is he law of X, X unde P pe. Hee µ is suppoed by he diagonal. Thus P pe = P x,x mdx whee m is he law of X unde P pe, equal o N ; 2λ coh λ 2. In his simple case, i is possible o explici he semi-maingale decomposiion of he bidge P x,x, since i solves he following s.d.e. { λ dx = db λx d + sinhλ x e λ X d 38 X = x. Indeed he addiional em in he dif of P x,x wih espec o he dif of P is equal o z log p, X,, x whee p, z,,. is he densiy of he Gaussian law P X./X = z. To compue his densiy i is sufficien o compue EX /X = z and EX 2/X = z, which come diecly fom he equaliy : X = e λ X + e λ s db s. This complees he descipion of he desinegaion of P pe ino bidges. Le us also menion he wok of Recoules who poved in [8] ha P pe is he law of he pocess soluion of { X dx = db λ sinhλ X anhλ d X N ; 2λ coh λ Remak ha equaion 39 is a andomized vesion fo X no moe deeminisic of equaion 38, which exacly eflecs a he level of he semi-maingale popey he above desinegaion P pe = P x,x N ; 2λ cohλ 2 dx. Fom he poin of view of enopy, Recoules emaked also ha P pe is, among Gaussian saionay peiodic pocesses, he unique one which minimizes he Kullback infomaion wih espec o he Bownian bidge wih iniial law N ; λ 2. 6 Sochasic Newon equaion fo ecipocal pocesses. In [23] one of he auhos poved ha a ecipocal diffusion saisfies a weak fom of a sochasic Newon equaion involving he sochasic deivaive inoduced by Nelson cf. [5]. We ecall below he definiion of his deivaive. Definiion 6. Le X, [, ] be an inegable pocess on [, ]. I is called fowad esp. backwad diffeeniable if and only if fo each [, [, esp. ], ] he limi lim h h EX +h X /F esp. lim h h EX X h / ˆF exiss in L Ω. These limis ae denoed especively by d + X and d X. The inegaion by pas fomula 2 enables us o pove he following saemen abou he exisence of a Nelson deivaive fo a ecipocal pocess. 2

21 Poposiion 6.2 Le PΩ be a ecipocal pocess in he class R P saisfying he assumpions of Poposiion 4.. Then he Nelson deivaives of he coodinae pocess d + X and d X exis fo almos all ], [. Poof : Fom Poposiion 4., saisfies he inegaion by pas fomula 2. So, like in he sep 2 of he poof of Theoem 4.3, we can see ha X +h X /F = h h +h β d whee β is given by 28. Thanks o Föllme s esul cf [8], Poposiion 2.5, i suffices o show ha fo all τ ], [, β d < +, in ode o deduce ha, fo almos evey ], [, d + X exiss and is equal o β. We can conclude using he assumpions on X and F, ogehe wih he inequaliy : Remak 6.3 : β d X X d + 2 F p, X p dp. Thee ae seveal definiions of he Nelson deivaive depending on he sense given o he limis : in L Ω o L 2 Ω o a.s.. We have chosen L Ω. Wih assumpions on he squae inegabiliy of X and F, we could also pove he analog of he above poposiion wih an L 2 Ω-limi. The exisence of he deivaives is valid only fo almos all. The assumpions of Poposiion 4. ae no song enough o gaanee he exisence of second ode Nelson deivaives. Neveheless, by assuming hei exisences, we obain he following Theoem 6.4 Le PΩ be a ecipocal pocess in he class R P saisfying he assumpions of Poposiion 4. and such ha fo all ], [, he fis and second ode deivaives d + X, d X, d + d + X, d d X exis. Then, fo allmos all ], [, d + d + X /X = d d X /X = F, X 4 Poof : Le ], [ and s ], [ and also u > and h > such ha + h + u. Le us se g [,+u] [+h,+h+u] and ϕ be a bounded egula funcion on R. Fom he inegaion by pas fomula 22, we deduce ha ϕx s X +u X /F + ϕx s X +h+u X +h /F +h = ϕx s +h+u +h F p, X p dpd + +u F p, X p dpd. When dividing by u and leing u, using also he dominaed convegence heoem, one obains +h ϕx s d + X +h d + X = ϕx s F p, X p dp fo any ], [, which can be ewien as ϕx s d +X +h d + X h /F = ϕx s h 2 +h F p, X p dp.

22 The l.h.s. has a limi when h ends o fom he assumpions on he exisence of he second deivaive. The limi of he.h.s. follows fom Föllme s esul [8], Poposiion 2.5 and holds fo almos all. Thus we have poved ha, fo any s <, ϕx s d + d + X = ϕx s F, X. I emains o le s incease o. The dominaed heoem applies since by definiion d + d + X L Ω and fo almos all, F, X L Ω fom assumpion 6. Remak 6.5 : This saemen is a paicula case of he esul in [23], Theoeme 8., in which such an equaion is poved in dimension d. Then, he equaion involves a supplemenay em whee appeas a ecipocal chaaceisics which vanishes in dimension. I is woh o noice ha when X is Makovian a songe fom of equaion 4 holds : d + d + X = d d X = F, X. An analogous siuaion also occus in dimension d. Theoem 6.4 implies ha 2 d +d + X + d d X /X = F, X. The em 2 d +d + X + d d X can be inepeed as an acceleaion in Sochasic mechanics. This is why such equaion may be called Newon equaion cf. [24]. Remak 6.6 : Kene in [3] has also poved wo esuls of second ode naue concening ecipocal pocesses. In he fis he esablishes wha he calls second ode Felle posulaes, which povide a momen esimae of infiniesimal second ode incemens of he fom X +h + X h 2X k /X h, X +h. The esimaes only depend on he ecipocal chaaceisics. In his second esul he gives a meaning o a second ode s.d.e. whose coefficiens ae he ecipocal chaaceisics. Fo deails and igouous saemens, we efe he eade o [3]. Aknowledgemens : The auhos ae vey gaeful o P. Caiaux fo fuiful discussions and suggesions on seveal echnical poins of his wok. Refeences [] J.-M. Bismu, Maingales, he Malliavin Calculus and hypoellipiciy unde geneal Hömande s Condiions, Z. Wasch. Vew. Geb , [2] J-P. Camichael, J-C. Masse and R. Theodoescu, Pocessus gaussiens saionnaies écipoques su un inevalle, C.R. Acad. Sc. Pais, Seie I [3] K.L. Chung and R.J. Williams, Inoducion o sochasic inegaion, Second Ediion, Bikhaüse 99 [4] J.M.C. Clak, A local chaaceizaion of ecipocal diffusions, Applied Soch. Analysis, Vol. 5, 45-59, Eds: M.H.A. Davis and R.J. Ellio, Godon and Beach, New Yok 99. [5] Ph. Couège and P. Renouad, Oscillaeus anhamoniques, mesues quasi-invaianes su CR, R e héoie quanique des champs en dimension, Aséisque 22-23, Soc. Mah. de Fance, Pais

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