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 7641 École Polyechnique Palaiseau Cédex, Fance oelly@cmapx.polyechnique.f Michèle Thieullen Laboaoie de Pobabiliés e Modèles Aléaoies UMR CNRS 7599 Univesié Pais 6, boie 188 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[, 1]; 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: 6G15-6G6-6H1-6J6. KEY-WORDS: Recipocal pocess, Inegaion by pas fomula, Sochasic bidge, Sochasic diffeenial equaion wih bounday condiions, Sochasic Newon equaion. 1

2 1 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 [24] 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 [11] poved ha he se of ecipocal pocesses is paiioned ino classes; each subclass is chaaceized by a se of funcions, called Recipocal Chaaceisics [4], [13]. The main esul we obain is ha, fo eal-valued pocesses, each class of ecipocal pocesses wih Recipocal Chaaceisics 1, 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 [21]. This funcional equaion is indeed an inegaion by pas fomula on he pah space C[, 1]; 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 [13]. I was achieved in he Gaussian case by Kene, Fezza and Levy in [15]. 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 1. Concening he geneal non Gaussian case, one of he auhos poved in [25] ha ecipocal pocesses saisfy a sochasic Newon equaion which involves Nelson deivaives, he ecipocal chaaceisics as well as a sochasic vesion of acceleaion. A he end of secion 4 of he pesen pape, we sudy he elaionship beween ou esul and he esul of [25]. The 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 Quanum Field heoy by Couège and Renouad [5] see also [23]. Sill a lo of poblems in his diecion emain open. 2

3 The pape is devided ino he following secions. 1. 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. 2 Noaions and famewok Le Ω = C[, 1]; R be he canonical - polish - pah space of coninuous eal-valued funcions on [, 1], endowed wih F, he canonical σ-field. Le X [,1] denoe he family of canonical pojecions fom Ω ino R. PΩ is he se of pobabiliy measues on Ω. We use equivalenly he noaion Qf o E Q f fo he inegal of he funcion f unde a pobabiliy measue Q. Le P PΩ denoe he Wiene measue on Ω saisfying P X = = 1. Moe geneally, fo x R, P x is he shifed Wiene measue saisfying P X = x = 1. We define now he space of smooh cylindical funcionals on Ω : S = {Φ, Φω = ϕω 1,..., ω n whee ϕ is a bounded C -funcion fom R n in R wih bounded deivaives and 1... n 1}. Clealy S L 2 Ω; P. On S we define he deivaion opeao D in he diecion g L 2, 1 by whee D g Φω = = D Φω = n i=1 n i=1 ϕ i ω 1,..., ω n gd x i gd Φωd ϕ x i ω 1,..., ω n 1 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 1,2 as he closue of S fo he following nom : Φ 2 1,2 = E P Φ 2 + E P D Φ 2 d. I is well known see fo example [1] ha he opeao D also called Malliavin deivaion is he dual opeao on D 1,2 of he sochasic inegaion opeao δ defined on Ω by δgω = gdω : g L 2, 1, Φ D 1,2, E P D g Φ = E P Φ δg 1 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 < 1, 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 = {Q PΩ, s < 1, Q./F s ˆF = P./X s, X } 2 whee he fowad esp. backwad filaion F [,1] esp. ˆF [,1] is given by F = σx s, s, esp. ˆF = σx s, s 1. 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 Q is a Makovian field in he sense ha, fo s < 1, F s ˆF and σx ; s ae independen unde Q condiionnally o σx s, X. Neveheless, a ecipocal pocess is no necessaily a Makov pocess. Jamison gave in [11] he following descipion of he subse R M P whose elemens ae he Makovian pocesses of R P : R M P = {Q R P, ν, ν 1 σ-finie measues on R, Q X, X 1 1 dx, dy = p, x, 1, yν dxν 1 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 [11] : R P = {Q PΩ, m PR 2, Q = P /X = x, X 1 = ymdx, dy}. 4 R 2 Remak ha fom he above definiion 4 any ecipocal pocess Q in R P is a mixue of bidges of P. 3 Chaaceizaion of RP, he ecipocal class associaed o he Bownian moion 3.1 Dualiy unde he Bownian bidge We ecalled in he above equaliy 1 he dualiy beween Malliavin deivaive D and sochasic inegaion δ unde he Wiene measue P. In fac, 1 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 1 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 1 o a smalle space han L 2, 1. So le us inoduce he funcion space L 2, 1 = { g L 2, 1, 4 g d = }.

5 I is he ohogonal subspace in L 2, 1 o he consan funcions. Le us sess he following emak: fo he chaaceizaion based on inegaion by pas fomula developed in he es of he pape, i is enough o conside he class of sep funcions g L 2, 1. Fo hese funcions, δg is ininsically and ivially defined; in paicula he sochasic inegal does no depend on he efeence pobabiliy measue on Ω. We have Poposiion 3.1 Le x, y R 2 and P x,y PΩ be he law of he Bownian bidge on [, 1] fom x o y. Then g sep funcion in L 2, 1, Φ S, P x,y D g Φ = P x,y Φ δg. 6 Poof : The dualiy fomula 6 has been poved by Dive in [8] even fo he Bownian bidge on a Riemannian manifold. His poof elies on he absolue coninuiy of P x,y wih espec o P x on F τ, wih < τ < 1. Howeve fo he sake of compleeness, le us skech an alenaive poof of his dualiy. As noiced a he beginning of he secion 3.1, he dualiy P µ Φ δg = P µ D g Φ 7 holds fo any g sep funcion, Φ S and µ PR. Taking Φω = φ ω φ 1 ω 1 Φω fo φ, φ 1 C R, and Φ S, one obains fom 7 P µ φ X φ 1 X 1 P µ Φδg/X, X 1 = P µ φ X φ 1 X 1 P µ D g Φ/X, X 1 + P µ φ X φ 1 1X 1 Φ g d. So, fo g sep funcion in L 2, 1, he las em vanishes and his implies P X,X 1 Φδg = P µ Φδg/X, X 1 = P X,X 1 D g Φ fo a.s.x, X 1 unde P µ. By coninuiy of he map x, y P x,y he dualiy fomula 6 holds fo all x, y R 2. 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, 1 and L2, 1 defined by : g L 2, 1, αg = g gs ds. In fac, following Gosselin and Wuzbache [1], Poposiion 2.2, if X is a Bownian moion unde P, he image pocess of X unde he ansfomaion Θ : ω Θω = 1 dω s 1 s <1 is a Bownian bidge wih law P, ; he sochasic inegal δgθx = gdθx is well defined fo g L 2, 1 and moeove : δgθx = δαgx P a.s.. So, o deduce 6 fom 1 i is enough o emak ha, fo g L 2, 1 and Φ S, D g Φ Θ = D αg Φ Θ. 5

6 3.2 Chaaceizaion of he condiional pobabiliies The naual quesion is now o analyse if he dualiy unde a measue Q beween D and δ esed on all g, Φ L 2, 1 S chaaceizes he bidges of Q. The posiive answe is he objec of he following : Poposiion 3.3 Le Q PΩ such ha Qsup [,1] X < +. If g sep funcion in L 2, 1, Φ S, QD gφ = Q Φ δg 8 hen Q./X, X 1 = P X,X 1 Q a.s.. Poof : Fis, following he same agumen as in Poposiion 3.1, i is clea ha 8 also holds unde Q./X, X 1 Q a.s.. Fo simpliciy, le us denoe by Q x,y PΩ he law of he bidge of Q on [, 1] beween x and y, x, y R 2. Le g a fixed sep funcion on [, 1], and fo λ R, define ψλ = Q x,y expiλδ g. 9 By eceneing g, we also inoduce he sep funcion Now, emaking ha ψ is diffeeniable on R, we obain ψ λ = iq x,y δ g expiλδ g = iq x,y δg + y x g = g g d L 2, 1. 1 g d expiλδ g = ie iλy x g d Q x,y δg expiλδg + iy x g d ψλ. Fom 8, using he fac ha Φ = expiλδg S, we deduce ha fo Q X, X 1 1 a.a.x, y, Q x,y δg expiλδg = Q x,y D g expiλδg which is equivalen o So, ψ λ = Q x,y δg expiλδg = iλ g 2 d Q 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 = 1 is ψλ = exp λ2 g 2 d g d 2 + iλy x g d

7 Thus, fo Q X, X 1 1 almos all x, y, equaliy 11 holds ue fo all g in he following counable se of sep funcions : { p i= α i1 [si,s i+1 [, = s <... s p < s p+1 = 1, p N, s i, α i Q}. This se is dense in L 2, 1, so equaliy 11 holds also ue fo each g L 2, 1, since is boh sides ae L 2, 1-coninuous funcionals of g unde he assumpion ha Qsup [,1] 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 = x1 + B + y B 1 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 [22]. O one emaks ha ψ is associaed o a Gaussian pocess : by aking λ = 1 and p g = α i 1 [i 1, i [, = < 1 <... < p 1 < p = 1, i= i is clea ha Q x,y expiδ g is he exponenial of a bilinea fom in α i. Moeove, aking now g = 1 [s,], we obain he fis wo momens of his Gaussian pocess : implies Q x,y expiλδ1 [s,] = e λ2 2 s s2 +iλy x s Q x,y X = y + 1 x and CovX s, X = s1, 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 1 chaaceizes he se of Wiene measues {P µ, µ PR} PΩ see [21], Theoem 1.2. By esicing he class of es funcions g o hose wih vanishing inegal on [; 1], 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. Theoem 3.4 Le Q PΩ such ha Qsup [,1] X < +. The following wo asseions ae equivalen : i g sep funcion in L 2, 1, Φ S, QD g Φ = Q Φ δg ii Q RP, i.e. Q is a ecipocal pocess in he same class as he Bownian moion. Poof : By Poposiion 3.3, i implies he a.s. equaliy beween he bidges of Q and hose of P. Bu Q = Q /X = x, X 1 = y mdx, dy R 2 whee m = Q X, X 1 1.Then using he definiion of RP given in 4 we obain diecly asseion ii. Recipocally, if Q RP, he desinegaion 4 holds. So Q is a mixue in x, y of bidges P x,y. Bu, by Poposiion 3.1, unde each bidge he dualiy beween D and δ holds. This popey emains valid by mixing he undelying measue. So i holds. 7

8 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 12 X = x whee B is a Bownian moion and he dif b saisfies he following egulaiy assumpions : b C 1,2 [, 1] R ; R 13 K >,, x [, 1] R, x b, x K1 + x Since condiion 13 implies ha b is locally lipschiz coninuous unifomly on ime, boh condiions 13 and 14 ensue exisence and uniqueness of a song soluion o equaion 12 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 useful 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, [, 1], x, y R and belongs, as funcion of s, xesp., y, o C 1,3 ], 1] R ; Resp.C 1,3 [, 1[ R ; R.15 Le us now inoduce a space-ime funcion F defined on [, 1] R and deived fom b by : F, x = b, x + b, x x b, x b, x. 16 x2 This funcion ogehe wih he diffusion coefficien 1 due o he fac ha he maingale pa of X is a Bownian moion ae he so-called local ecipocal chaaceisics associaed o P cf [4] and [13]. The funcion F, as funcional of he dif, is invaian on he se R M P and moeove he pai 1, F chaaceizes compleely he ecipocal class R P see Theoem 1 in [4] when b is bounded and Theoem 4.7 in [26] unde less esicive assumpions. 4.1 An inegaion by pas fomula Le us now invesigae how he dualiy equaion i in Theoem 3.4 saisfied by evey ecipocal pocess in he Bownian class RP is peubaed when he efeence pocess admis a dif b. Poposiion 4.1 Le Q PΩ a ecipocal pocess in he class R P. Suppose moeove ha Q sup X 2 < + and Q [,1] Then he following inegaion by pas fomula is saisfied unde Q : g sep funcion in L 2, 1, Φ S, QD g Φ = Q Φ δg + Q Φ F, X 2 d < g F, X dd.18 8

9 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 1. We assume he following : β C 1,2 [, τ] R ; R and βτ, X τ L 2 P β F β, X L 2 d dp β whee F β = β + β x β 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 2 gd P β Φ βτ, X τ. 19 Poof of Lemma 4.2: Le us denoe by M β he densiy of P β wih espec o P x, M β τ = exp β, X dx 1 2 We denoe by M n,β he.v. defined by M n,β = exp χ n log M β β 2, X d. funcion wih bounded deivaive on R saisfying { χn 1 [ n 1,n+1] c = n + 11 ], n 1[ + n + 11 ]n+1,+ [ χ n 1 [ n,n] = Id.1 [ n,n]. whee χ n is a smooh bounded Such a cu-off fo M β appeas in [7]. Remak ha M n,β M β + 1. Le P n β PC[, τ]; R be he posiive measue wih Radon-Nikodym deivaive M n,β wih espec o he Wiene measue P x. By definiion of P n β, 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. 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 = g β, X + β, X x β, X d x β, X dx d 9

10 which, by Io s fomula, is equal o 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 1 [ n+1,n+1] log M β M β βτ, X τ gd M β 1 [ n+1,n+1] log M β g F β p, X p dpd. F β p, X p dpd. is in L 1 P x since he.v. ino paenhesis is in L 2 P β. Poof of Poposiion 4.1: Le us denoe by µ he law of X, X 1 unde Q. I is sufficien o pove ideniy 18 unde Q x,y fo µ-a.a. x, y, since i will emain ue by einegaion unde µ. Obviously, assumpion 17 emains ue unde Q x,y fo µ-a.a. x, y. In he sequel of he pesen poof, we fix such an x, y. Moeove, since Q R P, Q x,y coincides wih P x,y and is heefoe he law of a Bownian diffusion on each [, τ], τ < 1 whose dif β saisfies β, z = b, z + log p, z, 1, 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 1 P β is no moe equied. The emaining assumpions of Lemma 4.2 on β and F β F ae diec consequences of assumpions 15 and 17. Theefoe, fo all Φ S, F τ -measuable and all sep funcions g L 2, τ, one has Q x,y D g Φ = Q x,y Φ δg + Q x,y Φ g F, X dd. 2 Le us now fix Φ S, F 1 -measuable, and g a sep funcion in L 2, 1. These ae he esing objecs which we need in ode o pove 18. Since Φ S, hee exiss a funcion ϕ and a eal numbe τ < 1 such ha ΦX = ϕx, X 1,, X τ, y, Q x,y -a.s.. We also fix n lage enough so ha τ < 1 1 n and g is consan on [1 2 n ; 1[. Le us se g n = g1 [,1 2 n [ + n gd1 [ n,1 1 ]. n n By consucion g n L 2, 1 1 n since g L2, 1. Fom Lemma 4.2, we deduce he ideniy Q x,y D gn Φ = Q x,y Φ δg n + Q x,y 1 n Φ g n 1 n F, X dd. I emains o veify ha each em conveges when n ends o infiniy owads he coesponding em in 18 wien unde Q x,y. We have he followinginequaliies : 1

11 Q x,y D gn Φ D g Φ DΦ g n g 1 = 2 C n DΦ whee C is he consan value of g on [1 2 n, 1[. Q Φ x,y δg n g Φ Q x,y X 1 X 1 2 n which conveges o by a.s. coninuiy of pahs and dominaed convegence heoem hanks o assumpion 17. Q Φ x,y 1 n g n 1 n F, X dd g 1 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 4.1. Moe pecisely, ou main esul is o show ha he inegaion by pas fomula 18 chaaceizes he egula elemens of R P. Moe pecisely, ecall ha in he pevious secion, we inoduced he egulaiy assumpions 13 and 15 in ode o define he ecipocal chaaceisic F. In he same way, in ode o pove a convese saemen o Poposiion 4.1, we have o conside pobabiliies on Ω which a pioi saisfy he following se of egulaiy condiions which will be denoed by A in he sequel: - A1 < u, y, z hee exiss a densiy funcion q, z, u,., 1, y such ha QX u dw/ X = z, X 1 = y = q, z, u, w, 1, ydw - A2 x, y, u, w, q, x, u, w, 1, y is sicly posiive - A3 u, w, y,, z q, z, u, w, 1, y belongs o C 1,2 [, 1[ R ; R and fo all, z hee exiss a neighbohood V of, z and a funcion φ V u, w, 1, y such ha and sup qs, ξ, u, w, 1, y + z qs, ξ, u, w, 1, y + zz qs, ξ, u, w, 1, y φ V u, w, 1, y s,ξ V R φ Vu, w, 1, y F u, w dudw < +. Theoem 4.3 Le Q PΩ saisfying A and such ha Q sup X 2 < + and Q [,1] If he following inegaion by pas fomula is saisfied unde Q : F, X 2 d < g sep funcion in L 2, 1, Φ S, QD g Φ = Q Φ δg + Q Φ g F, X dd 22 hen Q is a ecipocal pocess in he class R P. Poof : The poof of his heoem divides in hee seps. Sep 1 : We fis pove ha X, [, 1] is a Q-quasi-maingale on [, 1]. 11

12 This amouns o veify ha sup Q n 1 QX i+1 X i /F i < + i= whee he supemum is aken ove all he finie paiions = < 1 <... < n = 1 of [, 1]. Le us fix such a paiion, and ake g i = 1 [i, i+1 ] + i+1 i 1 i 1 [i,1]. The inegaion by pas fomula 22, applied o g i and any Φ F i -measuable, implies ha, fo i n 1, Q X i+1 X i /F i We hus have he following inequaliy Q n 1 i= X1 X i+1 i = i+1 i Q /F i Q 1 i i + i+1 i Q F, X dd/f i. 1 i QXi+1 X i /F i n 1 i+1 i Q X 1 X i + 2Q 1 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 Q X 1 X s 1 s ds. The convegence of his inegal is a diec consequence of he following Lemma 4.4 Le Q PΩ saisfying he assumpions sup Q X 2 < + and Q [,1] F, X d 2 < If he inegaion by pas fomula 22 is saisfied unde Q fo all Φ S, hen i holds also fo he unbounded funcional defined by ΦX = X X s, s < 1. Moeove, hee exiss a posiive consan C such ha s [, 1], Q X 1 X s 2 C1 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 22 holds ue fo any sep funcion g L 2, 1 and Φ nx = χ n X X s. Due o he assumpions 23, he dominaed convegence heoem applies o each em and hen, 22 holds also fo ΦX = X X s. Fo poving he second asseion, le us se g = 1 1 s 1 [s,1] 1 and ΦX = X 1 X s fo s [, 1]. Taking = 1 in he fis asseion, one deduces he ideniy : s = Q X 1 X s 2 1 s + Q X 1 X s Q 1 1 s X 1 X s X 1 X s F, X dd 12 F, X dd.

13 We hus conclude ha Q X 1 X s 2 1 s sup Q X [,1] sup [,1] 1 Q X 2 2 Q F, X d which is finie by assumpion 23. Remaking ha assumpions 23 ae weake han assumpions 21, his complees he poof of sep 1. By Rao s heoem cf. [6] Chapie VII, since X, [, 1] is a coninuous Q-quasimaingale, i is hen a coninuous Q-semi-maingale. Sep 2 : We now idenify he local chaaceisics of he coninuous Q-semi-maingale X, [, 1]. - Le us denoe by A he bounded vaiaion pa of X. We fis pove ha fo any [, 1], he andom measue QdA/F on [, 1] is absoluely coninuous wih espec o Lebesgue measue, wih densiy β. saisfying A A β = Q /F + 1 s Q 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 Q gda /F = gβ d. Equaion 22 applied o Φ = Φ, F -measuable and o g = g 1 u Q gda /F = 1 u + 1 u u gd Q A u A /F u u gd u s u u u u g gd1 [,u] yields Q F p, X p /F dpd Q F p, X p /F dpds. 24 Assumpion 21 implies ha Q da s < + ; so we can apply Fubini s heoem o he l.h.s. of he above equaliy. Taking u = 1 in 24, we obain ha QdA/F is absoluely coninuous wih espec o Lebesgue measue on [, 1], and is densiy is given by β = Q A 1 A /F 1 QF p, X p /F dp s QF p, X p /F dpds. 25 Fom his expession we obain he coninuiy and even he a.s. deivabiliy of he funcion β fom [,1[ o L 1 Q. Moeove, fo all u >, using he expession given in 24, we also have β = Q A u A /F u u QF p, X p /F dp + 1 u u u Fo fixed, leing u end o, one obains fom 26 he desied fom fo β : β = Q A A /F + 1 s s QF p, X p /F dpds 26 QF p, X p /F dpds, < < 1. Fom he expession of QA/F, we now wan o deduce he value of A. Fis we pove he following equaliy as pocesses in L 1 d Q: β. = Qβ../F

14 Since s β s is coninuous fom [,1[ o L 1 Q, hen β = lim s Q As A s /F, and we have Qβ /F = Q lim QA s A s s /F /F As A = lim Q s s /F Bu As A Q s /F As A = Q s = s s s A A s /F β s 1 s β 1 s s u QF p, X p /F dpdu u QF p, X p /F dpdu = β s + β s β s 1 s s QF p, X p /F dpdu 1 s s u s QF 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 25, fo almos all, β is diffeeniable and β = QF, X /F. This complees he poof of 27. Now we conclude obseving ha he pocess A u A u β d u [,1] is boh a bounded vaiaion pocess and a coninuous Q-maingale due o 27. 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, [, 1] unde Q is he following : dx = dm + β, Xd whee M is a Q-maingale and β, X =: β X is given fo < 1 by β, X = Q X 1 X /F 1 QF p, X p /F dp s QF p, X p /F dpds. 28 Le us show ha he maingale M is in fac a Bownian moion. The assumpion 21 and fomula 28 imply ha sup [,τ] M L 2 Ω, τ [; 1[. So, following Meye s eminology, 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 Q /F d = τ h h in L 1 Q cf. [16], Theoems T 28 and T 29 p

15 Wih he same agumens as in he poof of Lemma 4.4 we can veify ha 22 holds also fo ΦX = Φ XX +h X, whee [, 1[, h >, and Φ is F -measuable, and fo g = 1 [,+h] h 1 [,1] 1 ; we obain X+h X 2 Q /F h = 1 h 1 + Q X +h X X 1 X /F 1 Q X +h X 1 +h F s, X s dsd/f h 1 1 +Q X +h X F s, X s dsd/f. 1 The.h.s. conveges in L 1 Q o 1 when h ends o unifomly in [, τ] hanks o assumpions 21 and Lemma 4.4, so Q is a Bownian semi-maingale. Sep 3: In he las sep, we show ha he coodinae pocess unde Q is ecipocal, and we idenify is ecipocal class. Since Q is he mixue of is bidges unde Q X, X 1 1, i is sufficien o pove ha fo Q X, X 1 1 -almos all x, y he bidge Q x,y belongs o he ecipocal class R P. Following he same agumen as in he poof of Poposiion 3.3, fo Q X, X 1 1 -almos all x, y, he inegaion by pas fomula 22 holds ue unde Q x,y. Le us fix such an x, y R 2 and s ], 1]. We now show ha Q x,y is a Makovian semi-maingale. Moe pecisely, we pove ha he law of X, [s, 1] is he same unde Q x,y./f s and Q x,y./x s. Le us denoe fo simpliciy Q x,y./f s by Q x,y F s and Q x,y./x s by Q x,y X s. These wo pobabiliies saisfy also equaion 22 fo es funcions g wih suppo in [s, 1]. By he same agumens as in Seps 1 and 2, we deduce ha X, [s, 1] is a Bownian semi-maingale unde boh pobabiliies whose difs a ime < 1, compued as in 28, ae especively given by Q x,y F s U, X/F and Q x,y X s U, X/F, whee Bu, fo s, U, X = y X 1 F u, X u du Q x,y F s./f = Q x,y X s./f = Q x,y./f. s F u, X u duds. 29 Then boh difs coincide a.s. which implies ha Q 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 1 Q x,y F u, X u /X = z du s Q x,y F u, X u /X = z duds. 3 By he same agumens as above, Q y =: Q./X 1 = y is a Makovian semi-maingale. Theefoe, Q x,y F u, X u /X = z = Q y F u, X u /X = x, X = z = Q y F u, X u /X = z = F u, wq, z, u, w, 1, ydw. Thanks o hypoheses A,, z β x,y, z belongs o C 1,2 [, 1[ R ; R and he ecipocal chaaceisics associaed o Q x,y ae 1, F x,y, whee F x,y is deived fom β x,y as was F fom b R 15

16 in 16. Le us now pove ha F x,y = F fo all x, y R. Fom 3 and assumpions 21, he pocess β x,y, X admis a fowad deivaive defined by lim Qx,y β x,y + h, X +h β x,y, X /F. h h Moeove his deivaive is equal o F, X. Indeed, lim Qx,y β x,y + h, X +h β x,y, X /F h h = lim Q x,y Q x,y U + h, X/F +h Q x,y U, X/F /F h h = lim Q x,y U + h, X U, X /F h h y = 1 2 X 1 2 lim Qx,y X +h X + 1 F p, X p dp/f h h1 h +h +Q x,y F p, X p dp F p, X p dpds/f s = F, X since all he ems of he.h.s. vanish excep lim h Q x,y 1 h +h F p, X pdp/f which ends o he desied expession. Since Q β x,y, X < + and Q 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 s fomula, ha is F, X = F x,y, X. The sic posiiviy of q, x,, z, 1, y assumed in A implies F = F x,y. This complees he poof of Theoem 4.3. Remak 4.5 : Le us make some commens abou he esuls of secion 4. If Q PΩ belongs o he class R P and saisfies he assumpions of Poposiion 4.1, we can see ha, as in sep 2 of he poof of Theoem 4.3, Q X +h X /F = 1 h h +h β d whee β is given by 25. Thanks o a esul of Föllme cf [9], Poposiion 2.5, we conclude ha fo almos evey [, 1[, he fowad Nelson deivaive defined as d + X := L 1 1 Ω lim h h EX +h X /F exiss and is equal o β. By symmey we also obain he exisence of d X := L 1 1 Ω lim h h EX X h / ˆF. Ou inegaion by pas fomula enables us o ecove a paicula case of Theoem 8.1 in [25]: if Q PΩ is a ecipocal pocess in he class R P, saisfies he assumpions of Poposiion 4.1 and is also such ha fo all ], 1[, he fis and second ode deivaives d + X, d X, d + d + X, d d X exis hen, fo allmos all ], 1[, Qd + d + X /X = Qd d X /X = F, X

17 This implies ha Q 1 2 d +d + X +d d X /X = F, X. The em 1 2 d +d + X +d d X can be inepeed as an acceleaion in Sochasic mechanics. This is why such an equaion may be called Newon equaion cf. [27]. Kene in [14] 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 QX +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 [14]. As coollay of Seps 1 and 2 of he above poof, we obain he fac ha any ecipocal pocess wih ecipocal chaaceisics 1, F saisfying assumpions 21 is a semi-maingale. 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 ; 1 λ. 32 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 33 X = X 1. 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 [12] sudied he momens of such Gaussian pocesses and elaed fileing poblems. Then, he fac ha he soluion of 33 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 22. 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 [19], 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 33 see [18] fo a descipion of such a geneal class. 17

18 The mehod of vaiaion of consans yields he following fom fo he unique soluion of 33: X = e λ X + e λ s e λ s db s = 1 e λ db e λ1+ s s + 1 e λ db s = ΨB 34 whee Ψ is he map on Ω defined by : e λ s Ψω = 1 e λ dω e λ1+ s s + 1 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 1 2 CovX s, X = 2λ sinh λ 2 =: R, s 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 [15], a genealisaion o he non saionay case is poved. Theoem 5.1 The law P pe of he soluion of 33 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 22 wih F, x = λ 2 x. Le g L 2, 1 and Φ S. By definiion,. gsds ΦX P pe D g Φ = P pe 1 lim ɛ ɛ ΦX + ɛ 1 = lim ɛ 1 = lim ɛ ɛ. ɛ P pe ΦX + ɛ gsds ΦX Φ P pe Φ P pe 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 35 X = X 1 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 35 is equal o ΨB ɛ in he same way as he soluion of 33 was equal o ΨB. We hus have P pe 1 D g Φ = lim ɛ ɛ P Eɛ g 1Φ Ψ 18

19 whee P is he Wiene measue and Eɛ g denoes he Gisanov densiy : Theefoe Eɛ g = exp ɛ gsdb s ɛ2 2 P pe 1 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 33. 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 1 gd =. 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 1 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 ; 1 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 + sinhλ1 x e λ1 X d 36 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, 1, x whee p, z, 1,. is he densiy of he Gaussian law P X 1./X = z. To compue his densiy i is sufficien o compue EX 1 /X = z and EX1 2/X = z, which come diecly fom he equaliy : X 1 = e λ1 X + e λ1 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 [2] ha P pe is he law of he pocess soluion of { X dx = db λ sinhλ1 X anhλ1 d X N ; 1 2λ coh λ

20 Le us noice ha equaion 37 is a andomized vesion, fo X no longe deeminisic, of equaion 36, which exacly eflecs a he level of he semi-maingale popey he above desinegaion P pe = P x,x 1 N ; 2λ cohλ 2 dx. Unde P pe, F is no degeneaed and he dif of X a ime in 37 is a funcion of X, X. So P pe is no Makovian while clealy P x,x is Makovian. 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 ; 1 λ 2. Aknowledgemens : The auhos ae vey gaeful o P. Caiaux fo fuiful discussions and suggesions on seveal echnical poins of his wok. The auhos also hank an anonymous Refeee fo seveal suggesions who helped hem o impove a fis vesion of his pape. Refeences [1] 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 199 [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 1991 [5] Ph. Couège and P. Renouad, Oscillaeus anhamoniques, mesues quasi-invaianes su CR, R e héoie quanique des champs en dimension 1, Aséisque 22-23, Soc. Mah. de Fance, Pais 1975 [6] C. Dellacheie and P.-A. Meye, Pobabiliés e Poeniel, Vol II. Théoie des Maingales, Hemann, Pais 198 [7] D. Deeude, Diffusions infini-dimensionnelles e champs gibbsiens, Thèse en pépaaion [8] B.K. Dive, A Cameon-Main ype quasi-invaiance heoem fo pinned Bownian moion on a compac Riemannian manifold, Tans. Am. Mah. Soc [9] H. Föllme, Time evesal on Wiene space, Sochasic Pocesses in Mahemaics and Physics, L. N. in Mah. 1158, Spinge [1] P. Gosselin and T. Wuzbache, An Io ype isomey fo loops in R d via he Bownian bidge, Séminaie de Pobabiliés XXXI, L. N. in Mah. 1655, Spinge [11] B. Jamison, Recipocal pocesses, Z. Wasch. Vew. Geb [12] H. Kwakenaak, Peiodic linea diffeenial sochasic pocesses, SIAM J. Conol Opimizaion

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