EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP WITH WHITE-NOISE DRIFT

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1 The Annal of Probability, Vol. 8, No. 1, EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP WITH WHITE-NOISE DRIFT By David Nualart 1 and Frederi Vien Univeritat de Barcelona and Univerity of North Texa We tudy the exitence and uniquene of the olution of a functionvalued tochatic evolution equation baed on a tochatic emigroup whoe kernel p t y x i Brownian in and t. The kernel p i uppoed to be meaurable with repect to the increment of an underlying Wiener proce in the interval t. The evolution equation i then anticipative and, chooing the Skorohod formulation, we etablih exitence and uniquene of a continuou olution with value in L. A an application we prove the exitence of a mild olution of the tochatic parabolic equation du t = x udt + v dt x u + F t x u W dt x where v and W are Brownian in time with repect to a common filtration. In thi cae, p i the formal backward heat kernel of x + v dt x x. 1. Introduction. The main goal of thi paper i to etablih the exitence and uniquene of olution for the following anticipative tochatic evolution equation: u t x = p t y x u yy 1.1 d d t + p t y x F y u y W d yy Here, the random field W = W t x t x i Gauian and centered with covariance min t Q x y, where Q i a bounded covariance function. For any <tlet F t be the σ-field generated by the family of random variable W r x W x r t x. We require p to be a tochatic kernel ee Definition 1 below. Thi mean that p t y x i meaurable with repect to the σ-field F t, the mapping y p t y x i a probability denity on and the following emigroup property i atified: p r y z p r t z xz = p t y x for any r t. On the other hand, we aume that F y u i F - meaurable and atifie the uual Lipchitz and linear growth condition Received May 1998; revied May Supported by DGICYT Grant PB Supported by NSF International Opportunitie Potdoctoral Fellowhip INT-9678 and NSF-NATO Potdoctoral Fellowhip DGE AMS 1991 ubject claification. Primary 6H15; econdary 6H7. Key word and phrae. Stochatic parabolic equation, anticipating tochatic calculu, Skorohod integral, tochatic emigroup. 36

2 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 37 with repect to the variable u, uniformly in the other variable. Thi implie that, even if u i adapted, the tochatic integral under the pace integral of 1.1 i anticipative. The integrand i the product of an adapted factor A =F y u y time a term B =p t y x, which i adapted to the future increment of the random field W. For integrand of thi type, Pardoux and Protter in [19] introduced a tochatic integral called the twoided tochatic integral, which i defined a the limit of Riemann um of the following type: A B W = lim A i B i+1 W i+1 W i i Later it wa proved in [16] that thi tochatic integral coincide with the Skorohod integral, which i an extenion of the Itô integral that can be interpreted a the adjoint of the derivative operator on the Wiener pace. Here we chooe thi type of tochatic integral in the formulation of the evolution equation 1.1. Thi choice i jutified by the concrete example of application to a tochatic partial differential equation SPDE. Equation 1.1 can be conidered a an example of the following abtract tochatic evolution equation of a random emigroup: 1. u t =T t u + t T t F u W d where u and W are procee taking value in a Hilbert or Banach pace. In thi equation T t t i a family of random linear operator on, atifying the backward flow property T t = T t u T u. In our cae, the emigroup i defined on the pace of bounded continuou function on by 1.3 T t f x = p t y x f yy Stochatic evolution equation with nonrandom emigroup have been extenively tudied ee [4] and reference therein. Only recently ha the quetion of uing random emigroup been addreed. In [1], the author tudy the exitence and uniquene of the olution of a tochatic evolution equation with a random emigroup T t that i F t -meaurable. It generator i the heat kernel of a econd-order elliptic differential operator whoe coefficient are random and adapted. We conider here the cae of a tochatic emigroup, that i, we aume that T t i F t -meaurable. Thi implie that the emigroup ha independent increment, and it infiniteimal generator may be, in general, a differential operator whoe coefficient are white-noie in time. For thi reaon, the kernel p t y x may be irregular like Brownian motion in the variable and t. A cla of uch emigroup ha been contructed in [7] uing tochatic flow. In comparion with the reult proved in [1], the F t -meaurablility hypothei on the emigroup allow u to get uitable etimate for the Skorohod integral in term of the firt derivative of p t y x, while in [1] two derivative are required, and the differentiability of p in the time variable i neceary.

3 38 D. NUALART AND F. VIENS Our motivation to tudy thi kind of equation i the analyi of a tochatic parabolic equation of the form 1.4 u dt x = x u t xt + v dt x x u t x +F t x u t x W dt x where the procee v t x and W t x are Brownian in time with repect to a common filtration. If thee random field are not patially mooth, the equation cannot have a meaning in the trong ene. One mut define a weaker ene of olution. Thi paper chooe to undertand 1.4 in the evolution ene: thi i imply a tochatic evolution equation like 1.1 above; the kernel p being the backward heat kernel of the formal operator + v. Several other weak ene for thi Cauchy problem have been invetigated in the literature. They include the o-called martingale problem of Stroock and Varadhan; a materful treatment can be found in the recent monograph chapter [14], in which the regularity condition on W and v are extremely weak, the trade-off being that one can only guarantee the exitence of the law of a olution, rather than a olution itelf a a function of the procee W and v. The determinitic notion of vicoity olution, which made it appearance in the theory of tochatic procee for it relation to the nonlinear Feynman Kac formula ee [18], ha been adapted to the tochatic etting, of which the mot recent treatment, in [3], appear to be quite general. The theory of white-noie analyi ha been ued to define ditribution-valued olution to pecial type of SPDE, via the o-called Wick product ee [5]. A olution of a tochatic PDE in the weak ene can be introduced uing tet function and integrating by part. Uually the evolution olution i alo a olution in the weak ene ee, for intance, [1]. Weak olution have een a recent renewal of interet in the etting of meaure-valued olution [11], [6]; alo ee the introductory remark in [7], a well a in the connection of SPDE with uperprocee. Thee work reveal that the weak ene i not tailored to deciding when an SPDE ha a function-valued olution. Mot recently, a very ucceful attempt to olve SPDE by mean of analytical method wa completed by Krylov in [8]. In thi monograph, the author give very weak aumption on coefficient imilar to W and v in 1.4, guaranteeing that the olution i in ome Sobolev pace of ditribution-valued procee. On the other hand, tochatic Sobolev embedding theorem are ued in [8] to obtain continuity reult. Our approach to contructing an evolution olution treat the cae of L uing the o-called factorization method, in the pirit of the work with nonrandom emigroup in [4]. In comparion with the analytical method ued in [8], thi approach provide an explicit integral expreion for the olution, and it ha ome advantage like the poibility of handling, without additional effort, equation on bounded domain with Dirichlet or Neumann boundary condition. Another interet of the evolution olution i found in the multiplicative linear cae F t x u =u in eeking a Feynman Kac formula. A forthcoming

4 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 39 article [] will how how thi formula come for free in the evolution etting, thank to the exitence of the kernel p alluded to above, and it repreentation in term of the tochatic Markov proce ϕ with generator + v. Thi tudy will ue thi formula to invetigate the olution Lyapunov exponent. Another approach to the Feynman Kac formula can be found in [13], for a different form Let u alo remark that the author of the preent paper, together with a collaborator, have hown in [1] that the approach ued in thi paper can be extended, in the one-dimenional cae, to the cae where W i a pace time white noie. We now explain the tructure of thi paper. Following the general cheme for evolution equation with nonrandom emigroup ee [4], we etablih in Section the exitence and uniquene of a olution with value in L for 1.1 Theorem 3 by a fixed point method. Thi theorem i a conequence of the etimate given in Propoition 4 which, under regularity and integrability condition v, vi and vii on the tochatic kernel p t y x, follow from the iometry property of the Skorohod integral and the emigroup property. Section 3 i devoted to etablihing the continuity of the olution a an L -valued proce. For thi we need the maximal inequality for the Skorohod integral tated in Propoition 6, which require lightly tronger L p integrability condition on p t y x [ v p, vi p and vii p for ome p>, and ue of Itô formula for the Skorohod integral following the approach introduced in [1]. In order to analyze 1.4 we contruct uch a tochatic kernel in Section 4, that i, a kernel p which atifie the forward Kolmogorov equation p d y = y p y + p y v d y y where v t x i a d-dimenional centered Gauian random field with covariance min t G x y, and we aume that the matrix G atifie the coercitivy aumption I 1 G x x > The contruction of thi kernel follow the approach developed by Kifer and Kunita in [7], baed on the backward tochatic flow ϕ t x aociated with v t x. Section 5 i devoted to howing that thi tochatic kernel atifie the condition v p, vi p and vii p, introduced in Section 3, provided the random field v t x atifie ome regularity and integrability condition in the variable x. A a conequence, thi prove the exitence and uniquene of the olution to 1.1 for thi particular tochatic kernel. Finally, in Section 6 we how that the olution u t x to 1.1 for thi kernel i alo a weak olution to 1.4, thereby jutifying the choice of the Skorohod formulation for eeking an adapted olution to the evolution equation Stochatic evolution equation with a tochatic kernel: exitence and uniquene of a olution in L. Fix a meaurable pace S with a finite meaure µ on it, a well a a time interval T. Conider the product pace T S equipped with the product meaure λ µ, where λ denote the Lebegue meaure on T. Let M = M A A B T

5 4 D. NUALART AND F. VIENS be a centered Gauian family of random variable, defined in ome probability pace F P, with covariance function given by E M A M B = λ µ A B Suppoe that F i generated by M. We will aume that the random field W t y appearing in 1.1 i of the form t W t y = a λ y M dλ S where a i a determinitic meaurable function verifying the following condition:.1 C a = up a λ y µ dλ < y S In thi way, the covariance function of W i Q x y = a λ x a λ y µ dλ S In principle, although µ i alway a poitive meaure, M and a may be complex valued. For notational implicity, we will aume that M and a are real valued. Condition.1 imply ay that Q x x =EW 1 x i a bounded function. Thi can be conidered a a weak form of patial ubhomogeneity. For each <t T we will denote by F t the σ-field generated by the random variable M A A t S. Thi definition coincide with the one given in the introduction in term of the random field W. We can develop a tochatic calculu of variation with repect to the Gauian family M following the line of [15]. The reference Hilbert pace i here H = L T S λ µ. For an element f H we can define the Gauian random variable M f = T S f λ M dλ. Let be the cla of mooth random variable of the form G = g M f 1 M f n where g i an infinitely differentiable function with bounded derivative of all order. For a uch a random variable we define it derivative a the random field on T S given by n g D λ G = M f x 1 M f n f i λ i=1 i Iterated derivative are defined in an obviou way. Then, for any integer k 1 and any real number p 1 the Sobolev pace k p i defined a the completion of with repect to the eminorm, G p k k p = p/ E D 1 λ T S j 1 D j λ j G d 1 µ dλ 1 d j µ dλ j j= For any real and eparable Hilbert pace V we denote by k p V the correponding Sobolev pace of V-valued random variable. Note that the derivative

6 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 41 operator preerve adaptedne and that, if F i F F t T -meaurable, then D r λ F = if r t. The Skorohod integral i defined a the adjoint of the derivative operator in L. That i, a quare integrable random field H λ i Skorohod integrable if for any G we have [ E D λ GH λ d µ dλ ] T S c G L The Skorohod integral of H, denoted by δ H, i then defined via the Riez repreentation theorem by the duality relationhip [ ]. E D λ GH λ d µ dλ = E Gδ H T S We will alo make ue of the notation δ H = T S H λm dλ. A random field H λ i aid to be adapted if H λ i F -meaurable for each λ. It hold that quare integrable adapted random field are Skorohod integrable and the Skorohod integral with repect to M coincide with the Itô tochatic integral, which can alo be defined by mean of the theory of martingale meaure ee, e.g., [1]. On the other hand, procee in the pace 1 H are alo Skorohod integrable. We will make ue of the following formula for the L -norm of the Skorohod integral of an L -valued proce. Henceforth we will denote by the norm in L. Lemma 1. Let H = H λ T λ S be an L -valued quare integrable random field uch that H λ x belong to 1 for almot each λ x, the proce D λ H r λ x 1 r r λ T S i Skorohod integrable for almot each λ x and d E H λ x D λ H r λ x M drλ.3 T S S d µ dλx < Then H i Skorohod integrable and E H λ M dλ T S = E H λ d µ dλ +E T S T S H λ x D λ H r λ x M drλ µ dλx S

7 4 D. NUALART AND F. VIENS Proof. By an approximation argument we can aume that H i an mooth elementary proce. In thi cae the formula i a traightforward conequence of the duality relationhip between the Skorohod integral and the derivative operator and the following iometry property: E H λ M dλ T S = E H λ d µ dλ T S T +E D λ H r λ D r λ H λ dr µ dλ µ dλ S S We will alo need the following Fubini theorem for the Skorohod integral, whoe proof i an immediate conequence of the definition of the Skorohod integral. Lemma. Let H λ θ λ T S θ be a meaurable random field parameterized by a meaure pace ν with finite meaure ν. Suppoe that E H λ θ ν dθ d µ dλ < T S H λ θ i Skorohod integrable for ν-almot all θ, and E δh θ ν dθ <. Then H λ θ ν dθ i Skorohod integrable and δ H θ ν dθ = δh θ ν dθ Conider a meaurable random field H y parameterized by T uch that H d y a λ y dy < a.. The integral H T y W d yy i defined by H y W d yy = H y a λ yy M dλ T T S provided the random field H y a λ yy i Skorohod integrable with repect to M. With thi definition, 1.1 can be written in term of M a follow: u t x = p t y x u yy.4 d t + p t y x F y u y a λ yy M dλ S Let u now introduce the kind of tochatic kernel we are going to deal with.

8 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 43 Definition 1. A random function p t y xefined for <t T, x y i called a backward tochatic kernel if the following condition are atified: i [Adaptedne] p t y x i F t -meaurable; ii p t y x ; iii p t y xy = 1; iv [Backward flow property] p r y z p r t z xz = p t y x for almot all y and for all x, <r<t T. Conider the following additional condition: v C b = up t y E p t y xx < vi p t y x belong to the Sobolev pace 1 for each x y and <t T and p t x belong to 1 L. Moreover, there exit a verion of the derivative uch that the following limit exit in L L for each λ t x: D λ p t x =lim ε D λ p ε t x Furthermore, p t y i left continuou in t T with value in L L. vii There exit contant c 1 c, uch that and up E a λ y p t y x D λ p t z x µ dλx dy z S c 1 t 1/ up E a λ y p t y x D λ p t z x µ dλz dx y S c t 1/ Taking into account the propertie of the derivative operator and the fact that p t x belong to 1 L, we can write the following formula for r<<t: D λ p r t y x =D λ p r ε y z p ε t z xz = p r ε y z D λ p ε t z xz and, letting ε tend to zero and uing hypothei vi we deduce.5 D λ p r t y x = p r y z D λ p t z xz We require the random function F T to be progreively meaurable and to atify the uual Lipchitz and linear growth

9 44 D. NUALART AND F. VIENS condition in the variable u. That i, we aume that F atifie the following condition: 1. F i meaurable with repect to the σ-field B t F t when retricted to t, for each t T.. F t y u F t y v K T u v, for all t T, y and u v. 3. F t y u K T 1 + u, for all t T, y and u. With thee preliminarie we are able to tate the main reult of thi ection. Theorem 3. Let u be a function in L. Let F y u be a random function atifying the above condition 1, and 3. Let p t y x be a tochatic kernel in the ene of Definition 1 atifying condition v, vi and vii. Then there exit a unique adapted L -valued olution to.4 uch that E T u d < For the proof of thi theorem we need the following etimate of a Skorohod integral of the form t p t y x φ y W d yy, where φ y i an d adapted quare integrable proce. Propoition 4. Let φ = φ y T y be an adapted random field uch that E T φ d <. Let p t y x be a tochatic kernel in the ene of Definition 1 atifying condition v, vi and vii. Then the L -valued random field { } 1 t p t y φ y a λ yy T λ S i Skorohod integrable with repect to M for almot all t T, and for ome contant C>, which depend on T, C a, C b, c 1 and c, it hold that t E p t y φ y a λ yy M dλ S C t t 1/ E φ d Proof. Denote by E the cla of mooth elementary adapted random field of the form.6 φ y = n G ik b k y 1 ti t i+1 i k=1 where G ik, b k K d,= t 1 < <t n+1 = T, and G ik i F ti - meaurable. Let φ be an adapted random field uch that E T φ d <. We can find a equence φ n of mooth elementary adapted random field in the cla E atifying lim n T E φ n φ d =

10 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 45 Therefore the equence of function t t t 1/ E φ n φ d converge to zero in L 1 T. Thi implie the exitence of a ubequence n i uch that for all t T out of a et of zero Lebegue meaure t lim t 1/ E φ ni φ d = i Recall that if a equence H n of Skorohod integrable procee converge in L H L to H, and δ H n converge in L L, then H i Skorohod integrable and δ H i the limit of δ H n. Hence, we can aume that φ i of the form.6. Set H λ x = p t y x φ y a λ yy and t t x = H λ x M dλ Notice that n t x = G ik i k=1 t i t i+1 t S S p t y x b k y a λ yy M dλ Let u how that the proce H λ x 1 t verifie the aumption of Lemma 1. By condition v we have t.7 E H λ d µ dλ C ac b E φ d < t S In order to check that the proce D λ H r λ x 1 r r λ T S i Skorohod integrable and condition.3 hold let u compute, uing.5, D λ H r λ x =D λ p r t y x φ r y a λ y dy d = D λ p r t y x φ r y a λ y dy d = p r y z D λ p t z xz φ r y a λ y dy d = p r y z φ r y a λ y dy D λ p t z xz Now we ue the fact that the random variable D λ p t x i F t -meaurable and belong to the pace L L. Thi fact, together with the particular form of the proce φ imply that D λ H r λ x i Skorohod integrable in S, and D λ p t z x can be factorized out of the Skorohod integral, obtaining D λ H r λ x M drλ = D λ p t z x zz S

11 46 D. NUALART AND F. VIENS A a conequence, d E H λ x D λ H r λ x M drλ d µ dλx t S S d = E H λ x D λ p t z x z z d µ dλx t S.8 t d d 1 E p t y x D λ p t z x x a λ y µ dλ S [ Eφ y + E z ] dz dy d t t c t 1/ E φ d + c 1 t 1/ E 1 The right-hand ide of.8 i finite becaue φ i mooth and elementary and, hence, condition.3 hold. A a conequence, Lemma 1 yield E t = E H λ d µ dλ.9 +E t S d H λ x t S Subtituting.7 and.8 into.9 yield t E t C ac b E φ d D λ H r λ x M drλ S t +c t 1/ E φ d + c 1 t d µ dλx t 1/ E d Thi expreion eaily implie the deired reult by a uitable generalization of Gronwall lemma. Proof of Theorem 3. Suppoe that u and v are two adapted olution to.4, uch that E T u d < and E T v d <. Set F y u = F y u y From Propoition 4 and the Lipchitz property of the function F, we obtain E u t v t t = E C t CK T S p t y z [ F y u F y v ] a λ yy M dλ t 1/ E F u F v d t t 1/ E u v d

12 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 47 and thi implie that u = v. The proof of the exitence can be done by the uual Picard iteration procedure. That i, we define recurively u t x = p t y x u yy and u n+1 t x = p t y x u yy t + p t y x F y u n y a λ yy M dλ S for all n. Uing u L and condition v, we obtain E u t <. It follow by induction, uing Propoition 4, that E n= u n+1 t u n t <, and the limit of the equence u n provide the olution. Under the condition that the initial Picard approximation u t i a continuou function from T into L, we can ue the previou proof to how that the olution to Equation.4 hare the ame continuity property. A ufficient condition for u continuity in L i lim up h y E p t+ h y x p t y x x = If u happen to be almot urely continuou from T into L, one might ak whether the olution to.4 till hare the ame property. Thi i true under additional condition on the tochatic kernel, a the reult of the next ection how. 3. Stochatic evolution equation with a tochatic kernel: continuity of the olution. In thi ection we will how that the olution u t of.4 obtained in the lat ection i continuou in time. For thi we need ome additional integrability condition on the kernel p t y x. Fixp and conider the following hypothee: p/ v p C p 1 = up E up p t y xx < y t T vi p Condition vi hold and for each K> and <δ<t,wehave t δ p/ E D λ p t y x dx dy µ dλr d < B K S where B K = y y K. Moreover, for every compact et K and for all, t p t i continuou in T with value in L 1 B K.

13 48 D. NUALART AND F. VIENS vii p There exit a contant C p uch that for all y z up E D λ p t z x D λ p t y x µ dλz dx y S C p t p/ The main ingredient in the proof of the continuity of the olution to the tochatic evolution equation.4 are the etimate for the Skorohod integral etablihed in the next two theorem. Thee theorem are analogou to Theorem 3. and Theorem 3.3 in [1] for the cae of a random emigroup T t which i F t -meaurable. For a given random field φ = φ y T y we will define the operator S t φ λ x = p t y x φ y a λ yy Uing condition.1 we have 3.1 S t φ L S C a d p t y x φ y dy dx Propoition 5. Fix p < 4 and α 1. Let φ = φ y T y be an adapted random field uch that E T φ p d <. Let p t y x be a tochatic kernel in the ene of Definition 1 atifying condition v p, vi p and vii p. Then the L -valued random field t α + S t φ λ T λ S i Skorohod integrable with repect to M for almot all t T, and for ome contant C 1 >, which depend on T, p, α, C a, C p 1 and C p, it hold that t E t α p p t y φ y a λ yy M dλ S 3. d t C 1 t α+ p/4 α 1 E φ p d Proof. A in the proof of Propoition 4, we can aume that φ i of the form.6. Fix t >t 1 in T, and define and B λ x =1 t1 t α S t 1 φ λ x X t x = t S B λ x M dλ for t t 1. Suppoe firt that p t 1 y x i an elementary backwardadapted proce of the form n 3.3 H ijk β j y γ k x 1 i i+1 i j k=1 p/

14 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 49 where H ijk, β j γ k K d,= 1 < < n+1 = t 1, and H ijk i F i+1 t 1 - meaurable. Applying Itô formula for Hilbert-valued Skorohod integral ee, for intance, Propoition.9 in [1], taking the mathematical expectation and uing the Cauchy Schwarz inequality a in the proof of Theorem 3. in [1], we obtain E X t p p p 1 E t X p B L S + B L S D B r λ M drλ S L S d Unlike the proof of Theorem 3. in [1], here we keep together the factor B L S and S D B r λ M drλ L. Then, Hölder inequality lead to the following etimate: S t E X t p E X p p /p A d where A = p p 1 t α { E t α B p /p L S 1 + E t α B p 1/p L S E D B r λ M drλ S p L S 1/p } Becaue φ i a imple proce, it i eaily een that by condition v p we have E X t p <. Then the firt lemma in [] implie that that i, 3.4 where E X t p p t A d p/ t E X t p C T p α t α 1 + t t α+ αp/ 1 1 =E S t 1 φ p L S and =E S D B r λ M drλ p L S

15 5 D. NUALART AND F. VIENS Condition v p and vi p guarantee that the right-hand ide of 3.4 i finite. A a conequence, we can approximate p t 1 y x by elementary backwardadapted procee of the form 3.3 in uch a way that 3.4 till hold. The term 1 can be etimated a follow uing 3.1 and aumption v p : 3.5 p/ 1 C p/ a E p t 1 y x φ y dy dx d C p/ a E [ φ p φ y ] E C p/ a C p 1E φ p p/ p t 1 y x dx dy In order to etimate the term we firt write, uing.5, D λ B r λ x = t r α D λ p r t 1 y x φ r y a λ y dy d = t r α p r y z D λ p t 1 z x dz φ r y a λ yy = t r α S r φ λ z D λ p t 1 z xz Now we ue the fact that the random variable D λ p t 1 z x i F t1 -meaurable. Thi implie that it can be factorized out of the Skorohod integral, and we obtain p E D B r λ M drλ S L S 3.6 p = E D p t 1 z Y zz L S where Y z = S t r α S r φ λ z M drλ Applying condition vii p we get 3.7 C p t 1 p/ E Y p Subtituting 3.5 and 3.7 into 3.4 yield t E X t p C t α E φ p d + t t α+pα/ t 1 p/4 E φ p E Y p 1/ d

16 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 51 for ome contant C which depend on T, p, α, C a, C p 1 and C p. If we take t = t 1,wehaveX t = Y t and we obtain t E X t p C t α+ p/4 α 1 E φ p d + t t α+ p/4 α 1 E X p d Applying Gronwall lemma we get the deired reult. Note that in the proof of Propoition 5 we have only ued, intead of v p, the weaker etimate, p/ up E p t y xx < t y Condition v p i required in the proof of the maximal inequality. Propoition 6. Fix <p<4. Let φ = φ y T y be an adapted random field uch that E T φ p d <. Let p t y x be a tochatic kernel in the ene of Definition 1 atifying condition v p, vi p and vii p. Then the L -valued random field, { } 1 t p t y x φ y a λ yy T λ S i Skorohod integrable with repect to M and there exit a contant C >, which depend on T, p, α, C a and C p 1 and C p uch that E up t T t S t C E φ p d p p t y φ y a λ yy M dλ Proof. We will make ue of the factorization method in order to handle the upremum in t. Fixα 1/p 1/. We can write 3.8 p t y x = C α t p r y z r α p r t z x t r α 1 dr dz where C α = in πα /π. By Propoition 5 for all r T a.e. the proce r α + S r φ λ z i Skorohod integrable. Uing 3.8 and Fubini

17 5 D. NUALART AND F. VIENS theorem for the Skorohod integral ee Lemma yield S t φ λ x M dλ t S t = C α r α S r φ λ z t S p r t z x t r α 1 dr dz M dλ t = C α r α S r φ λ z p r t z x M dλ r S t r α 1 dr dz The term p r t z x can be factorized out of the Skorohod integral and we obtain S t φ x λ M dλ 3.9 t S t = C α Y r z p r t z x t r α 1 dr dz where Y r z = r S r α S r φ λ z M dλ Applying condition v p and Hölder inequality we obtain 3.1 E up t T 1 π p E t S p S t φ λ M dλ t up t T t r α 1 1/ p Y r z p r t z xx dz dr T p/ C p α T E up Y r z p r t z xx dz dr t T T C p α T E Y r p Y r z up t r T T C p α T C p 1 E Y r p dr p r t z xx p/ dz dr

18 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 53 Finally, uing the etimate 3. yield p E up S t φ λ M dλ t T t S C p α T C p 1 C 1 T T C E φ p d r r α+ p/4 α 1 E φ p d dr We are ready to tate and prove our main reult. Theorem 7. Let u be a function in L. Let F y u be a random function atifying the above condition 1, and 3. Let p t y x be a tochatic kernel in the ene of Definition 1 atifying condition v p, vi p and vii p for ome p>. Then the L -valued olution to.4 ha a continuou verion and atifie 3.11 E up u t p < t T Proof. We firt have to how that the following two term atify etimate 3.11: A 1 t = p t y x u yy and t A t = p t y x F y u y a λ yy M dλ S The firt etimate follow from condition vi p and the econd from Propoition 6 and Theorem 3. On the other hand, we have to how that A 1 and A are continuou. Firt notice that if Z r y i a bounded random field with compact upport K in y, wehave Z r y p r t + δ y p r t y dy Z d K p r t + δ y x p r t y x y dx which tend to zero almot urely for all r by condition vi p. Taking into account condition v p, in proving the continuity of A 1 we can aume that u i a mooth function with compact upport, and in thi cae the continuity follow from property vi p by letting Z y =u y above. In order to how the continuity of A we write, uing 3.9, A t + δ A t 1 t π Y r y p r t + δ y p r t y t r α 1 dr dy

19 54 D. NUALART AND F. VIENS + 1 t+δ π Y r y p r t + δ y t r α 1 dr dy t = Ɣ 1 + Ɣ By the maximal inequality 3.1 we can aume that Y r x i a mooth elementary adapted proce. In thi cae, letting Z r y =Y r y above, it i clear from property vi p that the term Y r y p r t + δ y p r t y y converge to zero in L a δ tend to zero for each fixed r and ω. Then the convergence of Ɣ 1 to zero follow by the dominated convergence theorem and property v p. Finally, property v p alo implie that Ɣ converge to zero a δ tend to zero. 4. Stochatic emigroup generated by random partial differential operator. Suppoe that p t y x i a tochatic kernel in the ene of Definition 1. Set T t f x = p t y x f yy, where f i a bounded Borel d function on. Then T t define a tochatic emigroup of poitive operator. Let u recall, following [7], the definition and ome propertie of thi type of tochatic emigroup. Let b be the Banach pace of bounded continuou function on. Definition. A family T t <t T of random linear operator on b i called a tochatic emigroup if it atifie the following condition: i T t f for any f, T t 1 = 1, and T t f n for any equence f n in b ; ii T t u T u = T t for any <u<t; iii T t f i an F t -meaurable random variable for each f b and each <t. Then a tochatic kernel p t y x in the ene of Definition 1 give rie to a tochatic emigroup, provided T t f i continuou whenever f i continuou and bounded. Converely, if a tochatic emigroup i uch that the probability meaure induced by T t i abolutely continuou with repect to the Lebegue meaure, then we can find a verion of it denity and thi will produce a tochatic kernel. In thi ection we contruct a tochatic emigroup whoe infiniteimal generator i the random operator + v, where v t x i a d-dimenional Gauian field that i Brownian in time, and the differential v t xt = v dt x i interpreted in the backward Itô ene. Aume that v t x can be repreented a t v t x = g λ x M dλ S

20 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 55 where g S i a meaurable function, differentiable with repect to the variable x, and atifying the following condition: 4.1 C g = up g λ x + x g λ x µ dλ < x S Thi condition mean that both v 1 x and it derivative x v 1 x, which only need to exit in the L ene, have variance that are bounded in x. Set G ij x y = S g i λ x g j λ y µ dλ and G x =G x x. Let u introduce the following condition: H1 x =I 1 G x εi for each x d and for ome ε>. Thi i known a the coercivity condition. Let σ be a matrix uch that σσ =. Let b t bead-dimenional tandard Brownian motion with variance t defined on another probability pace Q. Conider the following backward tochatic differential equation on the product probability pace F P Q 4. ϕ t x =x t v dr ϕ t r x + t σ ϕ t r x b dr Applying Theorem and in [1] one can prove that 4. ha a olution ϕ = ϕ t x t T x which i a tochatic flow of homeomorphim. Thi mean that ϕ r ϕ t r x = ϕ t x for all <r<t x, a.. Moreover, ϕ t x i continuou in the three variable. Equation 4. can alo be written a t t 4.3 ϕ t x =x g λ ϕ t r x M drλ + σ ϕ t r x b dr S For each t T we introduce the random operator T t defined by T t f x =E Q f ϕ t x where f belong to b. In the equel we will denote by E the mathematical expectation with repect to the probabilitie P and P Q, and by E Q the expectation with repect to Q Propoition 8. of Definition. The operator T t form a tochatic emigroup in the ene Proof. It i not difficult to how that T t f belong to b for any f in b. Propertie i and iii are obviou. Property ii follow from the flow property T t f x =E Q f ϕ t x = E Q f ϕ u ϕ t u x = E Q E Q f ϕ u z z=ϕt u x =T t u T u f x

21 56 D. NUALART AND F. VIENS Let be the cla of function which are twice continuouly differentiable, and let b be the pace of function in which are bounded and have bounded partial derivative up to the econd order. The tochatic emigroup atifie the following forward Kolmogorov equation although time flow backward in thi equation, it mut be called the forward equation becaue it flow in the ame direction a that ued to define the flow ϕ: 4.4 T t f x =f x + t T t r v dr f x + t T t r f xr for any function f in the pace b. The econd ummand in the right-hand ide of the above equation ha to be undertood a the expectation with repect to the probability Q of a backward tochatic integral t t 4.5 T t r v dr f x =E Q v dr ϕ t r x f ϕ t r x Equation 4.4 follow eaily from Itô formula, f ϕ t x = f x t t f ϕ t r x v dr ϕ t r x f ϕ t r x σ ϕ t r x b dr d i j=1 d i j=1 t t f x i x j ϕ t r x G ij ϕ t r x r f x i x j ϕ t r x σσ ij ϕ t r x r A a conequence, integrating with repect to the probability Q we obtain t T t f x =f x +E Q v dr ϕ t r x f ϕ t r x t +E Q f ϕ t r x r which i 4.4. The following condition [tronger that 4.1] imply that the random operator T t map b into b : S up x S α g λ x µ dλ < α g λ x α g λ y µ dλ C x y δ for ome δ > and for any multiindex α = α 1 α d uch that α = α 1 + +α d. Indeed, under thee condition the mapping ϕ r x are twice continuouly differentiable in x, with almot urely Hölder-continuou econd derivative ee [9], Chapter 3 and 4. Henceforth we will aume con-

22 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 57 dition 4.6 and 4.7. Under thee condition, the tochatic emigroup T t alo atifie the following backward Kolmogorov equation: 4.8 T t f x =f x + t T r f x v dr x + t T r f xr for any f K d. Here the tochatic integral i an ordinary Itô integral. Proof of the backward Kolmogorov equation. Let = t <t 1 < < t n = t be a ubdiviion of the time interval t. Uing the emigroup property we can write n 1 T t f x f x = T ti+1 t i I T ti f x i= By uing a tandard localization argument, one can how that 4.4 actually hold for any tet function g b atifying the condition t E g ϕ t r x dr < t E g ϕ t r x r < We wih to ue 4.4 on the inteval t i t i+1 with the tet function g = T ti f. Thi i legitimate; a T ti f i independent of F ti t i+1, it may be conidered a determinitic; moreover, g i in b and atifie bound 4.9 and 4.1, which i proved by exploiting the following bound on the derivative of ϕ: for j = 1, 4.11 up E j ϕ t x < t T Thi lat fact i proved by uing Itô formula on the SDE atified by j ϕ t x. Therefore, where T t f x f x = A 1 = A = n 1 i= n 1 i= ti+1 t i ti+1 t i n 1 i= + ti+1 t i n 1 i= ti+1 t i T ti+1 r v dr T ti f x = A 1 + A + A 3 + A 4 T ti+1 r T ti f xr T ti+1 r I v dr T ti f x T ti+1 r I T ti f xr

23 58 D. NUALART AND F. VIENS n 1 A 3 = v t i+1 x v t i x T ti f x A 4 = i= n 1 i= T ti f x t i+1 t i The term A 1 and A converge to zero in L and the term A 3 and A 4 converge, repectively, to the lat two ummand in the right-hand ide of 4.8, a the meh of the partition tend to zero. Let u firt prove the convergence of A 1. Uing the expreion 4.5 we can write where and A 11 = A 1 = A 1 = n 1 i= n 1 i= n 1 i= E Q ti+1 t i = A 11 + A 1 E Q ti+1 Then we have n 1 E A 11 =E v dr ϕ ti+1 r x T ti f ϕ ti+1 r x ti+1 v dr x T ti f x t i t i v dr ϕ ti+1 r x [ T ti f ϕ ti+1 r x T ti f x ] ti+1 [ E Q v dr ϕti+1 r x T ti f x v dr x T ti f x ] t i up x d ti+1 i= k l=1 t i G x up t x G kl ϕ ti+1 r x [ k T ti f ϕ ti+1 r x k T ti f x ] [ l T ti f ϕ ti+1 r x l T ti f x ] dr E T t f x where π =up i t i+1 t i. Note that, by 4.11, up t x up t r < π E ϕ t r x x E T t f x f up E ϕ t x < t x and, imilarly up t E T t f <. Hence, E A 11 converge to zero a π. For the term A 1 we can write n 1 E A 1 =E d ti+1 [ Gkl ϕ ti+1 r x G kl ϕ ti+1 r x x +G kl x ] t i i= k l=1 k T ti f x l T ti f xr

24 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 59 up E T t f x G t x x y up t r < π E ϕ t r x x and again thi converge to zero a π. The convergence of A would follow by the ame argument. Let u how that A 3 converge to the econd ummand of the right-hand ide of 4.8: E A 3 t = E T r f x v dr x n 1 i= n 1 = E ti+1 i= k l=1 T G T G v dr x [ T ti f x T r f x ] t i d ti+1 t i G kl x [ k T ti f x k T r f x ] [ l T ti f x l T r f x ] dr up E T u f x T r f x r u π r u up r u π r u E [ f ϕ u x ] [ f ϕ r x ] To how that thi converge to zero a π, we write E f ϕ u x f ϕ r x E f ϕ u x f ϕ r x ϕ u x +E ϕ u x ϕ r x f ϕ r x f [E ϕ u x 4] 1/[E ϕ u x ϕ r x 4] 1/ + f E ϕu x ϕ r x Therefore, we only need to how that E ϕ u x ϕ r x 4 tend to zero when u r, uniformly in, and thi follow eaily uing Itô formula. In a imilar way one can how that A 4 converge in L to the lat ummand in the right-hand ide of 4.8. The next propoition how that the marginal probability denity p t y x =Q ϕ t x dy /dy exit and atifie the condition given in Definition 1. Thi reult ha been proved by Kunita in [1] ee alo [7], Theorem.4

25 6 D. NUALART AND F. VIENS under the condition that the random field v t x can be repreented a a finite linear combination of independent ordinary Brownian motion multiplied by -vector field on Our proof, which only require condition 4.1, i baed on the criterion of abolute continuity proved by Bouleau and Hirch [] and ue the technique of the partial Malliavin calculu ee [17]. Propoition 9. Suppoe that g atifie condition 4.1 and the coercivity condition H1. Let ϕ t x be the tochatic flow olution of the backward tochatic differential equation 4.3. Then, there i a verion of the the marginal denity p t y x =Q ϕ t x dy /dy which atifie condition i to iv of Definition 1. Proof. One can how that the random variable ϕ j t x, <t T, x, j = 1 d, belong to the Sobolev pace 1, with repect to the product meaure P Q. Let u denote by D k θ the derivative operator with repect to the Brownian motion b. We have the following linear equation for the derivative of ϕ t x, for θ t, 1 i k d: θ D k θ ϕi t x =σ ik ϕ t θ x g i λ ϕ t r x D k θ ϕ t r x M drλ d l=1 θ S σ il λ ϕ t r x D k θ ϕ t r x b l dr Let u denote by γ ϕt x the Malliavin matrix of the random vector ϕ t x, that i, γ ij d ϕ t x = t D k θ ϕi t x Dk θ ϕj t xθ k=1 By mean of the technique of the partial Malliavin calculu it follow that ee Theorem 4. in [17] almot urely the marginal law of ϕ t x i abolutely continuou with repect to the Lebegue meaure in for all <t T and x, if almot urely we have 4.13 det γ ϕt x > k=1 <t T x Condition 4.13 i an immediate conequence of 4.1. In fact, notice firt that we can chooe a verion of the derivative D θ ϕ t x which i continuou in θ t and in x. Then, let v be a unit vector in uch that d t v t d γ ϕt x v = D k θ ϕi t i x v dθ = i=1 Thi implie d i=1 D k θ ϕi t x v i = for each k and θ, and chooing θ = we get di=1 σ ik ϕ t x v i = which implie v =. Hence, 4.13 hold. Propertie i to iv of Definition 1 hold trivially due to the fact that T t i a tochatic emigroup.

26 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 61 The forward and backward Kolmogorov equation can be written a follow in term of the tochatic kernel p t y x : t p t y x f yy = f x + p r t y x v dr y f yy 4.14 d t + p r t y x f yy dr t p t y x f yy = f x + v dr x x p r y x f yy 4.15 d t + x p r y x f yy dr Equation 4.14 i the weak formulation of the following tochatic partial differential equation where the tochatic integral i interpreted a an Itô backward tochatic integral: p d y = y p y + p y v d y y ee, e.g., [9], Chapter 6. Formally, a announced in the introduction, we can ay that p t y x i the heat kernel of the operator + v On the other hand, 4.15 lead to the following evolution equation: 4.16 p t y x =q t y x t + v dr z z p r y z q r t z xz where q t y xenote the heat kernel; that i, q t y x = 4π t d/ x y exp 4 t 5. Etimate of a tochatic heat kernel. Let ϕ t x be the tochatic flow olution of 4.3, and denote by p t y x it aociated tochatic kernel. In thi ection we will how that, under uitable aumption, the tochatic kernel p t y x alo atifie condition v p, vi p and vii p for all p. Condition v p will be a conequence of the backward Kolmogorov evolution equation On the other hand, we will make ue of the technique of the Malliavin calculu in order to provide a priori etimate for the integral of the tochatic kernel p t y x in x and to how condition vi p and vii p. Propoition 1. We aume the coercivity condition H1. Suppoe that g i d + 1 time continuouly differentiable in the variable x, and the following integrability condition hold: up α g λ x 5.1 x S x 1 α 1 xd α d µ dλ <

27 6 D. NUALART AND F. VIENS for any multiindex α = α 1 α d with α d + 1, where α =α 1 + +α d. Then the tochatic kernel p t y x atifie p up E p t y xx < t T y for all p 1. Proof. We will denote by δ the divergence operator with repect to the Brownian motion b. Applying the integration-by-part formula of Malliavin calculu with repect to the Brownian motion b we deduce the following expreion for the tochatic kernel p t y x : 5. p t y x =E Q 1 ϕt x >y H t x where ϕ t x y mean ϕ i t x y i for each coordinate i = 1 d, and H t x i a random variable given by d H t x =δ γ 1 ϕ t x Dϕ t x d 1δ 1 δ γ 1 ϕ t x Dϕ t x δ γ 1 ϕ t x Dϕ t x Equation 5. follow from the duality relationhip between the derivative and divergence operator and from the fact that d z 1 z 1 d z>y = δ y z Let σ be a ubet of the et of indexe 1 d. Clearly d z 1 z 1 d z i <y i i σ z i >y i i σ = 1 σ δ y z where σ i the cardinality of σ. Hence, for every σ we can write the following alternative formula for the kernel p t y x : 5.3 p t y x = 1 σ E Q 1 ϕ i t x <y i i σ ϕ i t x >y i σ H i t x Conider the following decompoition: p t y xx = where Q σ denote the region Set 5.4 B t = σ 1 Q σ p t y xx Q σ = x x i <y i i σ x i >y i i/ σ t S g λ ϕ t r x M drλ + t σ ϕ t r x b dr

28 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 63 Then, ϕ t x =x+b t.ifx Q σ we ue the expreion for p t y x given in 5.3. In thi way we obtain for x Q σ, p t y x E Q 1 ϕ i t x >y i i σ ϕ i t x <y i σ H i t x E Q 1 x i y i > B i t i σ x i y i < B i t i σ H t x E Q 1 exp x i y i <exp B i t i σ exp x i y i <exp B i t i σ H t x exp d i=1 x i y i t E Q 1 exp B i t B i t H t x t i σ i σ Integrating the above expreion over Q σ, umming over σ, taking the mathematical expectation of the power p and uing the Cauchy Schwarz inequality yield p E p t y xx d p 1 t d/ p 1 d exp Q σ σ 1 i=1 x i y i dx t p up E exp B i t + B i t H t x p x Q σ t i σ i σ d p 1 t p/ up E exp σ 1 x Q σ H t x p p p B i t + B i t t i σ i σ 1/ Note that if M t = i σ B i t + i σ B i t, then the quadratic variation of thi martingale i t M t = g S i λ ϕ t r x g i λ ϕ t r x µ dλr i σ i σ d t + σ k=1 ik ϕ t r x σ ik ϕ t r x dr i σ i σ = d t

29 64 D. NUALART AND F. VIENS becaue σσ = = I 1 G. A a conequence, we obtain p E exp B i t + B i t exp 4p d t i σ i σ and p E p t y xx exp 4p d pd t p/ up H t x p p x We claim that for all p wehave 5.5 up t x H t x p C t d/ for ome contant C>. Thi inequality would complete the proof. In order to how 5.5 we proceed a follow. Set for j = 1 d, jhj 1 H j = δ γ 1 ϕ t x Dϕ t x and H = 1. Then H d = H t x. With thi notation we can write for j = 1 d, H j = jhj 1 j γ 1 ϕ t x δdϕ t x D γ 1 ϕ t x H j 1 Dϕ t x = A j H j 1 + DH j 1 B j where for j = 1 d, 5.6 A j = j j γ 1 ϕ t x δdϕ t x Dγ 1 ϕ t x Dϕ t x and 5.7 j B j = γ 1 ϕ t x Dϕ t x By Hölder inequality for the Sobolev norm in the Wiener pace, for each integer k and each real p 1 we can write for j = 1 d, H j k p C k p A j k p H j 1 k p + B j k p H j 1 k+1 p A a conequence, in order to prove the inequality 5.5 it uffice to how that for j = 1 d, 5.8 A j j d j+1 p c 1 t 1/ and for j = d, 5.9 B j j d j+1 p c 1 t 1/

30 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 65 From formula 5.6 and 5.7 and uing Hölder inequality it follow that, for each integer k, each real p 1 and each index j = 1 d,wehave A j k p C k p γ 1 ϕ t x k+1 p Dϕ t x k+1 p B j k p C k p γ 1 ϕ t x k p Dϕ t x k p Hence, in order to how 5.8 and 5.9, it uffice to check that for each p 1 and k = 1 d+ 1, 5.1 γ 1 ϕ t x k p c k p t 1 and 5.11 Dϕ t x k p c k p t 1/ Property 5.11 follow from the etimate up E D j 1 θ 1 D j k 5.1 θ k ϕ t x p < x <t T θ 1 θ k t 1 j 1 j k d Thi etimate can be eaily checked uing Burkholder inequality and condition 5.1. In fact, condition 5.1 together with the coercivity hypothei H1 imply that G and σ have d + 1 bounded derivative. In order to how the etimate 5.1, taking into account 5.11 and the formula for the derivative of the invere of a matrix, it uffice to how that for any p wehave 5.13 γ 1 ϕ t x p c t 1 and thi follow from 1 p 5.14 det γ ϕt x c t d The proof of inequality 5.14 require ome computation. Set We can write p E det γϕt x = Ɣ = inf v t γ ϕt v =1 x v py p 1 P det γ ϕt x < 1 dy y py p 1 P Ɣ d < 1 dy y We have for any unit vector v, d t v t d γ ϕt x v = v j D i θ i=1 ϕj t x dθ j=1 d t h+ d v j D i θ ϕj t x dθ i=1 j=1

31 66 D. NUALART AND F. VIENS ε d t h t h+ d v j G i j θ dθ i=1 j=1 where h 1 and G i j θ = θ + ε t h+ t h G θ dθ S θ g j λ ϕ t r x D i θ ϕ t r x M drλ σ jk λ ϕ t r x D i θ ϕ t r x b dr We decompoe the previou integral a follow: py p 1 P Ɣ d < 1 dy y 4/ε t = py p 1 P Ɣ d < 1 dy y ε + py p 1 P 4/ε t 4 ε t pd + t h t h+ G θ dθ < 1 dy y 1/d In order to etimate the integral we will take a value of h depending on y, that i, h = 4/ε t y 1/d. Notice that for y 4/ε t we have h 1. With thi value of h we obtain for q>pd, t h+ py p 1 P G θ dθ 1 dy 4/ε t y 1/d t h+ q py p 1+ q/d E G θ dθ dy 4/ε t C py p 1 q/d dy 4/ε t = C t d p q/d Thi complete the proof of the propoition. In a imilar way we can check condition vi p and vii p. Propoition 11. Suppoe that g i d + 3 time continuouly differentiable in the variable x, and the following integrability condition hold: α g λ x 5.15 up x S x 1 α 1 xd α µ dλ < d

32 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 67 for any multiindex α = α 1 α d with α d + 3, where α =α 1 + +α d. Then the tochatic kernel p t y x atifie condition vi p and 5.16 up y for all p 1. E D λ p t x z D λ p t y z µ dλz dx S C p t p Proof. We recall that D λ denote the derivative with repect to the Gauian meaure M. From the expreion 5. for the denity p t y x, it follow eaily that p t y x belong to the Sobolev pace 1 and it derivative for θ t i given by D θ λ p t y x =E Q 1 ϕt x y D θ λ H t x E Q 1 ϕt x y t x where t x i the random variable t x = d j=1 1Dϕt jdθ δ γϕt x x λ ϕ j t x H t x From 5.17 it i not difficult to how that property vi p hold. On the other hand, applying the operator D λ to 4.14 yield D λ p ε t y x f yy = p t y x f y g λ yy + D λ p r t y x v dr y f yy ε + D λ p r t y x f yy dr ε and letting ε tend to zero, we obtain D λ p t y x = div p t y x g λ y Hence, in order to how 5.16 we have to etimate the following quantitie: p A 1 = E x p t x z y p t y z z dx p A = E y p t y z z p A 3 = E y p t x z p t y zz dx p

33 68 D. NUALART AND F. VIENS The mot difficult term i the firt one. We will give ome idea about the etimation of thi term and for the other one can ue a imilar procedure. Uing the integration-by-part formula of Malliavin calculu yield x p t x z =E j Q 1 ϕt z x H j t z where H j t z i the random variable, H j jht t z =δ γ 1 ϕ t z Dϕ t z z Given two ubet σ τ of 1 d, define Q σ τ = { x z x i <z i i σ x i >z i i/ σ z j <y j j τ z j >y j j/ τ } If x z Q σ τ, we will write p t x z p t y z xj yk E Q 1 ϕ i t z <x i i σ ϕ i t z >x i σ H j i t z E Q 1 ϕ i t z >y i i τ ϕ i t z <y i τ H k i t z E Q 1 z i x i < B i t z i σ x i z i <B i t z i σ H j t z E Q 1 y i z i <B i t i τ z i y i < B i t i τ Ht k z exp d i=1 z i x i + y i z i t E Q exp M t z H j t z E Q exp N t z H kt z where B t z ha been defined in 5.4, M t z = 1 B i t z + B i t z t i σ i σ and N t z = 1 B i t z B i t z t i τ i τ Integrating on Q σ τ, umming with repect to the et σ τ, taking the expectation of the power p and uing Hölder inequality, we obtain, a in the proof

34 of Propoition 1, E EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 69 p p t x z p t y z xj yk dz dx [ EH j t z 4p ] EHt k z 4p 1/4 C t p up z It hold that H j t z 4p C t d/ 1/, and, a a conequence we obtain A 1 C t p. Let u how condition v p, uing the backward Kolmogorov equation. Propoition 1. Suppoe that g atifie condition 5.1 and the coercivity condition H1. Then for all p, p up E p t y xx < y up t T Proof. Integrating with repect to x all term in 4.16 and uing the integration-by-part formula yield t p t y xx = 1 + v dr z z p r y zz t = 1 div v dr z p r y zz t = 1 S div g λ z p r y zz M drλ Applying Doob maximal inequality and Burkholder inequality, we obtain for any t 1 >, E up t t 1 p t y xx p t1 p/ C p 1 + E divg λ z p r y zz µ dλr C p 1 + ke t1 S p r y zz dr p/ where k = up z S div g λ z µ dλ p/

35 7 D. NUALART AND F. VIENS p. Set t 1 =E up t t1 p t y xx We have proved that d t1 t 1 C p 1 + k rr Finally, Gronwall lemma allow u to conclude the proof. Notice that we need to aume that up t T E p t y xx p i finite, and thi property follow from Propoition Equivalence of evolution and weak equation. Suppoe that p t y x i the tochatic kernel introduced in Propoition 9. That i, p t y x i the marginal probability denity Q ϕ t x dy /dy of the backward tochatic flow ϕ driven by the vector field v + σb. By 4.15 we know that p t y x i the fundamental olution in the variable x t of the equation u dt x = x u t xt + v dt x x u t x The purpoe of thi ection i to how that the evolution olution to 1.1 obtained in Theorem 7 i a weak olution to the following tochatic partial differential equation: 6.1 du t = x u t xt + v dt x x u t x +F t x u t x W dt x Let u firt introduce the notion of a weak olution. Definition 3. Let u = u t x t T x be an adapted random field uch that E T u d <. Suppoe that the Gauian random field v t x and W t x atify condition 4.1 and.1, repectively. We ay that u i a weak olution to 6.1 if for every φ k d we have t φ x u t xx = φ x u xx + u t x φ xx t 6. u x div φ x v d x x t + u x W d x φ xx Under condition.1 the econd tochatic integral in 6. i well defined. Moreover, under condition 4.1 the firt tochatic integral in 6. i alo well defined a t u t x div φ x v d x x t = u x φ x div g λ x + φ x g λ x M dλx S Theorem 13. Aume the velocity field v atifie the coercivity condition H1, and condition If u i the evolution olution of 1.4, then it i a weak olution of 6.1 in the ene of Definition 3.

36 EVOLUTION EQUATION OF A STOCHASTIC SEMIGROUP 71 Proof. Aume that u i the evolution olution to 1.4. The evolution equation 1.4 can be written in the following way uing the tochatic emigroup T t : 6.3 u t x =T t u x + T t F u a λ x M dλ t S where F uenote the random function F y u y. We are going to ue the fact that if t, then T t atifie the backward Kolmogorov equation 4.8. Notice that thi equation hold for any function f in L. Indeed, uing the fact that the kernel p i in, we have that x T r f x = x p r y x f yy = dy f y x p r y x. A a conequence, we obtain u t x =u x t S t S t S t T r u x v dr x + t T r u xr F x u x a λ x M dλ t T r F u a λ x v dr x M dλ t T r F u a λ xr M dλ Fix a tet function φ in k d. Multiplying the above equation by φ, integrating with repect to x and uing integration by part yield u t φ = u φ d t φ x T r u x v dr xx d t φ x T r u x div v dr xx + + d t S d t S d t S t φ T r u r φ x F x u x a λ x M dλx t t φ x T r F u a λ x v dr x M dλx φ x T r F u a λ x div v dr x M dλx t + φ T r F u a λ r M dλ t S where enote the calar product in L. Now we apply Fubini theorem and we obtain