STOCHASTIC EVOLUTION EQUATIONS WITH RANDOM GENERATORS. By Jorge A. León 1 and David Nualart 2 CINVESTAV-IPN and Universitat de Barcelona

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1 The Annal of Probability 1998, Vol. 6, No. 1, STOCASTIC EVOLUTION EQUATIONS WIT RANDOM GENERATORS By Jorge A. León 1 and David Nualart CINVESTAV-IPN and Univeritat de Barcelona We prove the exitence of a unique mild olution for a tochatic evolution equation on a ilbert pace driven by a cylindrical Wiener proce. The generator of the correponding evolution ytem i uppoed to be random and adapted to the filtration generated by the Wiener proce. The proof i baed on a maximal inequality for the Skorohod integral deduced from the Itô formula for thi anticipating tochatic integral. 1. Introduction. In thi paper we tudy nonlinear tochatic evolution equation of the form (1.1 X t = ξ + A X + F X d + B X dw t T where W i a cylindrical Wiener proce on a ilbert pace U. The olution proce X = X t t T i a continuou and adapted proce taking value in a ilbert pace. The function F ω x and B ω x are predictable procee atifying uitable Lipchitz type condition and taking value in and L U, repectively. We will aume that A ω i a random family of unbounded operator on. A notion of weak olution for (1.1 can be introduced a uual (ee Definition 5.. In the cae where (1.1 i a coercive evolution ytem on a normal triple K K, we can interpret (1.1 a an evolution equation to be olved in K (ee [5] and [1]. In thi cae, the proof of exitence of a unique weak olution for (1.1 follow cloely the idea of Pardoux [11]. When A i a determinitic family of operator, in order to olve Equation (1.1 one look for a mild (or evolution olution, which atifie the evolution equation (1. X t = S t ξ + S t F X d + S t B X dw where S t t T i an evolution ytem determined by A t S t = d/dt S t. We refer to [1] for a baic account of thi theory. In the cae of a random family of operator A t, the correponding evolution ytem S t i alo random and t -meaurable (where t t T Received February Reearch partially upported by CONACYT Grant 35P-E967. Reearch partially upported by DGICYT Grant PB93-5. AMS 1991 ubject claification. 615, 67. Key word and phrae. Stochatic evolution equation, tochatic anticipating calculu, Skorohod integral. 149

2 15 J. A. LEÓN AND D. NUALART i the natural family of σ-field determined by W. A a conequence, the proce S t B X i not -meaurable, and the tochatic integral appearing in (1. i anticipative. That i, although both the olution proce X t and the random family of operator A t are adapted, the aociated tochatic evolution equation involve an anticipating integral. It i well known that a mild olution of (1., where the anticipating integral i interpreted a a Skorohod integral, i not a weak olution of (1.1 (ee [7] becaue a complementary term appear. We how in Section 5 (ee Propoition 5.3 that a mild olution of (1. where the tochatic integral i a forward integral i alo a weak olution to (1.1. Roughly peaking, the forward integral i defined a the limit (in probability of Riemann um defined taking the value of the proce on the left point of each interval. In the cae of real-valued procee, thi type of integral wa tudied, among other author, by Ruo and Valloi in [13]. The main difficulty in handling thi tochatic integral i to obtain uitable etimate for the L p -norm of the integral. One way to do thi, in the anticipating cae, conit in expreing the forward integral a the um of the Skorohod integral plu a complementary term. In Section 4 we obtain an expreion relating the forward and the Skorohod integral (Propoition 4. and we deduce an etimate for the L p -norm of the upremum of an indefinite forward integral (Theorem 4.4. Thi theorem i one of the main reult of thi paper and contitute the fundamental tool for olving the tochatic evolution equation (1.. The Skorohod integral i an extenion of the Itô integral to the cae of anticipating integrand, and it wa introduced by Skorohod in [14]. It turn out that thi generalization of the Itô integral coincide with the adjoint of the derivative operator on the Wiener pace. A a conequence, one can apply the technique of the Malliavin calculu (ee [8] in order to contruct a tochatic calculu for the Skorohod integral. Thi ha been done by Nualart and Pardoux in [1], among other. The Skorohod integral of ilbert-valued procee with repect to a cylindrical Wiener proce ha been tudied by Grorud and Pardoux in [3]. In Section we preent the baic fact on the Malliavin calculu with repect to a cylindrical Wiener proce. We need to introduce random variable with value in the pace of linear operator L G, where and G are real and eparable ilbert pace, and the correponding Sobolev pace D 1 L G are more general than the pace of ilbert Schmidt operator D 1 L G conidered in [3]. The baic etimate for the L p -norm of a Skorohod integral (that i ued in Section 4 in order to control the L p -norm of the forward integral i obtained in Section 3. We need to etimate a Skorohod integral of the form t S t dw, where S t t i an t -meaurable random evolution ytem on a ilbert pace and = T i an L U -valued adapted proce. We prove that ( t p T (1.3 E up S t dw C E p S d t T

3 STOCASTIC EVOLUTION EQUATIONS 151 auming that S t i twice-differentiable in the ene of the Malliavin calculu. The contant C depend on p, T and on the random evolution ytem S t. Thi etimate follow from the Itô formula for the Skorohod integral, uing ome idea introduced by u and Nualart [4]. The emigroup property of the ytem S t allow howing thi etimate uing only two derivative of S t. Inequality (1.3 plu the decompoition of the forward tochatic integral obtained in Section 4 allow u to deduce an etimate imilar to (1.3 for the forward integral (ee Theorem 4.4. Uing thi, we prove in Section 5 a reult on the exitence and uniquene of a mild olution to (1. (Theorem 5.4. Finally, Section 6 contain an example that atifie the aumption of our reult. Namely, a random evolution ytem generated by a family of random econd order differential operator.. Preliminarie. In thi ection we preent ome baic element of the tochatic calculu of variation with repect to a cylindrical Wiener proce. For a more detailed account on thi ubject we refer to [3]. Let U be a real and eparable ilbert pace. Suppoe that W i a cylindrical Wiener proce over U defined on a complete probability pace P. That i, W = W t h h U t T i a zero-mean Gauian family uch that E W t h 1 W h = t h 1 h U for all h 1, h U and t T. We will alo aume that the σ-field i generated by W. If u L T U we et W u = T j=1 u e j U dw e j, where e j j 1 i a complete orthonormal ytem on U. We will alo ue the notation W u = T u t dw t U. If U 1 and U are two real and eparable ilbert pace we will denote by U 1 U it tenor product which i iometric to the pace L U U 1 of ilbert Schmidt operator from U to U 1. Let K be a real and eparable ilbert pace. For any p 1 we can introduce the Sobolev pace D 1 p K of K-valued random variable in the following way. If F i a mooth K-valued random variable of the form m (.1 F = f j W u 1 W u m b j j=1 where u i L T U, b j K and f j C b Rm (f i an infinitely differentiable function uch that f i bounded together with all it partial derivative, then the derivative of F i defined a m m f j DF = W u x 1 W u m b j u i i j=1 So DF i a mooth random variable with value in L T L U K. Then D 1 p K i the completion of the cla of mooth K-valued random variable,

4 15 J. A. LEÓN AND D. NUALART denoted by K, with repect to the norm ( T p/ F p 1 p = E F p K + E D t F S dt For each p 1 the operator D i cloable from K L p K into the pace L p L T L U K and for F D 1 p K we have that DF L p L T L U K. More generally, for any natural n 1, the Sobolev pace D n p K i defined a the completion of K by the norm n ( F p n p = E F p K + p/ E D t1 D tj F T j L U j K dt 1 dt j j=1 In particular, given two real and eparable ilbert pace and G we can conider K = L G, and in thi cae, for any F in the pace D 1 p L G we have that DF L p L T L L U G becaue L U L G = L L U G. We want to introduce Sobolev pace of random variable with value in the pace L G of linear bounded operator from in G. Taking into account that L G i a noneparable Banach pace, we cannot ue the preceding contruction. For p 1, L p L G denote the pace of all function F L G uch that: (a For every h, F h i a G-valued integrable random variable and there exit an element EF L G uch that E F h = EF h for all h. That i, F i Bochner integrable (ee [1], page 4. (b F p L G dp <. For more detail on thi definition ee [1]. The following definition provide a natural way to define derivative of L G -valued random variable. In order to implify the expoition, we will retrict ourelve to the cae p =. Thi will be ufficient for the ubequent application of thee notion. Definition.1. Let F L L G. We ay that F belong to the Sobolev pace D 1 L G if the following condition hold: (a For every h, F h belong to D 1 G. (b There exit an element DF L T L L U G uch that for every h we have (. D t F h = D t F h for almot all t ω T. Remark. D 1 L G D 1 L G and for any F in D 1 L G we have DF belong to the pace L T L L U G. In general we have that the incluion D 1 L G D 1 L G i trict. For intance, if G = and F i the identity operator I on,

5 STOCASTIC EVOLUTION EQUATIONS 153 then I D 1 L becaue I i not a ilbert Schmidt operator, but I h =h D 1 for any h and DI =. We will make ue of the following technical lemma concerning the derivative operator. We will denote by G J real and eparable ilbert pace. Lemma.. If ϕ L J and F D 1 L G then we have Fϕ D 1 L J G and D Fϕ = DF ϕ. Proof. Let j k k 1 be a complete orthonormal ytem on J. Clearly Fϕ i a random element with value in L J G and Fϕ S F L G ϕ S which implie that Fϕ L L J G. On the other hand, for each k 1wehave Fϕ j k D 1 G and D Fϕ j k = DF ϕ j k. ence T E D Fϕ j k L U G d k=1 = E E T T D F ϕ j k L U G d k=1 which implie the reult. D F L L U G d ϕ L J < Lemma.3. Conider a mooth L J -valued random element ϕ and let F D 1 L G. Then Fϕ D 1 L J G, and (.3 D Fϕ = DF ϕ + F Dϕ Proof. Without lo of generality, we can aume that ϕ = Rb where b L J and R i a real-valued mooth random variable of the form R = f W u 1 W u m with u i L T U and f C b Rm. Clearly, the compoition Fϕ belong to L L J G. Let u firt prove that the right-hand ide of (.3 belong to L T L J L U G. We have that Dϕ i a bounded random element with value in L T L J L U given by Dϕ = b DR (i.e., for each j J, Dϕ j =b j DR. A a conequence, where Fb L L J G, and E T F Dϕ = Fb DR T F D ϕ L U L J G d = E D R U Fb L J G d C ϕ E Fb L J G <

6 154 J. A. LEÓN AND D. NUALART where C ϕ i a contant. On the other hand, DF ϕ =R DF b, and E T D F ϕ L J L U G d R b L J E T D F L L U G d < For any j J we have that Fϕ j =R Fb j belong to D 1 G and by Lemma. we can write (.4 D Fϕ j = Fb j DR + R DF b j ence, it uffice to how that the right-hand ide of (.3 applied to j coincide with the right-hand ide of (.4, and thi i true becaue DF ϕ j =R DF b j =R DF b j and F Dϕ j = Fb j DR. Lemma.4. Let A D 1 L G and F D 1. Suppoe that A L G M and F M for ome contant M>. Then AF D 1 G and (.5 D AF = DA F + A DF Proof. We can find a equence F n of -valued mooth random variable uch that F n M + 1, F n converge to F in L and DF n converge to DF in L T L U. Clearly AF L G, and AF n converge to AF in L G. By Lemma.3 (with J = R we deduce that AF n D 1 G, and (.6 D AF n = DA F n + A DF n Finally, from our hypothee we get that the right-hand ide of (.6 converge to that of (.5 in L T L U G a n tend to infinity, which complete the proof. Lemma.5. Let A D 1 L G and B D 1 L J. Suppoe that A L G M and B L J M for ome contant M>. Then AB D 1 L J G, and D AB = DA B + A DB Proof. Clearly AB L L J G. Fixj J. We know that Bj D 1 and Bj M j J. By Lemma.4 we have AB j D 1 G and (.7 D AB j = DA Bj +A DB j Finally notice that DA B + A DB i an element of the pace L T L J L U G and A DB + DA B j coincide with the right-hand ide of (.7.

7 STOCASTIC EVOLUTION EQUATIONS 155 For any ubinterval I T we denote by I the σ-field generated by the family of random variable W u, upp u I. Lemma.6. Let A D 1 L G, and uppoe that A i I -meaurable for ome ubinterval I T. Then D t A = for almot all t ω I c. Proof. Let h. Then, by hypothei, A h i an I -meaurable random element belonging to D 1 G. Thi implie that, for every ϕ L T uch that upp ϕ I c,= T ϕ D A h d. Thu, the fact that i eparable and D A h = DA h give the reult. In the equel e i i 1 will denote a complete orthonormal ytem on U. We will write D e F h = DF h e for any F D 1 L G, and for each h, e U. Notice that D e F belong to L T L G. (.8 Lemma.7. Let A D 1 L G uch that T E D e i A L G d < Then, the adjoint of A A, belong to D 1 L G and D e A = D e A for each e U. Proof. Clearly A belong to L L G. Let F D 1 G, g G and h. Then, it i not difficult to ee that F g G h D 1 and D F g G h =h DF g. ence A g h h = g A h G h D 1 and (.9 D A g h h =h D A h g = DA g h h Thi implie that A g belong to D 1, and D A g = DA g. Finally, we have to how that DA belong to L T L G L U. Thi follow from condition (.8: T E D A L G L U E T D e i A L G d < Thu the proof i complete. A in [3] we will denote by δ the adjoint of the derivative operator D acting on D 1. That i, the domain of δ i the pace of procee u in L T L U uch that T E D t F u t S dt c u F L for any mooth -valued random variable F. Then δ u i the element of L determined by the duality relationhip E T D t F u t S dt = E F δ u

8 156 J. A. LEÓN AND D. NUALART for any F D 1. The operator δ i alo called the -Skorohod integral. It i an extenion of the Itô tochatic integral of -valued adapted procee in the ene that L a T ; L U Dom δ, where L a T L U denote the pace of adapted procee in L T L U. We will make ue of the following property of the Skorohod integral. Propoition.8. Let A D 1 L G and let B be an L U -valued proce which belong to the domain of δ. Suppoe the following condition hold: (i AB L T L U G ; (ii Aδ B L G ; (iii E T De i A L G d < and B L 4 T L U. Then AB Dom δ G and (.1 δ G AB =Aδ B T D e i A B e i d Proof. Note firt that by condition (iii the right-hand ide of (.1 belong to L G. Let F be a mooth G-valued random variable. We can write T T E AB D F L U G d = E AB e i D e i F G d T = E T = E E B e i A D e i F d B e i D e i A F d T B e i D e i A F d where A D 1 L G i the adjoint of A, and we have ued Lemma.3 in order to compute D A F. Notice that, by Lemma.7, D e i A = D e i A. ence, we obtain E T AB D F L U G d = E δ B A F E T = E R F G where R denote the right-hand ide of (.1. D e i A B e i F G d

9 STOCASTIC EVOLUTION EQUATIONS 157 Remark. Condition (iii of Propoition.8 can be replaced by the following: (iii D e i A L G M< for all T and B L T L U for ome contant M>. The Sobolev pace D k L G for any integer k 1 are defined a in Definition.1, replacing U by U k and D by D k in (.. If F D k L G, and p, we define F p k ( k p = E F p L G + p/ E D j T 1 j j F p L L U j G d 1 d j j=1 Let u recall Itô formula for anticipating ilbert-valued procee (ee [3], Propoition 4.1. We will ue the notation L k p J =L p T D k p J for any p 1, k a poitive integer and J a real and eparable ilbert pace. For any B Dom δ we will write δ B = T B dw. Propoition.9. Let C and let X = X t t T be the tochatic proce defined by X t = X + where we have the following: (i X D 1 ; (ii A L 1 ; (iii B L 4 L U. Then with X t = X X t = D t X + A d + X A d + B dw X X B L U d D t A d + X B dw D t B dw + B t Remark. The hypothee of Propoition.9 are lightly more general than thoe in Propoition 4.1 of [3]. The validity of the Itô formula under thee more general aumption follow from the finite-dimenional Itô formula etablihed in [9] under thee kind of aumption. We will make ue of the following Fubini-type theorem for the Skorohod integral whoe proof i a traightforward conequence of the duality relationhip. Lemma.1. Let u t x be an L U -valued random field parameterized by t x T G, where G i a bounded d-dimenional rectangle. Sup-

10 158 J. A. LEÓN AND D. NUALART poe that u L T G, and for almot all x G the tochatic proce u x belong to the domain of δ. Suppoe alo that E G δ u x dx <. Then G u t x dx t T belong to the domain of δ and T ( ( T u t x dx dw t = u t x dw t dx G 3. An etimate for the Skorohod integral. Let U be real and eparable ilbert pace. Let W be a cylindrical Wiener proce over U on the time interval T. We will make ue of the notation = t T t. Definition 3.1. A random evolution ytem i a random family of operator S t t T on verifying the following propertie: (i S L i trongly meaurable; (ii S t i trongly t -meaurable for each t ; (iii For each ω, S t t i an evolution ytem in the following ene: (a S =I and S t r =S t S r for any r t T. (b For all h, t S t h i continuou from into. Let u introduce the following hypothee on a given random evolution ytem. (1 For each t, S t D L, and S t p p d < for all p. ( There i a verion of D r S t uch that for all ω and h, the limit D S t h =lim D S t ε h ε exit in L U and D S t belong to D1 L L U. (3 There i a contant M> uch that the following etimate hold for all t r: (3a S t L M; (3b D S t r L L U M; (3c D e i r D S t L L U M. Remark. Fix t> ε>r, ε>. From property (a of a random evolution ytem we have S t r =S t ε S ε r Suppoe that the random evolution ytem S t atifie the hypothee (1, ( and (3. Applying Lemma.5 and.6 yield D S t r =D S t ε S ε r Now letting ε and uing property (b in the definition of a random evolution ytem, ( and (3, we obtain D S t r =D S t S r G

11 STOCASTIC EVOLUTION EQUATIONS 159 Indeed, for any h we have D S t ε S ε r h D S t S r h S D S t ε S ε r h S r h S + D S t ε D S t S r h S D S t ε L L U S ε r S r h + D S t ε D S t S r h S and thi converge to zero a ε tend to zero due to hypothee ( and (3. Let u now prove the following theorem. Theorem 3.. Fix p and α 1. Let = t t T be an L U -valued adapted proce uch that E T p S d <. Let S t be a random evolution ytem atifying the hypothee (1, ( and (3. Then the L U -valued proce t α S t I t, T belong to the domain of δ for almot all t T, and we have t p (3.1 E t α S t dw C t α E p S d for ome contant C> which depend on T, p, α and on the evolution ytem S t. Proof. Let u denote by the cla of L U -valued elementary adapted procee of the form (3. = f ik W u i 1 W ui n b ki ti t i+1 k=1 where f ik C b Rn b k L U <t 1 < <t n+1 <Tand upp u i j t i. Let be an L U -valued adapted proce uch that E T p S d <. We can find a equence n of elementary adapted procee in the cla atifying T lim E n n p S d = Thi implie that T ( lim E t α n n p S d dt = By chooing a ubequence we have that for all t T out of a et of zero Lebegue meaure, lim E t α n i i p S d = ence, we can aume that i of the form (3..

12 16 J. A. LEÓN AND D. NUALART We are going to apply Itô formula to the fuction F x = x p on. Recall that F x =p x p x and F x =p p x p 4 x x + p x p I Fix t >t 1 in T, and define B = t α S t 1 I t1 From hypothei (1 it follow that B L q L U, for each q. A a conequence, we can apply Itô formula (Propoition.9 to the proce X t = B dw, and to the function F x = x p. In thi way we obtain, for each t t 1, X t p = p X p X B dw (3.3 ( F X B + D B r dw r B S d + 1 We claim that the Skorohod integral appearing in (3.3, that can be written a p X p B X dw, ha zero expectation. Thi might not be true becaue thi Skorohod integral i defined by localization. Neverthele, our aumption imply that the proce X p B X belong to L 1 U Dom δ. In fact, we have, by [3], Propoition 4.1, E T X p B X U d T C 1 E X p 1 d ( T T p 1 C (1 + E D θ B L U U dθ d < due to hypothee (1 and (. Notice that hypothei (1 implie that E X p d < for any p. On the other hand we have, T E T T D θ X p B X U U dθ d [( C E T ( + E T 1/ ( X 4 p d E T 1/ ( X 4 p 1 d E ( T T ( T 1/ D θ X L U dθ d 1/ ] D θ B L U U dθ d < where we ue the fact that X L 1 4 (ee [15], Theorem.1. Thu, we have proved that X p B X belong to L 1 U.

13 STOCASTIC EVOLUTION EQUATIONS 161 Notice that F x L p p 1 x p. ence, taking expectation in (3.3 yield E X t p p p 1 ( E X p B S + B S D B r dw r d S Uing the inequality a b a + b we obtain E X t p p p 1 E X p B S d p p 1 + E X p D B r dw r d S Now we ubtitute B by it definition and we ue the adaptability of and Lemma.3 and.6 to get E X t p p p 1 E X p t α S t 1 S d (3.4 + p p 1 E X p t r α D S t 1 r r dw r Applying ölder inequality to the expectation in the right-hand ide of (3.4 yield E X t p p p 1 E X p 1 /p t α E S t 1 p S /p d = + p p 1 E X p 1 /p ( E E X p 1 /p A d Then the lemma proved in [16] implie that t r α D S t 1 r r dw r p S S d /p d that i, E X t p ( p A d p/ E X t p { p 1 (3.5 + p 1 t α E S t 1 p S /p d ( E t r α D S t 1 r r dw r p S /p d} p/

14 16 J. A. LEÓN AND D. NUALART [ ( p/ p/ 1 p 1 p/ p/ t α E S t 1 p S /p d ( p ] + t p/ 1 E t r α D S t 1 r r dw r d M p p 1 p 1 p/ t α E p S d + p/ 1 p 1 p/ t p/ 1 p E t r α D S t 1 r r dw r Uing the remark at the beginning of thi ection, Propoition.8 and hypothei (3, we can write t r α D S t 1 r r dw r S = t r α D S t 1 S r r dw r S = D S t 1 t r α S r r dw r t r α D e i r D S t 1 S r r e i dr (3.6 M + M t r α S r r dw r S d t r α D e i r D S t 1 L L U S r r e i dr t r α S r r dw r + M t r α r S dr Subtituting (3.6 into (3.5 yield { E X t p C M T p t α E p S d p (3.7 + E t r α S r r dw r d ( p } + E t r α r S dr d S S

15 STOCASTIC EVOLUTION EQUATIONS 163 Applying ölder inequality [for the integral with repect to t r α dr] and Fubini theorem to the lat ummand in (3.7, and taking t 1 = t we get t p E t α S t dw { C M p T α t α E p S d + t α E p S d p } + E t r α S r r dw r d By Gronwall lemma we deduce t p (3.8 E t α S t dw C t α E p S d where C i a contant depending on T M, p and α. Fix t T, and take t = t+1/n. From (3.8 for t = t+1/n and letting n tend to infinity we deduce that t α S t I t, T belong to Dom δ and (3.1 hold. The proof of the theorem i complete. Let u introduce the following hypothei on a random evolution ytem S t verifying (1 and (. (3 Condition (3a and (3c hold, and moreover, we have (3b D e i r S t L M, for all t r and for ome contant M>. Notice that (3b i tronger than (3b, and it implie that D S t e i L M for all t. The following theorem provide an etimate of the L p norm of the maximum of a Skorohod integral, and it contitute the main reult of thi ection. Theorem 3.3. Fix p>. Let = t t T be an L U -valued adapted proce uch that E T p S d <. Let S t be a random evolution ytem atifying hypothee (1, ( and (3. Then the L U valued proce S t I t, T belong to Dom δ and we have ( t p T E up S t dw CE p S d t T for ome contant C> which depend on T, p and on the evolution ytem S t.

16 164 J. A. LEÓN AND D. NUALART Proof. We will make ue of the factorization method in order to handle the upremum in t. Fixα 1/p 1/. We can write (3.9 S t = C α S t r t r α 1 S r r α dr where C α = in πα/π. By Theorem 3. we know that for all r T a.e., the proce S r r α I r belong to Dom δ. Then applying Propoition.8 and uing hypothei (3 we obtain for almot all r t, (3.1 where r S t r t r α 1 S r r α dw = S t r t r α 1 Y r r t r α 1 D e i S t r S r r α e i d Y r = r S r r α dw By Fubini theorem for anticipating tochatic integral (ee Lemma.1 and uing (3.9 we obtain (3.11 S t dw ( = C α S t r t r α 1 S r r α dr dw ( r = C α S t r t r α 1 S r r α dw dr Subtituting (3.1 into (3.11 yield S t dw = C α t r α 1 S t r Y r dr C α t r α 1 (3.1 ( r D e i S t r S r r α e i d dr Applying ölder inequality to the right-hand ide of (3.1 and uing hypothei (3b yield t up S t dw t T M π M π up t T ( p 1 αp 1 T t r α 1 Y r dr + M S d 1 1/p ( T 1/p T T α 1/p Y r p dr + M S d

17 STOCASTIC EVOLUTION EQUATIONS 165 and hence, (3.13 ( E up t T C T p α (E p S t dw T T Y r p dr + E p S d From Theorem 3. we deduce (3.14 E Y t p C t α E p S d Finally, ubtituting (3.14 into (3.13 and uing Fubini theorem we deduce the deired etimation. 4. The forward integral. Let U and be two real and eparable ilbert pace and let W be a cylindrical Wiener proce over U on the time interval T. We will denote by e i i 1 and h i i 1 complete orthonormal ytem on U and, repectively. Definition 4.1. Let Y T L U be a meaurable proce uch that Y u L 1 T a.. for each u U. We ay that Y belong to Dom δ if T Y n = n Y e i W +1/n T e i W e i d converge in probability a n tend to infinity. The limit of the equence Y n i denoted by T Y dw and i called the forward integral of Y with repect to W. The forward integral ha been tudied by Ruo and Valloi in [13] in the cae of real-valued procee. From Definition 4.1 it follow that for any proce Y belonging to Dom δ and for any A uch that Y t ω =, dt dpa.e. on T A we have T Y dw = a.. on A The next propoition etablihe the relationhip between the forward and the Shorohod integral of a proce of the form S t I t T where S t i a random evolution ytem and i an adapted proce. Propoition 4.. Let = t t T be an L U -valued adapted proce uch that E T S d <. Let S t be a random evolution ytem atifying hypothee (1, ( and (3. Then for each t T,

18 166 J. A. LEÓN AND D. NUALART S t I t, T belong to Dom δ and (4.1 S t r r dw r = δ S t 1 t + D r S t r e i r e i dr In order to prove (4.1 we firt tate the following. Lemma 4.3. Let and S t be a in Propoition 4.. Then for each t T, and each poitive integer n 1, ( n t 1 t+1/n T S t e i e i d Dom δ 1/n + and (4. Proof. t+1/n T = ( r t S t e i e i d dw r r 1/n + S t δ 1 +1/n e i e i d t+1/n T r t By (3 and Propoition.8 we have S t δ 1 +1/n e i e i d = = + + r 1/n + D e i r S t e i d dr ( +1/n S t e i e i dw r d ( j=1 +1/n D e j r S t e i e i e j U dr d ( +1/n S t e i e i dw r d +1/n D e i r S t e i dr d Notice that S t D 1 L and I +1/n e i e i atify the aumption (i, (ii and (iii of Propoition.8. Indeed, we have for all r t, j=1 D e j r S t L M

19 STOCASTIC EVOLUTION EQUATIONS 167 due to (3b. Finally, Fubini theorem for the Skorohod integral (ee Lemma.1 allow u to conclude the proof of the lemma. Proof of Propoition 4.. Fix t T. We only need to prove that n = n n t S t e i W +1/n e i W e i d converge in probability a n tend to infinity to the right-hand ide of (4.1. Actually we will how the convergence in L. Uing (4. we have t+1/n T ( r t n = n S t e i e i d dw r r 1/n + t+1/n T r t + n D e i r S t e i d dr r 1/n + Applying Theorem 3. with α = and p = yield E n n t+1/n T ( r t t S t e i e i d dw r S t r r dw r r 1/n + t+1/n T ( ( n E n S t e i e i d dw r t r 1/n + r ( n + CE n S r e i e i d r dr r 1/n + S +1/n ( n E n S t e i e i d dr t r 1/n + S r + 4CE n S r d r dr r 1/n + S r ( + 4CE n S r e i e i d dr r 1/n + i=n+1 S Thi expreion can be etimated by t+1/n T T r M E S d + 4C n E S r r t r 1/n + S dr d + 4CM E T i=n+1 e i d = a 1 + a + a 3 Then term a 1 and a 3 clearly converge to zero a n tend to infinity, uniformly with repect to t T. The convergence to zero of a a n tend to infinity follow from the etimate a 8C M + 1 T E S d which allow u to approximate by a proce in C T L L U.

20 168 J. A. LEÓN AND D. NUALART In a imilar way we can write n n t+1/n T r t D e i r S t e i d dr r 1/n + D r S t r e i r e i dr n n + t+1/n T t i=n+1 + = 1 n + n + 3 n D e i r S t e i d dr r 1/n + D r S t r e i r e i dr r { n D e i r S t e i r 1/n D r S t r e i r e i } d dr Clearly n 1 and n tend to zero in L a n tend to infinity. The term n 3 can be etimated a follow: n n 3 r n D e i r S t e i r e i d dr r 1/n r [ + n D e i r S t D r S t r e i ] r e i d dr r 1/n ( r 1/ n D e i r S t L d dr r 1/n ( r 1/ n e i r e i d dr r 1/n r + n D r S t r e i S r I r e i d dr r 1/n M ( r 1/ t n r S d dr r 1/n + M ( r 1/ t n S r I r e i d dr r 1/n and thi converge to zero uniformly with repect to t in L a n tend to infinity.

21 STOCASTIC EVOLUTION EQUATIONS 169 Combining Theorem 3.3 and the expreion given in Propoition 4. for the forward integral, we deduce the following maximal inequality for the forward integral. Theorem 4.4. Fix p>. Let = t t T be an L U -valued adapted proce uch that E T p S d <. Let S t be a random evolution ytem atifying hypothee (1, ( and (3. Then the L U valued proce S t I t T belong to Dom δ and we have ( p T E up S t r r dw r C S p T E p S d t T for ome contant C S p T > depending on T, p and the random evolution ytem S t. Proof. By Propoition 4. we know that the L U -valued tochatic proce S t I t T belong to Dom δ and we have (4.3 S t r r dw r = S t r r dw r + D r S t r e i r e i dr Then the reult follow from Theorem 3.3 and hypothei (3b. A a conequence of Theorem 4.4, we have the following continuity reult. Corollary 4.5. Let and S t be a in Theorem 4.4. Then the -valued proce S t dw, t T ha a continuou modification. Proof. Fix α 1/p 1/ and et Y r = r S r r α dw We know that the proce Y r i well defined for almot all r in T. From Propoition 4. applied to the proce r α r we deduce that (4.4 where r Y r = Y r + Y r = D S r e i r α e i d r S r r α dw On the other hand, ubtituting the relation D e i S t r S r = D S t e i S t r D S r e i

22 17 J. A. LEÓN AND D. NUALART into (3.1 yield (4.5 S t dw = C α t r α 1 S t r Y r dr D S t e i e i d + C α t r α 1 S t r ( r D S r e i r α e i d dr = C α t r α 1 S t r Y r dr D S t e i e i d ence, from Propoition 4. we deduce (4.6 S t dw = C α t r α 1 S t r Y r dr By (4.6, we only need to how that the right-hand ide of thi equation i continuou in t. Fix t <t T. Then our hypothee on the evolution ytem S t and the dominated convergence theorem imply that t r α 1 S t r Y r dr t r α 1 S t r Y r dr t r α 1 S t r Y r dr t + S t t t I t r α 1 S t r Y r dr + S t t t t r α 1 t r α 1 S t r Y r dr converge to zero a t t. In a imilar way we how that the above expreion converge to zero a t t. 5. Stochatic evolution equation with a random evolution ytem. In thi ection we will tudy nonlinear tochatic equation of the form (5.1 X t = ξ + A X + F X d + B X dw t T where ξ i an -valued -meaurable random variable and W i a cylindrical Wiener proce over the ilbert pace U on the time interval T. We will aume the following condition on the coefficient A F and B.

23 STOCASTIC EVOLUTION EQUATIONS 171 (A.1 The mapping F T i T -meaurable, where T denote the predictable σ-field of T, F t x F t y C x y F t x C 1 + x for ome contant C> and for all x y. (A. The mapping B T L U i T -meaurable, B t x B t y S C x y B t x S C 1 + x for ome contant C> and for all x y. (A.3 A ω T ω i a random family of unbounded operator on uch that Dom A where i a dene ubet of. We aume that A y L T for all y, and there exit a random evolution ytem S t atifying hypothee (1, ( and (3 uch that S t A t y = d dt S t y for all y Definition 5.1. We ay that an adapted and continuou -valued proce X = X t t T uch that E up t T X t p < for ome p>ia mild olution to (5.1 if (5. X t = S t ξ + S t F X d + S t B X dw for each t T, where dw denote the forward integral (ee Section 4. Definition 5.. An adapted and continuou -valued proce X = X t t T uch that E up t T X t p < for ome p>i a weak olution to (5.1 if for each y and t T we have X t y = ξ y + + A y X d y F X d + B X y dw U Propoition 5.3. Under aumption (A.1, (A. and (A.3, any mild olution to (5.1 i a weak olution. Proof. For each n 1 we define X n t = S t ξ + S t F X d + n S t B X e i W +1/n e i W e i d

24 17 J. A. LEÓN AND D. NUALART Notice that X n t = S t Xn + S t r F r X r dr + n S t r B r X r e i W r+1/n e i W r e i dr We know, by Aumption (A.3, that for all y, x we have σ S r σ A r y x dr = S t σ y x y x ence, for all y, we obtain Ɣ n = n S r σ A r y (5.3 = n σ B σ X σ e i W σ+ 1 e i W n σ e i dr dσ S t σ y y B σ X σ e i W σ+1/n e i W σ e i dσ = X n t y S t X n y S t r F r X r dr y n y B r X r e i W r+1/n e i W r e i dr On the other hand, applying Fubini theorem we have r Ɣ n = n S r σ A r y (5.4 = n = = = B σ X σ e i W σ+1/n e i W σ e i dσ dr r A r y S r σ B σ X σ e i W σ+1/n e i W σ e i dσ r A r y X n r S r Xn S r σ F σ X σ dσ dr A r y X n r dr A r y S r X n dr r A r y S r σ F σ X σ dσ dr A r y X n r dr S t X n y + X n y σ A r y S r σ F σ X σ dr dσ dr

25 = STOCASTIC EVOLUTION EQUATIONS 173 A r y X n r dr S t X n y + X n y y S t σ F σ X σ dσ + Comparing (5.3 and (5.4 yield y F σ X σ dσ (5.5 X n t y = X n y + + n A r y X n r dr + y F r X r dr y B r X r e i W r+1/n e i W r e i dr We have that, by Propoition 4. with S I, the lat ummand in (5.5 converge in L a n tend to infinity to B r X r dw r y = B r X r y dw r U Then it uffice to how that up t T E X t X n t converge to zero a n tend to infinity. Thi i a conequence of the etimate ued in the proof of Propoition 4.. Theorem 5.4. Let S t be a random evolution ytem atifying hypothee (1, ( and (3 and let F and B atify (A.1 and (A., repectively. Then (5.1 ha a unique mild olution. Proof of uniquene. Aume that X and Y are two mild olution to (5.1. Then, for arbitrary t T and p> uch that ( ( E X r p + E Y r p < we have up r T up r T X t Y t p = S t r F r X r F r Y r dr p + S t r B r X r B r Y r dw r p p 1 S t r F r X r F r Y r dr + p 1 S t r B r X r B r Y r dw r p 1 M p C p T p 1 X r Y r p dr p + p 1 up S r I t r B r X r B r Y r dw r T p

26 174 J. A. LEÓN AND D. NUALART ence, from Theorem 4.4, we obtain E X t Y t p p 1 M p C p T p 1 E X r Y r p dr + p 1 C S p T E B r X r B r Y r p S dr Therefore, uing ypothei (A., we get E X t Y t p p 1 M p C p T p 1 E X r Y r p dr + p 1 C S p T C p E X r Y r p dr = p 1 C p M p T p 1 + C S p T E X r Y r p dr which, together with Gronwall lemma, implie E X t Y t p =, for arbitrary t T, and the proof of uniquene i complete. Proof of exitence. The proof of exitence i imilar to that for a determinitic evolution ytem. We begin an iteration procedure with X t = S t ξ and let u define, for n 1 and t T, X n t = S t ξ + S t r F r X n 1 r dr (5.6 + S t r B r X n 1 r dw r Uing induction on n, it i eay to prove that aumption (A.1 and (A., Theorem 4.4 and Corollary 4.5 imply that X n i an adapted and continuou -valued proce uch that up t T E X n t p < Computation imilar to thoe in the firt tep of thi proof and Theorem 4.4 yield (5.7 n= E up X n+1 t t T X n t p < Therefore, from the Borel-Cantelli lemma, the equence X n n N i uniformly convergent in T, for almot all ω. Denote the limit by X t. Since X i the uniform limit of a equence of adapted and continuou -valued procee, it i alo adapted and continuou. The etimate (5.7 implie that X belong to L p T and that X n n N alo converge to X in L p T. Finally, from (5.6 and Theorem 4.4, it i eay to how that X i a mild olution of (5.1 and o the proof i complete. Remark. The exitence of a mild olution till hold if we uppoe that condition (A.1, (A. and (A.3 are true locally. That i, we aume that for all n, (A.1 and (A. are atified for any x y with x n and y n,

27 STOCASTIC EVOLUTION EQUATIONS 175 and with ome contant C n, and on the other hand, we alo aume that the random evolution ytem S t atifie (1, ( and (3 locally. Thi mean that there exit a equence k k N and a equence S k k N uch that k, and for each k, S = S k on k a.., and S k t i a random evolution ytem atifying condition (1, ( and (3. 6. Stochatic partial differential equation with random generator. Let O be a domain in R n and conider the ilbert pace = L O. A in the previou ection, W will be a cylindrical Wiener proce over a ilbert pace U on the time interval T. In thi ection we will firt provide ufficient condition for a random operator on L O given by a random kernel f x y ω to be in D 1 L. Lemma 6.1. Let f O O R + be a meaurable function uch that the following hold: (i f x L O for all x O; (ii up x O O f x y dy < and up x O O f y x dy <. Then the mapping L O L O given by g x = f x y g y dy i a bounded linear operator uch that ( 1/ ( L f x y dy up x O O O up y O O f x y dx 1/ Proof. Thi lemma i an immediate conequence of Fubini theorem and Schwarz inequality: g x dx = f x y g y dy dx O O O ( ( f x y dy f x y g y dy dx O O O ( ( f x y dy f x y dx g L O up x O O up y O Lemma 6.. Let f O O R + be a random meaurable function verifying the following condition: (i f x L O for every x O a..; (ii there exit two nonnegative random variable M 1, M uch that f z y dy M 1 a.., up z O up z O O O f y z dy M and E M p 1 <, E Mp < for ome p. O a..

28 176 J. A. LEÓN AND D. NUALART Then the random operator ω on defined by ω g x = f x y ω g y dy belong to the pace L p L. O Proof. Firt notice that by Lemma 6.1 for each ω a.., ω i a bounded linear operator on = L O and L M 1 M 1/ a.. Then the reult follow from the fact that f i meaurable and we have E p L E Mp 1 E Mp 1/ < We can tate a ilbert-valued verion of Lemma 6. whoe proof would be identical. Lemma 6.3. Let G be a real and eparable ilbert pace. Conider a meaurable function F O O G verifying the following condition: (i F x L O G for every x O a..; (ii there exit two nonnegative random variable M 1 and M uch that F y z G dy M 1 a.. up z O up z O O O F z y G dy M and E M p 1 <, E Mp <. Then the random operator from to L O G = L G defined by ω g x = F x y ω g y dy belong to the pace L p L L O G and O a.. L L O G M 1 M 1/ Lemma 6.4. Let f O O R + be a meaurable mapping verifying the hypothee of Lemma 6.. Aume, in addition, that f x y D 1 for each x y O, and that there exit a verion of the derivative D r f x y which i meaurable from T O O into U and verifie the following: (i Df x L T O U for all x O; (ii up z O O D rf x z U dx a 1 r a.., up z O O D rf z x U dx a r a.., where a 1 r and a r are nonnegative meaurable procee uch that E T a 1 r dr <, E T a r dr <. Then the random operator ω L O L O defined by (6.1 g x = f x y g y dy O

29 STOCASTIC EVOLUTION EQUATIONS 177 belong to D 1 L and for all r ω almot everywhere, D r ω i the operator in L L U given by the kernel D r f x y. Remark. Notice that for all r T a.e., the kernel D r f x y verifie the aumption of Lemma 6.3. Proof. By Lemma 6. we know that L L. According to Definition.1, in order to how that D 1 L we have to how that condition (a and (b of thi definition are atified. For (a we mut how that for every g L O, g belong to D 1. From condition (i of Lemma 6.4 it follow that g x D 1 for each x O, and D g x = O Df x y g y dy. Furthermore, uing condition (ii we get T E D r g x U dx dr O T ( E D r f x y U g y dy dx dr O O ( T ( E up D r f x y U dy ( x O up y O O O ( T E a 1 r a r dr { ( T E a 1 r dr E D r f y x U dx dr g L O g L O ( T } 1/ a r dr g L O < Thi implie that g D 1 (ee [15], Theorem 3.1. For (b, clearly D r g = ˆD r g, where ˆD r i the random operator belonging to the pace L L U aociated with the kernel D r f x y. ence, it uffice to how that ˆD r belong to the pace L T L L U. Thi follow from the fact that D r g x U dx a 1 r a r g L O O which implie ˆD r L L U a 1 r a r 1/ We can alo how a verion of Lemma 6.4 for kth differentiable operator. Lemma 6.5. Let f O O R + be a meaurable mapping verifying the hypothee of Lemma 6.. Aume that f x y D k for each x y O and for ome integer k 1, and there exit verion of the derivative D j r 1 r j f x y,

30 178 J. A. LEÓN AND D. NUALART 1 j k, which are meaurable from T j O O into U j and verify the following: (i D j f x L T j O U j for all x O, 1 j k; (ii up z O up z O O O D j r 1 r j f x z U j dx a 1 j r 1 r j D j r 1 r j f z x U j dx a j r 1 r j where a 1 j and a j are nonnegative meaurable random field uch that E T a j 1 j r dr < and E T a j j r dr <, for each j = 1 k Then the random operator ω on L O given by (6.1 belong to D k L, and for all r 1 r j ω a.e. D j r 1 r j ω i the operator belonging to L L U j given by the kernel D j r 1 r j f x y. Conider now a random econd order differential operator of the form (6. A t = a ij x t + b x i x i x t + c x t j x i i j=1 The coefficient a ij, b i and c are meaurable function from O T in R. Let u introduce the following hypothee on the random operator A t. (A1 For each x t O T, a ij x t, b i x t and c x t are t - meaurable (adaptability. (A The matrix a ij 1 i j n i ymmetric and uniformly elliptic. That i, there exit contant <c 1 c < uch that c 1 ξ a ij x t ξ i ξ j c ξ for all ξ R n i j=1 (A3 The coefficient a ij b i and c are continuou and uniformly bounded in O T, and, in addition they verify the following ölder continuity property: a ij x t a ij y K x y α + t α/ b i x t b i y t K x y α c x t c y t K x y α for ome contant <K<, α> and for all x y O, t T. Furthermore a ij t i of cla C 1 with uniformly bounded partial derivative. (A4 For each x t O T we have that a ij x t, a ij / x i x t, b i x t and c x t belong to D, and the derivative D r a ij x t U D a ij r x t x D r b i x t U D r c x t U i U

31 STOCASTIC EVOLUTION EQUATIONS 179 are bounded by a nonnegative proce r uch that ( T p E r dr < for all p. We alo aume that D a r 1 r a ij x t U U ij D r 1 r x t x i U U D r 1 r b i x t U U D r 1 r c x t U U (A4 are bounded by a nonnegative proce r 1 r uch that ( p E r 1 r dr 1 dr < for all p T We aume that the following quantitie are uniformly bounded: { D e k r a ij x t a + De k ij r x t x i k=1 up x t + D e k r b i x t + D e k r } c x t { up D e k r D a ij x t U + D e k r D a ij x t x i k=1 x t + D e k r D b i x t U + De k r D c x t U In what follow we will aume that O = R n. The cae of a bounded domain O with Dirichlet or Neuman boundary condition would be treated in a imilar way. Suppoe that A i a random econd order differential operator verifying hypothee (A1, (A and (A3 with O = R n. We will denote by Ɣ x t y the fundamental olution of Ɣ t = A tɣ t> (6.3 lim Ɣ x t y =δ x y t (For detail, ee [6]. Condition (A and (A3 imply that there exit contant c 1 c > uch that Ɣ x t y c 1 t n/ x y (6.4 exp ( c t (6.5 Ɣ x t y x c 1 t n+1 / exp ( i x y c t U }

32 18 J. A. LEÓN AND D. NUALART Propoition 6.6. Suppoe A t i a random econd order differential operator verifying hypothee (A1, (A and (A3. Let Ɣ x t y be the fundamental olution of (6.3. For any t>, t T let S t be the random operator on L R n given by (6.6 S t g x = Ɣ x t y g y dy n R Put S t t =Id. Then S t t T i a random evolution ytem on L R n in the ene of Definition 3.1. Proof. By contruction (ee [], the random kernel Ɣ x t y ω i a meaurable mapping from R n R n R + for each t>. From (6.4 we deduce Ɣ x t L R n for all x R n t> and (6.7 Ɣ x t y dy c 1 π n/ R n (6.8 Ɣ x t y dx c 1 π n/ R n ence, by Lemma 6.1, S t i a bounded linear operator on L R n. Moreover the mapping t ω S t ω i trongly meaurable from in L, and S t i t trongly meaurable from in L. Condition (iiia of Definition 3.1 clearly hold, and the continuity property (iiib i alo known (ee [6]. Propoition 6.7. Let A t be a random econd order differential operator verifying hypothee (A1, (A, (A3, (A4 and (A4. Then the random evolution ytem S t given by (6.6 verifie hypothee (1, ( and (3. The proof will be done in everal tep. Proof of (1. that By Lemma 6. and the etimate (6.7 and (6.8, we deduce S t L c 1 π n/ So S t L L and the norm of S t i uniformly bounded. In order to how that S t belong to D L for t>we will make ue of Lemma 6.5. We have to how that Ɣ x t y D for each x y R n, t>and that condition (i and (ii of Lemma 6.5 for j = 1 hold. Let u firt how that Ɣ x t y D 1. We recall that Ɣ x t y i the fundamental olution of Ɣ t = A tɣ t> lim Ɣ x t y =δ x y t

33 STOCASTIC EVOLUTION EQUATIONS 181 In the equel we will write Ɣ t x y for Ɣ x t y. Uing the characterization of the pace D 1 given by Sugita in [15] we can how that Ɣ t x y i RAC (ray abolutely continuou, and the derivative D r Ɣ t x y verifie t D rɣ t x y =A t D r Ɣ t x y + D r A t Ɣ t x y for r t. ence, Rn { n D r Ɣ t x y = Ɣ t τ x ξ D r a ij ξ τ Ɣ τ ξ y ξ i j=1 i ξ j (6.9 + D r b i ξ τ Ɣ τ ξ y ξ i } + D r c ξ τ Ɣ τ ξ y dτ dξ Integrating by part thi can be written a Rn { ( aij Ɣτ D r Ɣ t x y = Ɣ t τ x ξ D r ξ τ ξ y ξ i j=1 i ξ j + D r b i ξ τ Ɣ τ ξ y ξ i (6.1 } + D r c ξ τ Ɣ τ ξ y dτ dξ Rn i j=1 Ɣ t τ ξ i x ξ Ɣ τ ξ y D ξ r a ij ξ τ dτ dξ j From (6.4, (6.5 and (6.1 we obtain the following etimate: ( D r Ɣ t x y U C t n/ x y exp c t { [ n up D a ij r ξ τ ξ R n ξ + D r b i ξ τ ] U i U (6.11 i j=1 τ 1/ + up ξ R n D r c ξ τ U + up ξ R n i j=1 C t n/ exp ( D r a ij ξ τ U t τ 1/ τ 1/ } dτ x y c t r for ome contant c C >. ence, condition (i and (ii of Lemma 6.5 hold for j = 1.

34 18 J. A. LEÓN AND D. NUALART For the econd derivative we have t D r 1 r Ɣ t x y =A t D r 1 r Ɣ t x y + D r A t D r1 Ɣ t x y + D r1 A t D r Ɣ t x y + D r 1 r A t Ɣ t x y ence, Rn D r 1 r Ɣ t x y = Ɣ t τ x ξ { n D r a ij ξ τ D r1 Ɣ τ ξ y ξ i j=1 i ξ j + D r b i ξ τ D r 1 Ɣ τ ξ y +D ξ r c ξ τ D r1 Ɣ τ ξ y i + D r1 a ij ξ τ D r Ɣ τ ξ y ξ i j=1 i ξ j + D r1 b i ξ τ D r Ɣ τ ξ y +D ξ r1 c ξ τ D r Ɣ τ ξ y i + D r 1 r a ij ξ τ Ɣ τ ξ y ξ i j=1 i ξ j + D r 1 r b i ξ τ Ɣ τ ξ y ξ i } + D r 1 r c ξ τ Ɣ τ ξ y dξ dτ Integrating by part and uing the etimate (A4 and (6.11 we get D r 1 r Ɣ t x y U U (6.1 C t n/ x y exp ( r c t 1 r + r 1 r ence condition (i and (ii of Lemma 6.5 hold for j =. Furthermore, ( p/ S t p p = E S t p L + E D r S t L L U dr ( + E 1 + E { C and ypothei (1 hold. p/ D r 1 r S t L L U U dr 1 dr p/ p r dr + E r dr t p/} + E r 1 r dr 1 dr <

35 STOCASTIC EVOLUTION EQUATIONS 183 Proof of (. Fix an element h = L R n. Then D r S t h = rɣ t x y h y dy R n for r t where D r Ɣ t x y i given by formula (6.9. Let u define D S t a the operator given by the kernel D Ɣ t x y, where Rn t { ( D Ɣ aij Ɣτ t x y = Ɣ t τ x ξ D ξ τ ξ y ξ i j=1 i ξ j + D b i ξ τ Ɣ τ ξ y ξ i } + D c ξ τ Ɣ τ ξ y dτ dξ Rn i j=1 Ɣ t τ ξ i x ξ Ɣ τ ξ y D ξ a ij ξ τ dτ dξ j The difference D S t ε D S t i the operator in L L U given by the kernel Rn { ( aij Ɣτ ε Ɣ t τ x ξ D ξ τ ξ y ε ξ i j=1 i ξ j + D b i ξ τ Ɣ } τ ε ξ y +D ξ c ξ τ Ɣ τ ε ξ y dτ dξ i Rn Ɣ t τ ε i j=1 Rn + Ɣ t τ x ξ Rn i j=1 ξ i { Ɣ t τ ξ i x ξ Ɣ τ ε ξ y D ξ a ij ξ τ dτ dξ j i j=1 D ( aij ξ i ξ τ ( Ɣτ ε ( Ɣτ ε ξ j Ɣ τ ξ j ξ y ξ y + D b i ξ τ Ɣ τ ξ i ξ i } + D c ξ τ Ɣ τ ε Ɣ τ ξ y dτ dξ ( Ɣτ ε x ξ Ɣ τ ξ y D ξ j ξ a ij ξ τ dτ dξ j Let u denote thi kernel by ε t x y. We have for any h R n ε t x y h y dy U { } C τ + ε 1/ + + t τ 1/ τ + ε 1/ dτ ε

36 184 J. A. LEÓN AND D. NUALART R n t + x y ε n/ exp ( h y dy c t + ε { } + C τ 1/ + + t τ 1/ τ 1/ dτ R n Ɣ R n ε z y h y dy h z t n/ exp ( A a conequence, R n R n ε t x y h y dy U dx x z c t dz C { ( ε + ε + t 1/ ε h + R n Ɣ R n ε z y h y dy h z } dz and thi converge to zero a ε tend to zero. On the other hand, D S t belong to D1 L L U (ee firt tep of the proof of Propoition 6.7. Proof of (3. We have already een that S t L c 1 π n/. We have D e j r S t L = up h 1 R n (( up x R n De j r Ɣ t x y h y dy dx R n De j r Ɣ t x y dy ( up y R n De j r Ɣ t x y dx 1/ ence, the boundedne of j=1 D e i r S t L follow from (6.9 and ypothei (A4. Finally, let u how that k=1 De k r D S t L L U i bounded. We have, for r t, Rn D e k r D Ɣ t x y = Ɣ t τ x ξ { n D e k i j=1 + D e k r a ij ξ τ D Ɣ τ ξ y ξ i ξ j r b i ξ τ D Ɣ τ ξ i ξ y +D e k r c ξ τ D Ɣ τ ξ y

37 STOCASTIC EVOLUTION EQUATIONS i j=1 D a ij ξ τ D e k D b i ξ τ De k ξ i i j=1 D e k r Ɣ τ ξ i ξ j r Ɣ τ ξ y D e k r D a ij ξ τ Ɣ τ ξ i ξ j ξ y r D b i ξ τ Ɣ τ ξ y ξ i } + D e k r D c ξ τ Ɣ τ ξ y dτ dξ ξ y +D c ξ τ D e k r Ɣ τ ξ y Again uing (A4, integration by part and (6.9 we how the boundedne of the expreion and k=1 k=1 up De k x R n r D Ɣ t x y U dy up De k y R n r D Ɣ t x y U dx Remark. Theorem 5.4 together with Propoition 6.7 allow u to deduce the exitence of a unique mild olution for tochatic partial differential equation of the form u t = A tu + f t x u +g t x u Ẇ t x t T x D (6.13 u x =ϕ x where D R n i a bounded domain with mooth boundary, f g are continuou function on T D R which are Lipchitz and have linear growth in the lat variable, uniformly with repect to the firt two variable, and Ẇ i a Wiener proce in L D whoe covariance operator i bounded on D D. ere A t i a econd order operator of the form (6. with random and adapted coefficient atifying aumption (A.1, (A., (A.3, (A.4 and (A.4. The above method allow handling tochatic partial differential equation of the form (6.13 without motononicity or coercivity aumption on the coefficient. Acknowledgment. Thi work wa done while Jorge León wa viiting the Univerity of Barcelona. e i grateful for it hopitality.

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