Stochastic heat equation with white-noise drift

Transcription

1 Stochastc heat equaton wth whte-nose drft by lsa Alos, Davd Nualart * Facultat de Matematques Unverstat de Barcelona Gran Va 585, 87 Barcelona, SPAIN Freder Vens Deartment of Mathematcs Unversty of North Texas P.O. Box 358 Denton, TX , USA * Suorted by the DGICYT grant n o PB96 87.

2 Introducton The urose of ths aer s to establsh the exstence and unqueness of a soluton for antcatve stochastc evoluton equatons of the form u(t; x) = (; t; y; x) u (y) dy t (.) (s; t; y; x) F (s; y; u(s; y)) dw s;y ; where W = fw (t; x); t [; T ]; x g s a zero mean Gaussan random eld wth covarance (s^t) (jxjjyj jx yj). We asume that (s; t; y; x) s a stochastc semgrou measurable wth resect to the - eld fw (r; x) W (s; x); x ; r [s; t]g. The stochastc ntegral n quaton (.) s antcatve because the ntegrand s the roduct of an adated factor F s; y; u(s; y) tmes (s; t; y; x), whch s adated to the future ncrements of the random eld W. We nterret ths ntegral n the Skorohod sense (see [5]) whch concdes n ths case wth a two-sded stochastc ntegral (see [4]). The choce of ths noton of stochastc ntegral s motvated by the concrete examle handled n Secton 5, where (s; t; y; x) s the backward heat kernel of the random oerator d _v(t; x) d dx dx ; _v(t; x) beng a whte-nose n tme. In ths case, u(t; x) turns out to be (see Secton 6) a weak soluton of the stochastc artal d F (t; x; W : (.) A stochastc evoluton equaton of the form (.) on d erturbed by a nose of the form W (ds; y)dy, where W s a random eld wth covarance (s ^ t) Q(x; y); Q beng a bounded functon, has been studed n [3].Followng the aroach ntroduced n ths aer we establsh n Theorem 4. the exstence and unqueness of a soluton to quaton (.) wth values n L M (). Here L M () means the sace of real-valued functons f such that e M(x) jf(x)j dx < where M > and. Ths theorem s a consequence of the estmates of the moments of Skorohod ntegrals of the form t (s; t; y; x) (s; y) dw s;y ; obtaned n Secton 3 by means of the technques of the Mallavn calculus. Prelmnares For s; t [; T ], s t, we set = [; t] and s = [s; t]. Consder a Gaussan famly of random varables W = fw (A); A B(I T ); (A) < g, de ned on a comlete robablty sace, wth zero mean, and covarance functon gven by W (A) W (B) = (A \ B) ; where denotes the Lebesgue measure on I T. We wll assume that F s generated by W and the P -null sets. For each s; t [; T ], s t we wll denote by F s;t the -algebra

3 generated by fw (A); A [s; t] ; (A) < g and the P -null sets. We say that a stochastc rocess u = fu(t; x); (t; x) I T g s adated f u(t; x) s F ;t -measurable for each (t; x). Set H = L (I T ; B(I T ); ) and denote by W (h) = I T hd W the Wener ntegral of a determnstc functon h H. In the sequel we ntroduce the basc notaton and results of the stochastc calculus of varatons wth resect to W. For a comlete exoston we refer to [, ]. Let S be the set of smooth and cylndrcal random varables of the form F = f W (h ); : : : ; W (h n ) ; (.) where n ; f Cb (n ) (f and all ts artal dervatves are bounded), and h ; : : : ; h n H. Gven a random varable F of the form (.), we de ne ts dervatve as the stochastc rocess fd t;x F; (t; x) I T g gven by D t;x F = W (h ); : : : ; W (h n ) h (t; x); (t; x) I T : More generally, we can de ne the terated dervatve oerator on a cylndrcal random varable F by settng D n t ;x ;:::;t n;x n F = D t ;x D tn;x n F : The terated dervatve oerator D n s a closable unbounded oerator from L () nto L (I T ) n for each n. We denote by D n; the closure of S wth resect to the norm de ned by n kf k n; = kf k L () X kd F k L ((I T ) ) : = If V s a real and searable Hlbert sace we denote by D n; (V ) the corresondng Sobolev sace of V -valued random varables. We denote by the adjont of the dervatve oerator D. That s, the doman of (denoted by Dom ) s the set of elements u L (I T ) such that there exsts a constant c satsfyng (D t;x F ) u(t; x) dt dx c kf k L () ; I T for all F S. If u Dom ; (u) s the element n L () characterzed by (u) F = (D t;x F ) u(t; x) dt dx ; F S : I T The oerator s an extenson of the ItÙ ntegral (see Skorohod [5]), n the sense that the set L a(i T ) of square ntegrable and adated rocesses s ncluded n Dom and the oerator restrcted to L a(i T ) concdes wth the ItÙ stochastc ntegral de ned n [6]. We wll make use of the notaton (u) = I T u(t; x) dw t;x for any u Dom.

4 We recall that L ; := L (I T ; D ; ) s ncluded n the doman of, and for a rocess u L ; we can comute the varance of the Skorohod ntegral of u as follows: (u) = u (t; x) dt dx D s;y ; u(t; x) D t;x u(s; y) dt dx ds dy : I T I T I T We need the followng results on the Skorohod ntegral: Prooston. Let u Dom and consder a random varable F D ; such that F I T u(t; x) dt dx <. Then F u(t; x) dw t;x = F I T u(t; x) dw t;x I T (D t;x F ) u(t; x) dt dx ; I T (.) n the sense that F u Dom f and only f the rght-hand sde of (.) s square ntegrable. Prooston. Consder a rocess u n L ;. Suose that for almost all (; z) I T, the rocess fd ;z u(s; y) l [;] (s); (s; y) I T g belongs to Dom and, moreover, I T I D ;z u(s; y) dw s;y d dz < : Then u belongs to Dom and we have the followng exresson for the varance of the Skorohod ntegral of u: (u) = u (s; y) ds dy I T (.3) u(; z) D ;z u(s; y) dw s;y d dz : I T I We make use of the change-of-varables formula for the Skorohod ntegral: Theorem.3 Consder a rocess of the form X t = u(s; y) dw s;y, where () u L ;, () u L (I T ), for some >, () I T u (s; y) ds dy < N, for some ostve constant N. Let F : be a twce contnuously d erentable functon such that F s bounded. Then we have F (X t ) = F () F (X s ) u(s; y) dw s;y F (X s ) u (s; y) ds dy (.4) I t F (X s ) u(s; y) D s;y u(r; z) dw r;z ds dy : I s Notce that under the assumtons of Theorem.3 the rocess X t has a contnuous verson (see [, 5]) and, moreover, ff (X s ) u(s; y); (s; y) I T g belongs to Dom. 3

5 3 stmates for the Skorohod ntegral We denote by C a generc constant that can change from one formula to another one. Let (s; t; y; x) be a random measurable functon de ned on f s < t T; x; y g. We wll assume that the followng condtons hold: (H) For all s < t T; x; y ; (s; t; y; x) s F s;t -measurable. (H) (s; t; y; x), for each s < t T; x; y. (H3) For all s < t T; x ; (s; t; y; x) dy =. (H4) For each s [; T ]; x; y ; (s; t; y; ) s contnuous n t (s; T ] wth values n L (). (H5) For all s < r < t T; and x; y (s; t; y; x) = (s; r; y; z) (r; t; z; x) dz : (H6) For all s < t T; x; y ; (s; t; y; x) D ; and (s; t; ; x) belongs to D ; L (). Moreover, there exsts a verson of the dervatve such that the followng lmt exsts n L ; L () for each s; z; t; x D s;z (s; t; ; x) = lm "# D s;z (s "; t; ; x) : (3.) (H7) For all s < t T; x; y ; there exsts a nonnegatve random varable V (s; t; x) F s;t -measurable and > such that jx yj (s; t; y; x) V (s; t; x) ex (t s) and satsfyng that for all, there exsts a ostve constant C ; such that kv (s; t; x)k L () C ; jt sj : (H8) For all s < t T; x; y; z ; and there exsts a nonnegatve random varable U (s; t; x) F s;t -measurable a constant > and a nonnegatve measurable determnstc functon f(y; z) such that () jd s;z (s; t; y; x)j U (s; t; x) ex () su y f (y; z) dz C f, () ku (s; t; x)k L () C ; jt sj, jx yj (t s) f(y; z), for some ostve constants C ; ; C f >. The followng lemma s a straghtforward consequence of the above hyotheses: 4

6 Lemma 3. Under the above hyotheses we have that for all r < s < t T; x; y; z ; D s;y (r; t; z; x) = D s;y (s; t; u; x) (r; s; z; u) du : (3.) Proof. Takng nto account the roertes of the dervatve oerator and usng hyotheses (H), (H5) and (H6) we have that D s;y (r; t; z; x) = D s;y = (r; s "; z; u) (s "; t; u; x) du (r; s "; z; u) D s;y (s "; t; u; x) du : Now, lettng " tend to zero and usng hyotheses (H), (H4), (H6) and (H8) we can easly comlete the roof. We are now n a oston to rove our estmates for the Skorohod ntegral. For all M >, we wll denote by L M (IT ) the sace of rocesses = f(s; y); (s; y) I T g such that e Mjyj j(s; y)j ds dy < : I T Theorem 3. Fx > 4; [; 4 4 ) and M >. Let = f(s; y); (s; y) IT g be an adated rocess n L M (IT ). Assume that (s; t; y; x) s a stochastc kernel satsfyng hyotheses (H) to (H8). Then, for almost all (t; x) I T, the rocess f (t s) (s; t; y; x) (s; y) l [;t] (s); (s; y) I T g belongs to Dom, and e Mjxj (t s) (s; t; y; x) (s; y) dw s;y dx t C (t s) 4 e Mjyj j(s; y)j dy ds ; (3.3) for some ostve constant C deendng only on,, T, M,,, C ;, C ; and C f. Proof. Let us denote by S a the sace of smle and adated rocesses of the form (s; y) = m X ;j= F j l (t ;t ] (s) h j (y) ; where = t < t < : : : < t m = T; h j C K () and the F j are F ;t -measurable functons n S. Let be an adated rocess n L M (IT ). We can nd a sequence n of rocesses n S a such that lm n T e Mjyj j n (s; y) (s; y)j dy ds = : 5

7 We can easly check that ths mles the exstence of a subsequence n k such that for almost all t [; T ] t lm (t s) 4 e Mjyj j n k (s; y) (s; y)j dy ds = : k On the other hand, usng the fact that < 4 T A : = lm k j n k (s; y) and hyothess (H7) we have that e Mjxj (t s) (s; t; y; x) (s; y)j ds dy dx dt T C; lm k j n k (s; y) e Mjxj (t s) jx yj ex (t s) (s; y)j ds dy dx dt Notce that ex = C; lm k ex Mjxj e M jyj ex I T j n k (s; y) (s; y)j Mjxj jx yj (t s) Mjxj I T s jx yj dt dx ds dy : (t s) dx = ex x (t s) (t s) Mjx yj dx K t s e M jyj ; x dx (t s) where K = e M T 8 : Then A C;K lm k I T e M jyj j n k (s; y) (s; y)j ds dy = : Then, choosng a subsequence (denoted agan by n k ) we have that for almost all (t; x) I T lm (t s) (s; t; y; x) j n k (s; y) (s; y)j ds dy = : k Ths allows us to suose that S a. Fx t > t n [; T ] and de ne B x (s; y) = (t s) (s; t ; y; x) (s; y) X(t; x) = B x (s; y) dw s;y ; t [; t ] : Denote F (x) = jxj. Let F N be the ncreasng sequence of functons de ned by F N (x) = jxj y ( ) z ^ N dz dy : 6

8 Suose rst that (s; t; y; x) s an elementary backward-adated rocess of the form nx H jk j (y) k (z) l (s ;s ](s) ; ;j;k= where H jk S; j ; k CK (); = s < : : : < s n = t, and H jk s F s ;t - measurable. Then we can aly ItÙ s formula (see Theorem.3) for the functon F N and the rocess B x (s; y), obtanng that for all t < t, F N X(t; x) = F N X(s; x) B x (s; y) F N I s X(s; x) B x(s; y) ds dy D s;y B x (r; z) dw r;z ds dy : (3.4) Usng hyotheses (H), (H7) and (H8), Lemma 3. and the fact that s smle and adated we can easly check that, for all (s; t; y; x) satsfyng the hyotheses of the theorem, and for all t < t () Bx(s; y) ds dy < ; () B x (s; y) dw s;y <, () I s D s;y B x (r; z) dw r;z ds dy < : Ths allows us to deduce that (3.4) stll holds for every (s; t; y; x) satsfyng hyotheses (H) to (H8) and for all t < t. We can easly check that FN (x) ( ( ) ) F N(x) : Then we have that F N X(t; x) C ( F N X(s; x) Hˆlder s nequalty gves us that F N X(t; x) C ( t t F N X(s; x) F N X(s; x) B x (s; y) I s F N X(s; x)) Alyng the lemma of [7], g. 7 we obtan that F N X(t; x) C ( t t B x(s; y) ds dy ) D s;y B x (r; z) dw r;z dyds : Bx(s; y) dy B x (s; y) D s;y B x (r; z) dw r;z dy I s Bx(s; y) dy ds B x (s; y) D s;y B x (r; z) dw r;z dy I s ) ds : ds ) ds : 7

9 Fatou s lemma gves us that, lettng N tend to n nty t jx(t; x)j C ( t Bx(s; y) dy B x (s; y) D s;y B x (r; z) dw r;z dy I s ds ) ds (3.5) =: C I I : We have that I t t C ; C (t s) (t s) t t (t s) (t s) (s; t ; y; x) (s; y) dy jx yj ex (t s) V ds (s; t ; x) (s; y) dy jx yj ex (s; y) dy (t s) ds jx yj ex j (s; y)j dy ds : (t s) ds (3.6) On the other hand, usng Lemma 3. yelds B x (s; y) D s;y B x (r; z) dw r;z dy I s = (t s) (s; t ; y; x) (s; y) (t I s = (t s) (s; t ; y; x) (s; y) h (t I s = (t s) (t I s r) [D s;y (r; t ; z; x)] (r; z) dw r;z r) dy (r; s; z; u) D s;y (s; t ; u; x) du (r; z) dw r;z h (s; t ; y; x) (s; y) r) (r; s; z; u) (r; z) dw r;z D s;y (s; t ; u; x) du dy : Let us denote Y (s; u) := I s (t r) (r; s; z; u) (r; z) dw r;z. Notce that X(t ; x) = Y (t ; x). We have roved that B x (s; y) D s;y B x (r; z) dw r;z dy I s h = (t s) (s; t ; y; x) (s; y) dy D s;y (s; t ; u; x) Y (s; u) du dy ; 8

10 and then I t (t s) (s; t ; y; x) (s; y) h D s;y (s; t ; u; x) Y (s; u) du dy ds : We have that = h (s; t ; y; x) (s; y) D s;y (s; t ; u; x) Y (s; u) du (s; t ; y; x) D s;y (s; t ; u; x) (s; y) Y (s; u) du dy V (s; t ; x) U (s; t ; x) f(u; y) ex j(s; y) Y (s; u)j du dy C jt sj 3 4 f(u; y) ex j(s; y) Y (s; u)j du dy : Alyng Schwartz nequalty we obtan jy xj (t s) jy xj (t s) dy jx uj (t jx uj (t s) h (s; t ; y; x) (s; y) D s;y (s; t ; u; x) Y (s; u) du dy C (t s) 3 4 Y (s; u) f jx uj (u; y) e (t s) 4 du dy (s; y) e C (t s) 5 8 C (t s) 5 8 C (t s) 3 8 (s; y) e jx uj (t s) jy xj (t Y (s; u) e Y (s; u) e jy xj (t s) dy s) 4 du dy jx uj (t s) du jx uj (t s) du jy (s; y)j e 4 (s; y) e jy jx yj (t s) dy j(s; y)j e xj (t s) dy s) gg) 4 jx yj (t s) dy : 9

11 Ths yelds I C t (t s) 3 4 jx yj ex ( _ )(t s) j (s; y)j jy (s; y)j dy ds : (3.7) Puttng (3.6) and (3.7) nto (3.5) and usng the fact that < 4 4 jx(t; x)j C t (t s) 3 4 j (s; y)j jy (s; y)j dy jx yj ex c (t s) ds we obtan that where c = max ( ; ). Now we make t tend to t and use Fatou s lemma to obtan jy (t ; x)j C t (t s) 3 4 Usng an teratve rocedure we have that jy (t; x)j C t (t s) 3 4 e jx yj c (t s) j (s; y)j jy (s; y)j dy ds : e jx c (t s) j (s; y)j dy ds ; for all t t. Fnally for any xed t [; T ) lettng the arameter t n the de nton of Y (t; x) to converge to t and ntegratng wth resect to the measure e Mjxj dx leads to the desred result. Let us now consder the followng addtonal condton over the stochastc kernel (s; t; y; x): yj (H9) M There exsts a constant C M > such that su rt su sr e Mjxj (r; s; y; x) dx C M e Mjyj : We wll denote by L M () the sace of functons f : such that e Mjxj jf(x)j dx <. Theorem 3.3 Fx > 8 and M >. Let = f(s; y); (s; y) I T g be an adated rocess n L M (IT ). Assume that (s; t; y; x) s a stochastc kernel satsfyng condtons (H) to (H8) and (H9) M. Then for all t [; T ], the rocess f(s; t; y; x) (s; y) l [;t] (s); (s; y) I T g belongs to Dom for almost all x, and the stochastc rocess = n t = o (s; t; y; ) (s; y) dw s;y ; t [; T ]

12 osseses a contnuous verson wth values n L M (). Moreover, su e Mjxj (s; t; y; x) (s; y) dw s;y tt I dx t C e Mjyj j(s; y)j ds dy ; I T (3.8) for some ostve constant C deendng only on T,, M, C ;, C ;, C f,, and C M. Proof. Usng the same arguments as n the roof of Theorem 3. we can assume that the rocess s smle. Fx < < 4 4 and de ne Y (r; u) = (r I r s) (s; r; y; u) (s; y) dw s;y : As (s; t; y; x) = C s (t r) (r s) (s; r; y; u) (r; t; u; x) dr du ; wth C = sn t s easy to show that t (x) = C (t r) (r; t; u; x) Y (r; u) dr du : Then we have that for any t < t and ; 4 4 e Mjxj j t (x) t (x)j dx e Mjxj (t r) (r; t ; u; x) Y (r; u) dr du dx t e Mjxj (t r) I [(r; t ; u; x) (r; t; u; x)] Y (r; u) dr du dx t C ; (t t) e Mjxj (r; t ; u; x) Y (r; u) du dr dx C ; t e Mjxj [(r; t ; u; x) (r; t; u; x)] Y (r; u) du dr dx C ; (t t) jy (r; u)j su e Mjxj (r; s; u; x) dx dr du I T sr C I T e Mjxj l [;t] (r) j(r; t ; u; x) (r; t; u; x)j jy (r; u)j du dr dx : By domnated convergence, and usng hyothess (H4) both summands n the above exresson converges to zero as jt tj. On the other hand, takng t = we obtan (3.8). We wll also need the followng L -estmate for the Skorohod ntegral. Theorem 3.4 Fx M >. Let = f(s; y); (s; y) I T g be an adated random eld n L M (IT ). Assume that (s; t; y; x) s a stochastc kernel satsfyng condtons (H) to (H8). Then, for almost all (t; x) I T, the rocess f(s; t; y; x) (s; y) l [;t] (s); (s; y) I T g

13 belongs to Dom and we have that e Mjxj (s; t; y; x) (s; y) dw s;y dx t C (t s) 3 4 e Mjxj j(s; y)j dy ds ; for some ostve constant C deendng only on T, M, C ;, C ;, C f, and. (3.9) Proof. Usng the same arguments as n the roof of Theorem 3. we can assume that S a. Fx (t; x) I T and de ne B t;x (s; y) = (s; t; y; x) (s; y) l [;t] (s) X(t; x) = B t;x (s; y) dw s;y : By the sometry roertes of the Skorohod ntegral (Prooston.) we have that e Mjxj jx(t; x)j dx = e Mjxj jb t;x (s; y)j ds dy dx I h t e Mjxj B t;x (s; y) D s;y B t;x (r; z) dw r;z ds dy dx (3.) I s = I I : By hyothess (H7) we have that I e Mjxj jv (s; t; x)j ex ;jx yj (t s) j(s; y)j ds dy C; (t s) j(s; y)j ex jx yj Mjxj (3.) (t s) dx ds dy C; K (t s) e Mjyj j(s; y)j ds dy: On the other hand, usng the same arguments as n the roof of Theorem 3. t s easy to show that I = t e Mjxj (s; t; y; x) (s; y) D s;y (s; t; u; x) X(s; u) du dx dy ds = t 3 e Mjxj (s; t; y; x)d s;y (s; t; u; x) (s; y) X(s; u) dx dy du ds e Mjxj jv (s; t; x)j ju (s; t; x)j h ex jx yj (t s) C ; C ; e Mjxj (t s) 3 j(s; y)j e C e Mjxj (t s) 5 4 C t (t s) 3 4 jx uj (t s) f(u; y) j(s; y) X(s; u)j dy du jx uj (t s) jx(s; u)j f(u; y) e jx yj (t s) du dy ds dx jx yj dx ds dx jx uj (t s) du dy ( e _ )(t s) jx(s; y)j j(s; y)j dy e Mjyj jx(s; y)j j(s; y)j dy ds : ds dx (3.)

14 Now, substtuttng (3.) and (3.) nto (3.) and usng an teraton argument the result follows. Usng the same arguments t s easy to show the followng result. Corollary 3.5 Let = f(s; y); (s; y) I T g be an adated rocess n L (I T ). Assume that (s; t; y; x) s a random functon satsfyng hyotheses (H) to (H8). Then, for almost all (t; x) I T, the rocess f(s; t; y; x) (s; y) l [;t] (s); (s; y) I T g belongs to Dom and (s; t; y; x) (s; y) dw s;y dx C t (t s) 3 4 j(s; y)j dy ds ; (3.3) for some ostve constant C deendng only on T, C ;, C ;, C f, and.. 4 xstence and unqueness of soluton for stochastc evoluton equatons wth a random kernel Our urose n ths secton s to rove the exstence and unqueness of soluton for the followng antcatng stochastc evoluton equaton u(t; x) = (; t; y; x) u (y) dy (s; t; y; x) F (s; y; u(s; y)) dw s;y ; (4.) where (s; t; y; x) s a stochastc kernel satsfyng condtons (H) to (H8) and (H9) M ; u : s the ntal condton and F : [; T ] s a stochastc random eld. Let us consder the followng hyotheses. (F) F s measurable wth resect to the - eld B([; t] ) F ;t, when restrcted to [; t], for each t [; T ]. (F) For all t [; T ]; x; y; z for some ostve constant C. jf (t; y; x) F (t; y; z)j C jx zj ; (F3) M For all t [; T ]; x ; jf (t; x; )j h(x) ; for some h L M (). We are now n a oston to rove the man result of ths aer. Theorem 4. Fx M > and > 8. Let u be a functon n L M (). Consder an adated random eld F (s; y; x) satsfyng condtons (F) to (F3) M and a stochastc kernel (s; t; y; x) satsfyng hyotheses (H) to (H8) and (H9) M. Then, there exsts an unque adated random eld u = fu(t; x); (t; x) I T g n L M (IT ) that s soluton of (4.). Moreover, 3

15 () fu(t; ); t [; T ]g s contnuous a.s. as a rocess wth values n L M () and e Mjxj ju(t; x)j dx) C; (4.) su tt for some ostve constant C deendng only on T,, M, C ;, C ;, C f,, and C M. () If, moreover, u and h belong to L (), then u L (I T ). Proof of exstence and unqueness. Suose that u and v are two adated solutons of (4.) n L M (IT ), for some M >. Then, for every t [; T ] we can wrte e Mjxj ju(t; x) v(t; x)j dx = e Mjxj (s; t; y; x) F (s; y; u(s; y)) By Theorem 3.4 and the Lschtz condton on F we have that t e Mjxj ju(t; x) v(t; x)j dx (t s) 3 4 e Mjyj ju(s; y) Alyng an teraton argument we obtan that t e Mjxj ju(t; x) v(t; x)j dx C e Mjyj ju(s; y) F (s; y; v(s; y)) dw s;y dx : v(s; y)j dy ds : v(s; y)j dy ds ; from where we deduce that e Mjxj ju(t; x) v(t; x)j dx =. Consder now the Pcard aroxmatons 8 >< u (t; x) = (; t; y; x) u (y) dy >: u n (t; x) = (; t; y; x) u (y) dy (s; t; y; x) F (s; y; u n (s; y)) dw s;y : By hyothess (H), u (t; x) s adated. On the other hand, usng hyotheses (H3) and (H9) M we have that e Mjxj ju (y)j e Mjyj ju (y)j dy : (; t; y; x) u (y) dy dx e Mjxj (; t; y; x) dx dy Now, usng nducton on n and Theorem 3.4 t s easy to show that u n s adated and belongs to L M (IT ). Usng a recurrence argument we can easly show that X n= e Mjxj ju n (t; x) 4 u n (t; x)j dx < ;

16 and the lmt u of the sequence u n rovdes the soluton. Proof of () Usng the same arguments as n the roof of the exstence we can see that the soluton u belongs to L M (IT ). Now we have to show that the followng two terms are a.s. contnuous n L M (): A (t) = (; t; y; x) u (y) dy A (t) = (s; t; y; x) F (s; y; u(s; y)) dw s;y : In order to rove the contnuty of A, note that hyothess (H9) M mles that, for all ' and n L M () e Mjxj (; t; y; x) '(y) (y) dy dx su tt su j'(y) tt (y)j e Mjyj j'(y) (y)j dy : e Mjxj (; t; y; x) dx dy Hence, we can assume that u s a smooth functon wth comact suort. In ths case e Mjxj (; t "; y; x) (; t; y; x) u (y) dy dx jju jj e Mjxj j(; t "; y; x) (; t; y; x)j dx dy ; K whch tends to zero by hyotheses (H4) and (H9) M. The contnuty of A s an mmedate consequence of Theorem 3.3. Fnally, usng a recurrence argument t s easy to rove that the Pcard aroxmatons u n satsfy that X n= from where (4.) follows. su e Mjxj ju n (t; x) tt u n (t; x)j dx < ; Proof of exstence n L (I T ). Usng hyothess (H7) we have that (; t; y; x) u (y) dy dt dx ju (y)j (; t; y; x) dx dt dy I T I T T ju (y)j dy : Usng now nducton on n and Corollary 3.5 t s easy to show that I T ju n (t; x)j dt dx < and that u n s a Cauchy sequence n L (I T ). Ths mles that u belongs to L (I T ). For every ; " > and K > we denote by W ;" (K) the set of contnuous functons f : [ K; K] such that jjfjj ;";K := jf(x) f(z)j [ K;K] jx zj " dx dz < : 5

17 Notce that f f W ;" (K), then f s Hˆlder contnuous n [ K; K] wth order "=. Now our urose s to rove that, under some sutable hyotheses, the soluton u(t; ) belongs to W ;" (K), for some ; " > and all K >. Theorem 4. Fx > 4 and M >. Let u be a functon n L M (). Consder an adated random eld F (s; y; x) satsfyng hyotheses (F) to (F3) M and a stochastc kernel (s; t; y; x) satsfyng (H) to (H8) and (H9) M. Then, the soluton u(t; x) constructed n Theorem 4. belongs a.s., as a functon n x, to W ;" (K), for all > 8; " < 3 and K >. Proof. We have to show that the followng two terms belong to W ;" (K): B (x) = (; t; y; x) u (y) dy B (x) = (s; t; y; x) F s; y; u(s; y) dw s;y : Usng Mnkowsk s nequalty we have that jb (x) B (z)j [ jx zj " dx dz K;K] = jx zj " [(; t; y; x) (; t; y; z)] u (y) dy dx dz ; [ K;K] " dx jx zj k(; t; y; x) (; t; y; z)k ju (y)j dy dz : [ K;K] Takng nto account estmate (5.) and the same arguments as n [4], g. 7 t s easy to show that 8 s < t T ; x; y; z ; [; ] and ; k(s; t; y; x) (s; t; y; z)k h K jx zj (t s) () jy xj ex ex c(t s) for some K; c >. Ths gves us that, takng = jb (x) B (z)j [ jx zj " dx dz K;K] Ct Ct Ct [ K;K] jx zj " [ K;K] jy zj c(t s) ; jy xj ex ju (y)j dy dx dz c t jy xj ex ju (y)j dy dx c t e Mjyj ju (y)j dy < ; (4.3) whch gves us that B (x) belongs a.s. to W ;" (K). On the other hand, as n the roof of Theorem 3.3 we can wrte for (; 4 4 ) B (x) = C (t r) (r; t; u; x) Y (r; u) dr du ; 6

18 where Y (r; u) := (r I r s) (s; r; y; u) F (s; y; u(s; y)) dw s;y : Ths gves us that jb (x) B (z)j [ K;K] = C jx zj [ K;K] jx zj " dx dz " (t r) [(r; t; u; x) (r; t; u; z)] Y (r; u) dr du dx dz : Usng Mnkowsk s nequalty and the estmate (4.3) we obtan that jb (x) B (z)j [ jx zj " dx dz K;K] C jx zj " (t r) 3 ju xj ex ky (r; u)k dr du dx dz [ K;K] c(t r) " t C (t r) 3 ju xj ex ky (r; u)k du dr# dx [ K;K] c(t r) " t = C (t r) 3 [ K;K] Usng Holder s nequalty we obtan ju xj ex c(t r) jb (x) B (z)j [ jx zj " dx dz K;K] C t C t (t r) ky (r; u)k du dx dr# : e Mjuj jy (r; u)j du dr e Mjuj jy (r; u)j dr du ; rovded >. Fnally, from the roof of Theorem 3. and the facts that u L () and < 4 4 t s easy to show that e Mjuj jy (r; u)j dr du <, whch allows us to comlete the roof. We have made use of the followng condtons > " ; > ; < 4 4 : We can easly check that thanks to the fact that > 8 we can take and such that these nequaltes hold. 5 stmates for the heat kernel wth whte-nose drft In ths secton, followng the aroach of [3] we construct and estmate the backward heat kernel of the random oerator _v(t; x) d dx dx d, where v = fv(t; x); t 7

19 [; T ]; x )g s a zero mean Gaussan eld whch s Brownan n tme. The d erental _v(t; x)dt := v(dt; x) s nterreted n the backward ItÙ sense. More recsely, we assume that v can be reresented as v(t; x) = g(x; y) dw s;y ; (5.) where g : s a measurable functon, d erentable wth resect to x, satsfyng the followng condton su g(x; (x; y) dy < : (5.) Set G(x; y) = g(x; z) g(y; z) dz and let us ntroduce the followng coercvty condton: (C) P (x) := G(x; x) " > ; for all x and for some " >. Let b = fb(t); t [; T ]g be a Brownan moton wth varance t de ned on another robablty sace (W; G; Q). Consder the followng backward stochastc d erental equaton on the roduct robablty sace ( W; F G; P Q): t t q ' t;s (x) = x g(' t;r (x); y) dw r;y (' t;r (x)) db r : (5.3) s s Alyng Theorems 3.4. and 4.5. n [7] one can rove that (5.3) has a soluton ' = f' t;s (x); s t T; x g contnuous n the three varables and verfyng for all s < r < t; x. Then we have the followng result ' r;s ' t;r (x) = ' t;s (x) ; (5.4) Prooston 5. Let v be a Gaussan random eld of the form (5.) where the functon g sats es the coercvty condton (C) and assume that g s three tmes contnuously d erentable n x and sats es su x 3X k= Then there s a verson of the densty jg (k) (x; y)j dy < : (s; t; y; x) = Q(' t;s(x) dy) dy whch sats es condtons (H) to (H8) and (H9) M for each M >. Proof. Let us denote by b and D b the dvergence and dervatve oerators wth resect to the Brownan moton b. Alyng the ntegraton-by-arts formula of Mallavn calculus wth resect to the Brownan moton b we obtan (s; t; y; x) = Q l f't;s (x)>yg H t;s (x) ; (5.5) 8

20 where H t;s (x) = b D b ' t;s (x) kd b ' t;s (x)k Hyothess (H) follows easly from the exresson (5.5) because ' t;s (x) s F s;t -measurable. The fact that y 7 (s; t; y; x) s the robablty densty of ' t;s (x), whch has a contnuous verson n all the varables x; y ; s < t T, mly (H), (H3) and (H4). Hyothess (H5) s a consequence of the ow roerty (5.4). Alyng the dervatve oerator to (5.5) yelds where D r;z (s; t; y; x) = Q l f't;s (x)>yg D r;z H t;s (x) Q l f't;s (x)>yg t;s(x) ; t;s(x) = b D b ' t;s (x) kd b ' t;s (x)k D r;z ' t;s (x) H t;s (x) : : (5.6) Then hyothess (H6) follows easly from quaton (5.6). Condtons (H7), (H8) and (H9) M wll be roved n the followng lemmas. Lemma 5. The stochastc kernel (s; t; y; x) sats es condton (H7) wth the constant = K, for any K < 4. Proof. By (5.5) the kernel can be exressed as (s; t; y; x) = Q l fbt;s(x)>y xg H t;s (x) (5.7) = Q l f Bt;s>x yg H t;s (x) ; (5.8) where B t;s (x) = ' t;s (x) x. Snce B and B have the same dstrbuton, t s su cent to consder the exresson n (5.7) and assume that x y. Usng the trval bound for any a ; k > we obtan l fb>ag ex (K B ) (t s) ex (K a ) (t s) yj (s; t; y; x) e K jx (t s) V (s; t; x) ; where V (s; t; x) = Q ex (K B t;s(x) ) (t s) jh t;s (x)j : We only need to calculate jv (s; t; x)j. By Schwartz s nequalty jv (s; t; x)j ex (K B t;s(x) ) (t s) jh t;s (x)j : 9

21 Note that, f we x t and let s vary, B t;s (x) becomes a backward martngale wth quadratc varaton hb t; (x) s = t s = (t s): t g (' t;r (x); y) dy dr (' t;r (x))dr s Ths gves us that B t; (x) s a Brownan moton, and then, for any K < 4 K ex (t s) B t;s(x) = : (5.9) 4K On the other hand, t s known (see [3], roof of Prooston, (5.5)) that jh t;s (x)j C (t s) ; and now the roof s comlete. Lemma 5.3 The stochastc kernel (s; t; y; x) sats es condton (H8) wth the constant = K, for any K < 4. Proof. We exress D s;z (s; t; y; x) as n [3] as D s;z (s; t; y; h (s; t; y; x) g(y; (s; t; y; x) g(y; (s; t; y; (y; z) : Snce satsfy condton (H8) () and sats es the bound (H7), we only need (s; t; y; x) jx yj U (s; t; x) ex ; (5.) (t s) where jju (s; t; x)jj L r () C (t s). Now takng the (5.5) for and ntegratng by arts (s; t; y; x) = Q l fbt;s(x)>y xg Ht;s(x) nsde the formula where H t;s(x) = b D b ' b;s (x) kd b ' t;s (x)k H t;s(x) : The roof of Prooston n [3] ndcates that khb;s (x)k q C q (t s) for all q. Therefore, the estmates on ex K B t;s (x) (t s) from the roof of Lemma 5. yeld the lemma. Lemma 5.4 The stochastc kernel (s; t; y; x) sats es condton (H9) M for all M >.

22 Proof. By quaton (4.6) n [3] we know that (s; t; y; x) dx = q(s; t; y; x) dx h (s; r; y; z) q(r; t; z; x) dz dw r;y ; s where q(s; t; y; x) := (t s) Is t h ex jy xj 4 t s e Mjxj (s; t; y; x) dx = : Ths gves us that e Mjxj q(s; t; y; x) dx (s; r; y; z) e Mjxj q(r; t; z; x) dx dz =: T T : Notce that e Mjxj q(s; t; z; x) dx = e Mjzj t r = e Mjzj M M t r t r q(r; ; z; ) d : dw r;y (r; ; z; x) dx d e Mjxj q(r; ; z; x) dx d Fubn s stochastc theorem allows us then to wrte " T = g(z; # Is (s; r; y; z) e Mjzj dz dw dr;dy " t g(z; s (s; r; y; z) e Mjxj q(r; ; z; x) dz dw r;y #d " t g(z; # (s; r; y; z) q(r; ; z; ) dz dw r;y d s From (4.6) n [3] t follows that I s T = h Is t t s t s (s; r; y; z) e Mjzj dz h e Mjxj (s; ; y; x) q(s; ; y; x) h (s; ; y; ) q(s; ; y; ) d : dw r;y dx d

23 Usng ntegraton-by-arts formula t follows that h T = M g(z; y) (s; r; y; z) e Mjzj sg(z) dz Is t t s t s s h It s easy to show that for all (z; y) (s; r; y; z) e Mjzj dz h e Mjxj (s; ; y; x) q(s; ; y; x) h (s; ; y; ) q(s; ; y; ) d : dw r;y dw r;y dx d T s T j(s; ; y; )j d s C M;T e Mjyj : q(s; ; y; ) d e Mjxj (s; ; y; x) dx Then t follows that su tt I T s I T s e Mjxj (s; t; y; x) dx T C M;T (e Mjyj s C M;T e Mjyj ; C M;T ( (z; y) (s; r; y; z) e Mjzj dz dr dy ) g(z; y) (s; r; y; z) e Mjzj sg(z) dz dr dy (s; r; y; z) e Mjzj dz dr ) whch gves us (H9) M. Now the roof s comlete. 6 quvalence of evoluton and weak solutons Assume the notatons of Secton 5. By (4.5) n [3] we know that (s; t; y; x) s the fundamental soluton (n the varables t and x) of the equaton du t (t; x) dt (t; x) : (6.) Our urose n ths secton s to study the followng stochastc artal d erental equaton d u t (t; x) dt (t; x) F (t; x; u(t; W ; (6.) wth ntal condton u :. Let us ntroduce the followng de nton.

24 De nton 6. Let u = fu(t; x); (t; x) g be an adated rocess. We say that u s a weak soluton of (6.) f for every CK () and t [; T ] we have (x) u(t; x) dx = (x) u (x) dx (x) u(s; x) ds dx I t It (x) (x; y) dw s;y dx (6.3) (x) u(s; x) g(x; y) dw s;y dx (x) u(s; x) dw s;x : Now we have the followng result. Theorem 6. Under the hyotheses of Theorem 4.-), the soluton u = fu(t; x); (t; x) I T g of (.) s a weak soluton of (6.). Proof. Suose that u s the soluton of (.). Let fe k ; k g be a comlete ortonormal system n L (). For all m and (t; x) I T we de ne u m (t; x) = (; t; y; x) u (y) dy mx k= (s; t; z; x) F s (z) e k (z) dz e k (y) dw s;y ; (6.4) n where F s (z) := F (s; z; u(s; z)). The stochastc rocess u m (t; x) s well-de ned because o (s; t; z; x) F s(z) e k (z) dz e k (y) l (s; y) belongs to the doman of for each k. Ths roerty can be roved by the arguments used n the roofs of Theorems 3. and 3.4. By (4.8) n [3] we know that for all s < t T; x and f L () Ths gves us that (s; t; y; x) f(y) dy = f(x) t s t t u (t;x) = u (; r; y; x) u (y) dy (; r; y; x) u (y) dy v(dr; x) mx (s; x) e k (x) e k (y) dw s;y k= mx h t k= s mx h t k= (s; r; y; x) f(y) (s; r; y; x) f(y) dy v(dr; (s; r; z; x) F s(z) e k (z) dz dr e k (y) (s; r; z; x) F s(z) e k (z) dz v(dr; x) e k (y) dw s;y : 3

25 Let be a test functon n CK (). Usng ntegraton by arts formula and Fubn s theorem t s easy to obtan that u m (t; x) (x) dx = (x) u (x) dx (x) (; r; y; x) u (y) dy dr dx (x) (; r; y; x) u (y) dy v(dr; x) dx (x) (; r; y; x) u (y) dy dv v(dr; x) dx mx k= mx k= mx k= (x) (x) (x) F s (x) e k (x) dx e k (y) dw s;y " t " t I r I r # (s; r; z; x) F s (z) e k (z) dz e k (y) dw s;y dr dx # (s;r;z;x) F s(z) e k (z) dz e k (y) dw s;y v(dr;x) dx mx k= (x) " t I r # (s; r; z; x) F s (z) e k (z) dz e k (y) dw s;y dv v(dr; x) dx : Ths gves us that Notce that u m (t; x) (x) dx = mx k= (x) u (x) dx (x) F s (x) e k (x) dx e k (y) dw s;y (x) u m (r; x) dr dx It (x) u m (r; x) g(x; y) dw r;y I t It (x) u m (x; y) dw r;y dx dx : mx lm (x) F s (x) e k (x) dx e k (y) dw s;y (y) F s;y dw s;y m k= t X = lm (x) F s (x) e k (x) dx ds = : m k=m (6.5) 4

26 In order to comlete the roof t su ces to show that for any smooth and cylndrcal random varable G S we have lm G u m (t; x) (x) dx = G _ u(t; x) (x) dx ; m lm G (x) u m (r; x) dr dx = G (x) u(r; x) dr dx ; m lm G (x) u m (r; x) g(x; y) dw r;y dx m = G (x) u(r; x) g(x; y) dw r;y dx ; and lm G m = G It (x) u m (x; y) dw r;y dx It (x) (x; y) dw r;y dx : These convergences are easly checked usng the dualty relatonsh between the Skorohod ntegral and the dervatve oerator. eferences [] AlÚs,., LeÛn, J.A. and Nualart, D.: Stochastc heat equaton wth random coe - cents. Probab. Theory el.felds, to aear. [] AlÚs,. and Nualart, D.: An extenson of ItÙ s formula for antcatng rocesses. Journal of Theoretcal Probab., (998). [3] da Prato, G. and abczyk, J.: Stochastc equatons n n nte dmensons. Cambrdge Unversty Press, 99. [4] Fredman, A.: Partal d erental equatons of arabolc tye. Prentce-Hall, 964. [5] Hu, Y. and Nualart, D.: Contnuty of some antcatng ntegral rocesses. Statstcs and Probablty Letters 37, 3 (998). [6] Kfer, Y. and Kunta, H.: andom ostve semgrous and ther random n ntesmal generators. In: Stochastc Analyss and Alcatons (I.M. Daves, A. Truman, K.D. lworthy, eds.), World Scent c 996, [7] Kunta, H.: Stochastc ows and stochastc d erental equatons. Cambrdge Unversty Press, 99. 5

27 [8] Kunta, H.: Generalzed solutons of a stochastc artal d erental equaton. Journal of Theoretcal Probab. 7, (994). [9] LeÛn, J.A. and Nualart, D.: Stochastc evoluton equatons wth random generators. Ann. Probab. 6, (998). [] Mallavn, P.: Stochastc calculus of varatons and hyoelltc oerators. In: Proc. Inter. Sym. on Stoch. D. q., Kyoto 976, Wley, (978). [] Nualart, D.: The Mallavn Calculus and elated Tocs. Srnger-Verlag, 995. [] Nualart, D. and Pardoux,.: Stochastc calculus wth antcatng ntegrands. Probab. Theory el. Felds 78, (988). [3] Nualart, D. and Vens, F.: voluton equaton of a stochastc semgrou wth whtenose drft. Prernt. [4] Pardoux,. and Protter, Ph.: Two-sded stochastc ntegrals and calculus. Probab. Theory el. Felds 76, 5 5 (987). [5] Skorohod, A.V.: On a generalzaton of a stochastc ntegral. Theory Probab. Al., 9 33 (975). [6] Walsh, J.B.: An ntroducton to stochastc artal d erental equatons. Lecture Notes n Mathematcs 8, (984). [7] aka, M.: Some moment nequaltes for stochastc ntegrals and for solutons of stochastc d erental equatons. Israel J. Math. 5, 7 76 (967). 6