Equivalence and Hölder-Sobolev Regularity of Solutions for a Class of Non Autonomous Stochastic Partial Differential Equations

Transcription

1 Equivalence and Hölde-Sobolev Regulaity of Solutions fo a Class of Non Autonomous Stochastic Patial iffeential Equations Mata Sanz-Solé and Piee-A. Vuillemot Facultat de Matemàtiques, Univesitat de Bacelona UMR-CNRS 75, Univesité Heni-Poincaé, Nancy Abstact In this aticle we investigate a class of non autonomous, semilinea, stochastic patial diffeential equations defined on a smooth, bounded, convex domain of R d and diven by an infinite-dimensional noise; this noise is coloed elative to the space vaiable and white elative to the time vaiable. Unde an appopiate integability condition egading the covaiance opeato of the associated Wiene pocess, we intoduce thee notions of solution fo them and pove thei indistinguishability. We then pove the existence, the uniqueness and the pointwise boundedness of the moments, along with the spatial Sobolev egulaity and the joint space-time Hölde egulaity of such solutions. Moeove, we show how to weaken some equiements egading the covaiance opeato in ode to genealize the notions we alluded to above by intoducing a fouth type of solution, whose existence and egulaity popeties we also analyze in detail. Ou esults epesent a peliminay step towad the analysis of the suppot and the smoothness popeties of the coesponding laws. 1 Intoduction and Outline Solutions to cetain stochastic patial diffeential equations may be consideed as andom vaiables taking thei values in a suitable functional space. As such, thei laws ae identified with pobability measues on that space and it thus becomes natual to investigate the suppot and the smoothness popeties of these laws. Recently, many woks have been devoted to 1

2 these questions both in the hypebolic and in the paabolic case, paticulaly when the solutions ae jointly space-time Hölde continuous see, fo instance, [3], [37], [4], [41]. In this case, the functional space in question is typically a Banach space of Hölde continuous functions defined on some pat of Euclidean space. In the woks mentioned above, the deteministic pat of the equations is autonomous; moeove, the diving noise is eithe white elative to both the space- and the time-vaiable, o coloed elative to the space vaiable and white elative to the time vaiable; in addition, the space vaiable may vay ove the entie Euclidean space R d, which makes tools fom Fouie analysis eadily applicable to investigate existence and egulaity questions as these elate to the spatial coelations of the noise. One notable exception to this is the pape [3], in which the authos pove a suppot theoem in a Banach space of Hölde continuous functions fo the law of the solution to a one-dimensional, autonomous, semilinea, initial- Neumann bounday value poblem diven by a space-time white noise and defined on a bounded inteval. In this case, the authos analysis elies heavily on the existence of the coesponding paabolic Geen s function and on vey efined estimates fo it. In this aticle we investigate the indistinguishability and the joint spacetime Hölde continuity popeties of solutions to a class of non autonomous, semilinea, stochastic patial diffeential equations as a peliminay step towad the analysis of the suppot and the smoothness popeties of thei laws, this analysis being defeed to a sepaate publication. As we shall see, the complication in this case will stem fom the fact that the equations ae non autonomous, defined on a bounded domain of R d whee d is abitay, and fom the fact that thee ae a numbe of a pioi non equivalent possibilities to define a notion of solution fo them as is the case fo deteministic patial diffeential equations. We can define the class of poblems we shall investigate in the following way hee and below, we use the standad notations fo the usual Banach spaces of diffeentiable functions, of Hölde continuous functions and of Lebesgue integable functions defined on egions of Euclidean space: fo d N let R d be open, bounded, convex and assume that the bounday is of class C α fo some α, 1 see, fo instance, [17], [18], [31] and [47] fo a definition of this and elated concepts. Let C be a linea, self-adjoint, positive, non degeneate tace-class opeato in L ; this implies that C is an integal tansfom whose geneating kenel we denote by κ. In the sequel we wite e j j N fo an othonomal basis of L consisting of eigenfunctions of the opeato C and λ j j N fo the sequence of the coesponding eigenvalues. Let W., t t R a complete stochastic basis Ω, F, F t t R be an L -valued Wiene pocess defined on, P, stating at the oigin and

3 having the covaiance opeato tc. Recall that this means W., t R has independent Gaussian incements W., s t W., t of aveage zeo and covaiance opeato sc fo all s, t R, as well as continuous tajectoies almost suely. Moeove, witing.,. fo the usual scala poduct in L we have EW., s, v W., t, ˆv = s tcv, ˆv = s t dxdyκx, yvxˆvy 1 fo all s, t R and all v, ˆv L ; we also assume that W., t t R is F t t R -adapted and that the incements W., s t W., t ae F t -independent fo each s, t R. Finally, thee is anothe impotant popety of the Wiene pocess W., t R that we shall invoke below, namely, its Fouie decomposition W., t = λ 1 j e j.b j t in L whee B j t t R j N denotes a sequence of one-dimensional, independent, standad Bownian motions see, fo instance, [14]. Let T R and let us conside the following class of eal, paabolic, Itô initial-bounday value poblems: dux, t = divkx, t ux, t gux, tdt hux, tw x, dt, ux, = ϕx, x, ux, t nk x, t, T ], =, x, t, T ]. 3 In the peceding equations, the function k is matix-valued and the last elation stands fo the conomal deivative of u elative to k; moeove, we denote by n the unit oute nomal vecto to and we assume that the functions k and n satisfy the following hypothesis: K The enties of k satisfy the symmety elation k i,j. = k j,i. fo evey i, j {1,..., d}. Moeove, thee exists a constant β 1, 1] such that k i,j C α,β [, T ] fo each i, j and, in addition, we have k i,j,xl := C α, α [, T ] fo each i, j, l; futhemoe, thee exists a positive k i,j x l constant k such that the inequality k a kx, ta, a R d holds fo all 3

4 a R d and all x, t [, T ], whee. and.,. R d denote the Euclidean nom and the Euclidean scala poduct in R d, espectively. Finally, we have x, t d i=1 k 1α 1α, i,jx, tn i x C [, T ] fo each j and the conomal vecto-field x, t nkx, t := kx, tn is outwad pointing, nowhee tangent to fo evey t [, T ]. Regading the dift-nonlineaity g, the noise-nonlineaity h and the initial condition ϕ we have the following hypotheses, espectively: L The functions g, h : R R ae Lipschitz continuous. I We have ϕ C α ; moeove, ϕ satisfies the conomal bounday condition elative to k. Relations 3 define a class of non autonomous, semilinea, stochastic initialbounday value poblems diven by an infinite-dimensional noise which depends on both the space vaiable x and the time vaiable t. By vitue of 1, this noise is coloed with espect to x and white with espect to t, all popeties of its spatial coelations being completely encoded in the geneating kenel κ. Poblems of the fom 3 that involve a spatially coloed noise ae quite elevant to the mathematical analysis of a vaiety of physical pocesses in which the scale of the spatial coelations of the noise is much lage than that of its time coelations; paticula cases of them as well as thei deteministic countepats have been used ove the yeas to model, fo instance, cetain migation phenomena in population dynamics and population genetics see, fo instance, [4], [5], [9], [1], [11], [3], [49] and thei efeences. Futhemoe, thee ae many possible ways to define a notion of solution fo them and it is not a pioi evident to know which notions lead to indistinguishable, jointly space-time Hölde continuous pocesses. Accodingly, we shall oganize the emaining pat of this aticle in the following way: in Section, we state and discuss ou main esults concening indistinguishability, existence, uniqueness and Hölde egulaity, afte having intoduced fou notions of solution; two of these ae vaiational notions while the thid and fouth one involve a family of evolution opeatos defined though the deteministic, paabolic Geen s function associated with the pincipal pat of 3, whose existence and egulaity popeties ae ensued by Hypotheses K and I; afte having poved the equivalence of the fist thee notions in Section 3, we use the popeties of the Geen s function to pove the existence, the uniqueness, the pointwise boundedness of the moments and the joint space-time Hölde continuity of such solutions, as the fundamental heat kenel estimates fo the Geen s function tun out to be the most appopiate tools that allow us to do so. Ou poof of these popeties also shows that those solutions exhibit Sobolev egulaity in the space vai- 4

5 able, and in fact bings about the equivalence between two theoies hitheto unelated fo models as geneal as 3, namely, the vaiational theoy developed in [3], [4] and the Geen s function theoy initiated in [5]. In Section 3, we also show how to weaken some equiements concening the covaiance opeato C in ode to pove the existence and the egulaity of a solution of the fouth type, and establish an analogy between those weakened equiements and the so-called spectal measue conditions that have been intoduced ecently to analyze some classes of autonomous stochastic patial diffeential equations defined on the whole of R d see, fo instance, [1], [5], [33], [44], [46] and thei efeences. Finally, we efe the eade to [45] fo a shot announcement of the above and elated esults, and to [9], [35], [36], [39] and thei efeences fo othe ecent esults about existence, uniqueness and egulaity poved by completely diffeent methods. Statement and iscussion of the Main Results In the emaining pat of this aticle we wite. s fo L s -noms,. 1, fo the nom in the usual Sobolev space H 1 of functions on and C [, T ] ; L fo the space of all continuous mappings fom the inteval [, T ] into L endowed with the unifom topology. We wite c fo all ielevant, positive constants that occu in the vaious estimates unless we specify the constants othewise. The fist notion we intoduce is that of a vaiational solution tested with functions that depend only on the space vaiable. In addition to K, L and I above, this equies the following hypothesis egading the basis e j j N and the eigenvalues λ j j N of the opeato C: C We have e j L fo each j and λ j e j <. 4 Since we can ewite the eigenvalue equation Ce j = λ j e j as e j x = 1 dyκx, ye j y λ j fo almost evey x, and since e j = 1 fo each j, we can easily infe fom the peceding elation and fom Schwaz inequality that e j L 5

6 fo each j if we impose, fo instance, the integability condition x dy κx, y L. In this context, we emak that Hypothesis C defines a esticted set of tace-class covaiance opeatos since Condition 4 implies λ j := T C < by vitue of the existence of the continuous embedding L L. efinition 1. We say that the L -valued andom field u 1 ϕ., t t [,T ] defined on Ω, F, F t t [,T ], P is a vaiational solution of the fist kind to Poblem 3 if the following conditions hold: 1 u 1 ϕ., t t [,T ] is pogessively measuable on [, T ] Ω. We have u 1 ϕ L, T Ω; H 1 L Ω; C[, T ] ; L and consequently T E u 1 ϕ., τ 1, T u 1 = E ϕ., τ u 1 ϕ., τ < 5 as well as 3 The integal elation dxvxu 1 ϕx, t = E sup u 1 ϕ., t <. 6 t [,T ] dxvxϕx dx vx, kx, τ u 1 ϕx, τ R d dxvxgu 1 ϕx, τ dxvxhu 1 ϕx, τw x, 7 holds a.s. fo evey v H 1 and evey t [, T ], whee we have defined the stochastic integal by dxvxhu 1 ϕx, τw x, := λ 1 j v, hu 1 ϕ., τe j B j. 6

7 Fom the peceding definition and fom the above hypotheses, we easily infe that each tem in Equation 7 is well defined and finite a.s.; in paticula, ou definition of the stochastic integal with espect to W., t t R as an infinite sum of one-dimensional, independent Itô integals is based on the Fouie decomposition and epesents a eal-valued, squae integable andom vaiable. In ode to see this we invoke successively the isomety popety of Itô s integal, Schwaz inequality, Hölde s inequality between L 1 and L along with Hypothesis L to obtain E λ 1 j v, hu 1 ϕ., τe j B j v λ j E dx hu 1 ϕx, τ e j x c v λ j e j 1 sup E u 1 ϕ., t t [,T ] < 8 as a consequence of Hypothesis C and Relation 6. Vaiational solutions such as u 1 ϕ have been used in a numbe of situations see, fo instance, [4], [9], [1], [11] and the poof of thei existence and thei uniqueness fo poblems such as 3 can be taced to athe standad monotonicity and compactness aguments [3], [4], [43]. Relation 7, howeve, does not seem to be suitable fo the investigation of the joint Hölde continuity popeties of u 1 ϕ as it only defines this andom field implicitly. A peliminay step towad getting an explicit elation fo vaiational solutions in tems of the Geen s function associated with the pincipal pat of 3 can consist in testing them with functions that depend on both the space and the time vaiable. Fo evey t, T ], let us wite H 1, t fo the Sobolev space of all eal-valued functions v L, t that possess distibutional deivatives v xj L, t fo evey j {1,..., d}, along with a distibutional time-deivative v τ L, t. We denote the nom of H 1, t by v 1,,t =,t,t dx vx, τ d,t dx v xj x, τ dx v τ x, τ. 9 The following definition equies exactly the same fou hypotheses as above. 7

8 efinition. We say that the L -valued andom field u ϕ., t t [,T ] defined on Ω, F, F t t [,T ], P is a vaiational solution of the second kind to Poblem 3 if the fist two conditions of efinition 1 hold, and if the integal elation dxvx, tu ϕx, t = dxvx, ϕx dxv τ x, τu ϕx, τ dx vx, τ, kx, τ u ϕx, τ R d dxvx, τgu ϕx, τ dxvx, τhu ϕx, τw x, 1 holds a.s. fo evey v H 1, t and evey t [, T ], whee x vx, L and x vx, t L denote the Sobolev taces of v on and {τ R: τ = t}, espectively, and whee we have defined the stochastic integal as in efinition 1. Again, we see that evey tem in Equation 1 is well defined and finite a.s., and that the stuctue of 1 is identical to that of 7 up to the appeaance of the tem that involves the patial deivative v τ. It tuns out that these two notions of solution ae equivalent, which, togethe with the emak following 8, immediately implies the existence and the uniqueness of a vaiational solution of the second kind to 3; moe pecisely we have the following esult whose complete poof we give in Section active@pefix active@cha 3. Theoem 1. Assume that the above hypotheses hold; then, an L - valued andom field is a vaiational solution of the fist kind to 3 if, and only if, it is a vaiational solution of the second kind; in fact, thee exists a unique vaiational solution of the second kind to 3 and we have u 1 ϕ., t = u ϕ., t a.s. as equalities in L fo evey t [, T ]. We can actually pove Theoem 1 unde much weake conditions concening the egulaity of k and ϕ, but we shall efain fom doing so in view of the fact that Hypotheses K and I ae cucial egading the fomulation of the vaiational solutions in tems of the Geen s function G associated with the pincipal pat of 3. Recall that unde Hypotheses K and I, the function G : [, T ] [, T ] \ {s, t [, T ] : s t} R is continuous, 8

9 twice continuously diffeentiable in x, once continuously diffeentiable in t and satisfies the fundamental heat kenel estimates [ ] x µ t ν Gx, t; y, s ct s d µ ν x y exp c 11 t s whee µ = µ 1,..., µ d N d, ν N and µ ν with µ = d µ j see, fo instance, [17]. This allows us to define the following notion of mild solution to Poblem 3. efinition 3. We say that the L -valued andom field u 3 ϕ., t t [,T ] defined on Ω, F, F t t [,T ], P is a mild solution to Poblem 3 if the fist two conditions of efinition 1 hold, and if the elation u 3 ϕ., t = dyg., t; y, ϕy dyg., t; y, τgu 3 ϕy, τ dyg., t; y, τhu 3 ϕy, τw y, 1 holds a.s. fo evey t [, T ] as an equality in L, whee fo t = we have dyg., ; y, ϕy := lim t dyg., t; y, ϕy = ϕ. and whee we have defined the stochastic integal as above. The poof that each tem on the ight-hand side of 1 defines an L -valued function a.s. is complicated by the existence of the singulaity on the time-diagonal in G; fo the fist tem the statement follows fom the fact that ϕ is bounded and fom 11 fo µ = ν =, since the ight-hand side of 11 then extends to a Gaussian measue on R d, a fact that we shall use often in the sequel and efe to as the Gaussian popety of G. Fo the emaining pat of the agument we estict ouselves to the analysis of the stochastic tem; owing to the isomety popety of Itô s integal, Hypotheses C, L and the Gaussian popety we just alluded to, we fist have E λ 1 j Gx, t;., τ, hu 3 ϕ., τe j B j ce λ j e j E 1 dy Gx, t; y, τhu 3 ϕy, τ dy Gx, t; y, τ u 3 ϕy, τ 9

10 ce 1 dy Gx, t; y, τ u 3 ϕy, τ 13 fo evey x, whee we obtained the vey last estimate by applying Schwaz inequality elative to the finite measue dy Gx, t; y, τ on in ode to contol the singulaity of G. We then integate both sides of 13 with espect to x; though epeated applications of Fubini s theoem and by using the Gaussian popety once again along with 6 we obtain E dx c λ 1 j Gx, t;., τ, hu 3 ϕ., τe j B j 1 sup E u 3 ϕ., τ τ [,T ] <, which poves that x λ 1 t j Gx, t;., τ, hu 3 ϕ., τe j B j L a.s.. Ove the yeas, thee have been seveal esults in vaious contexts that establish elationships between diffeent kinds of vaiational solutions and thei mild fomulations, both in the deteministic and in the stochastic case see, fo instance, [], [8], [14], [16], [], [3], [5]. In paticula, the case of semilinea, non autonomous, stochastic evolution equations diven by semimatingales has been analyzed in [3] fom a vey abstact viewpoint. Howeve, none of the above woks has dealt with stochastic eaction-diffusion equations such as 3. Moeove, following [5], seveal notions of mild solutions that involve Geen s functions, Geen s distibutions o moe geneal semi-goup aguments have been used to investigate the existence and the egulaity popeties of solutions to seveal classes of hypebolic and paabolic stochastic patial diffeential equations see, fo instance, [6], [7], [1], [13], [14], [], [44], [46] and thei efeences. In this pespective, we next state a esult which, togethe with Theoem 1, establishes the existence and the uniqueness of a mild solution to 3. Theoem. Assume that the above hypotheses hold; then, an L - valued andom field is a vaiational solution of the second kind to 3 if, and only if, it is a mild solution; in fact, thee exists a unique mild solution to 3 and we have u ϕ., t = u 3 ϕ., t a.s. as equalities in L fo evey t [, T ]. As a consequence of Theoems 1 and, which pove the equivalence of the above thee definitions, it is fom now on legitimate to call solution to 1

11 Poblem 3 an L -valued andom field u ϕ., t t [,T ] that solves 3 in the sense of any of the thee notions we have intoduced. It tuns out that such a solution enjoys seveal impotant boundedness and egulaity popeties, as stated in the following esult. Theoem 3. Assume that the above hypotheses hold; then thee exists a unique solution to Poblem 3 such that x u ϕ x, t H 1 a.s. fo evey t [, T ], which satisfies the elation sup E u ϕ x, t < 14 x,t [,T ] fo evey [1,. Moeove, thee is a vesion of u ϕ x, t x,t [,T ] such that u ϕ.,. C β 1,β [, T ] a.s. fo evey β 1, α and evey β, α d. In the peceding statement we emak that both β 1 and β ae independent of the exponent β of Hypothesis K; moeove, β 1 is always independent of d, wheeas β depends explicitly on the dimension but only fo d 3; we shall see that the latte phenomenon is inheent in the pesence of the stochastic tem in 3. As testified by the many efeences we have quoted in this aticle, a significant pat of the ecent liteatue on stochastic patial diffeential equations is based on notions of mild solution which diffe fom ous in that they do not have a built-in equiement fo H 1 -egulaity. In ode to investigate this point in detail, we conclude this section by intoducing a fouth type of solution fo 3; we also state two existence and egulaity theoems fo it which hold unde conditions weake than C; the fist of these is the following: C d Thee exists s d, such that e j L s fo each j and λ j e j s <. 15 We emak that Hypothesis C implies Hypothesis C d. efinition 4. We say that the eal-valued andom field u 4 ϕx, t x,t [,T ] defined on Ω, F, F t t [,T ], P is a stong solution to Poblem 3 if the following conditions hold: 1 u 4 ϕ is pogessively measuable on [, T ] Ω. 11

12 We have sup x,t [,T ] E u 4 ϕx, t < fo evey [1,. 3 The elation u 4 ϕx, t = dygx, t; y, ϕy dygx, t; y, τgu 4 ϕy, τ dygx, t; y, τhu 4 ϕy, τw y, 16 holds a.s. fo evey x, t [, T ], whee G satisfies the same popeties as in efinition 3. We note the change of viewpoint in the peceding definition: we conside u 4 ϕ along with each tem on the ight-hand side of 16 as eal-valued andom fields indexed by x, t [, T ], and no longe as andom fields taking values in some functional space; futhemoe, we assume the boundedness of the moments fom the outset. Fom the peceding definition, it is then immediate that the fist two tems on the ight-hand side of 16 ae finite a.s.. The same is tue fo the stochastic tem by vitue of C d ; in ode to see this let s 1, d d be the dual exponent of s ; then, by using the isomety popety of Itô s integal, Hypothesis L, Schwaz inequality elative to the measue dy Gx, t; y, τ on, the Gaussian popety, of efinition 4 and Hölde s inequality, we obtain E c c λ 1 j Gx, t;., τhu 4 ϕ., τ, e j B j λ j E c c c λ j λ j e j s dy Gx, t; y, τe j y 1 u 4 ϕ y, τ dy Gx, t; y, τ e j y 1 dy Gx, t; y, τ s s t τ d d s dy t τ d exp [ c ] 1 s x y t τ t τ d d s < 17 1

13 since 1 d d s >. Wheeas Hypotheses K, L, I and C d allow us to pove the existence of a unique stong solution to 3, they do not suffice to imply the existence of a Hölde continuous vesion; fo this we need to stengthen C d in the following way: C d η Thee exist η, 1, s d 1 η, such that e j L s fo each j and Ou next esult is then the following. λ j e j s <. 18 Theoem 4. Assume that Hypotheses K, L, I and C d hold; then, thee exists a unique stong solution u 4 ϕx, t x,t [,T ] to 3. Moeove, if Hypothesis C d η holds, thee is a vesion of u 4 ϕx, t x,t [,T ] such that u 4 ϕ.,. C γ 1,γ [, T ] a.s. fo evey γ 1, α and evey γ, α d. η Finally, we note that we can weaken C d and C d η even futhe by intoducing the following two hypotheses, which now elate the covaiance opeato of the Wiene pocess to the diffeential opeato in the pincipal pat of active@pefix active@cha 3. H We have sup x,t [,T ] λ j dy Gx, t; y, τ e j y <. 19 H η Thee exists η, 1 such that sup x,t [,T ] t τ η λ j dy Gx, t; y, τ e j y <. Indeed, in the next section we show that C d implies H, that C d η implies H η and that we can still pove the existence and the Hölde egulaity of a stong solution to 3 unde Hypotheses 19 and ; howeve, this is at the expense of having to assume κx, y fo almost all x, y ; in fact, unde this additional estiction we notice that the thid tem on the ight-hand side of 16 is still finite a.s.: fom the isomety popety of Itô s 13

14 integal, Paseval s elation elative to the othonomal basis e j j N and the self-adjointness of C, we get E λ 1 j Gx, t;., τhu 4 ϕ., τ, e j B j E = E = E c λ j Gx, t;., τhu 4 ϕ., τ, e j C 1 Gx, t;., τhu 4 ϕ., τ, e j CGx, t;., τhu 4 ϕ., τ, Gx, t;., τhu 4 ϕ., τ dydz Gx, t; y, τ κy, z Gx, t; z, τ 1 sup E u 4 ϕy, τ < 1 y,τ [,T ] by vitue of Hypothesis L, Schwaz inequality applied to the expectation functional, of efinition 4 and the fact that we have dydz Gx, t; y, τ κy, z Gx, t; z, τ = λ j dy Gx, t; y, τ e j y < because of 19. The last esult of this section is then the following. Theoem 5. Assume that Hypotheses K, L, I, H hold and that κx, y fo almost all x, y ; then, thee exists a unique stong solution to 3. Moeove, if Hypothesis H η holds, thee is a vesion u 4 ϕx, t x,t [,T ] of this solution such that u 4 ϕ.,. C γ 1,γ [, T ] a.s. fo evey γ 1, α and evey γ, α d. η We devote the emaining pat of this aticle to poving the above five theoems; in paticula, we show that it is pecisely conditions 19 and that play a simila ôle in ou analysis of 3 as the spectal measue conditions we efeed to at the vey end of Section 1 play in the ecent woks we quoted thee. 14

15 3 Poof of the Main Results We begin by obseving that evey vaiational solution of the second kind to Poblem 3 is tivially a vaiational solution of the fist kind. Theefoe, we can educe the poof of Theoem 1 to that of the convese statement. Let p : [, T ] R be a polynomial in x and t, that is a finite sum of the fom px, t = µ,ν c µ,νx µ t ν whee c µ,ν R, whee µ and ν have the same meaning as in the peceding section, and whee x µ = x µ 1 1 xµ... xµ d d fo x = x 1,..., x d. Ou fist auxiliay esult towad the poof of Theoem 1 is the following. Poposition 1. Assume that the same hypotheses as in Theoem 1 hold and let u 1 ϕ., t t [,T ] be a vaiational solution of the fist kind to Poblem active@pefix active@cha 3. Then the integal elation dxpx, tu 1 ϕx, t = dxpx, ϕx dxp τ x, τu 1 ϕx, τ dx px, τ, kx, τ u 1 ϕx, τ R d dxpx, τgu 1 ϕx, τ λ 1 j p., τ, hu 1 ϕ., τe j holds a.s. fo evey polynomial p and evey t [, T ]. B j 3 The poof of the peceding poposition elies on seveal lemmas. Let us fist intoduce the anisotopic Sobolev space H 1,, T of all eal-valued functions v L, T that possess distibutional deivatives v xj L, T fo evey j {1,..., d}, whose nom we denote by v 1,,T ; =,T dx vx, τ d,t dx vxj x, τ. 4 While the H 1, t s ae the basic spaces of test functions fo vaiational solutions of the second kind, H 1,, T is the fundamental space in which the andom field u 1 ϕ lives since Relation 5 immediately implies that u 1 ϕ.,. H 1,, T a.s.. The peceding emak fist leads to the following integability popeties, whose poofs ae elementay and theefoe omitted. 15

16 Lemma 1. Assume that the same hypotheses as in Theoem 1 hold and let u 1 ϕ., t t [,T ] be as in Poposition 1. Then we have and x, τ vx, kx, τ u 1 ϕx, τ R d L 1, T a.s. fo evey v H 1. x, τ vxgu 1 ϕx, τ L 1, T The peceding lemma now leads to the following identity. Lemma. Assume that the same hypotheses as in Theoem 1 hold and let u 1 ϕ., t t [,T ] be as in Poposition 1. Then, fo any eal-valued function χ C 1 [, T ] satisfying χ =, the identity χ τ = χt dxvxu 1 ϕx, τ χτ χτ dxvxu 1 ϕx, τ dx vx, kx, τ u 1 ϕx, τ R d dxvxgu 1 ϕx, τ λ 1 j χτ v, hu 1 ϕ., τe j B j 5 holds a.s. fo evey v H 1 and evey t [, T ]. Poof. We may assume t > and then stat out fom Relation 7 at t = σ, multiply both sides by χ σ and integate with espect to σ on the inteval, t; we obtain dσχ σ = χt dxvxu 1 ϕx, σ dσχ σ dxvxϕx σ dx vx, kx, τ u 1 ϕx, τ R d 16

17 dσχ σ dσχ σ σ dxvxgu 1 ϕx, τ σ λ 1 j v, hu 1 ϕ., τe j B j 6 a.s. fo evey v H 1 and evey t [, T ]. Owing to the esult of Lemma 1 we may then integate by pats the second, thid and fouth tems on the ight-hand side of 6; in this way, by invoking Itô s fomula to handle the stochastic tem and by taking into account the fact that χ is non-andom and satisfies χ = we get χ τ = χt χt χt χt dxvxu 1 ϕx, τ dxvxϕx χτ χτ dx vx, kx, τ u 1 ϕx, τ R d dx vx, kx, τ u 1 ϕx, τ R d dxvxgu 1 ϕx, τ dxvxgu 1 ϕx, τ λ 1 j v, hu 1 ϕ., τe j B j χτ λ 1 j v, hu 1 ϕ., τe j B j 7 a.s. fo evey v H 1 and evey t [, T ]. We then goup togethe all tems containing χt and use Relation 7 once again to obtain 5. The peceding consideations now allow us to pove Relation 3. 17

18 Poof of Poposition 1. We fist split the polynomial p as px, t = px, p x, t := µ c µ, x µ c µ,ν x µ t ν 8 µ,ν ν and we obseve that x px, H 1 since is bounded. We may then choose v. = p., in Relation 7, so that we have dxpx, tu 1 ϕx, t = dxpx, u 1 ϕx, t dxp x, tu 1 ϕx, t = dxpx, ϕx dx px,, kx, τ u 1 ϕx, τ R d dxpx, gu 1 ϕx, τ λ 1 j p.,, hu 1 ϕ., τe j B j dxp x, tu 1 ϕx, t 9 a.s. fo evey t [, T ]. In ode to deal with the last tem of the peceding expession, we conside the tem that contains the patial deivative p τ in 3, which we ewite as = dxp τ x, τu 1 ϕx, τ = c µ,ν µ,ν ν dxp τ x, τu 1 ϕx, τ ντ ν 1 dxx µ u 1 ϕx, τ. 3 The next, cucial obsevation is that the integal contibution in the vey last tem of 3 is exactly equal to the left-hand side of 5 when we 18

19 choose τ χτ = τ ν and x vx = x µ thee. Since these two functions obviously satisfy the hypotheses of Lemma, we may then ewite the vey last tem of 3 by means of Relation 5 fo these choices of χ and v. By substituting the esulting expession in 3 and by esumming ove µ and ν we obtain dxp x, tu 1 ϕx, t = dxp τ x, τu 1 ϕx, τ dx p x, τ, kx, τ u 1 ϕx, τ R d dxp x, τgu 1 ϕx, τ λ 1 j p., τ, hu 1 ϕ., τe j B j 31 a.s. fo evey t [, T ]. We finally eplace the last tem on the ight-hand side of 9 by the ight-hand side of 31 and goup togethe all tems of the esulting expession by means of Relation 8. We can now easily extend the validity of Relation 3 by means of Weiestass appoximation theoem. Indeed, fo evey t, T ] let C 1 [, t] be the space of all eal, once continuously diffeentiable functions v defined on [, t], endowed with the C 1 -topology induced by the nom v C 1,t = max vx, τ x,τ [,t] d max x,τ [,t] v xj x, τ max v τ x, τ. 3 x,τ [,t] We then have the following esult. Poposition. Assume that the same hypotheses as in Theoem 1 hold and let u 1 ϕ., t t [,T ] be a vaiational solution of the fist kind to Poblem 3. Then the integal elation dxvx, tu 1 ϕx, t = dxvx, ϕx dxv τ x, τu 1 ϕx, τ dx vx, τ, kx, τ u 1 ϕx, τ R d 19

20 dxvx, τgu 1 ϕx, τ λ 1 j v., τ, hu 1 ϕ., τe j holds a.s. fo evey v C 1 [, t] and evey t [, T ]. B j 33 Poof. Relation 33 clealy holds fo t =, so that we may assume t >. Let v C 1 [, t]; on the one hand, by the classic Weiestass appoximation theoem, thee exists a sequence of polynomials p n n N such that the estimate v p n C 1,t < 1 34 n holds fo evey n N see, fo instance, [8]. On the othe hand, we have dxp n x, tu 1 ϕx, t = dxp n x, ϕx dxp n,τ x, τu 1 ϕx, τ dx p n x, τ, kx, τ u 1 ϕx, τ R d dxp n x, τgu 1 ϕx, τ λ 1 j pn., τ, hu 1 ϕ., τe j B j 35 a.s. fo evey n N and evey t [, T ] by the statement of Poposition active@pefix active@cha 1. We now show that Relations 34 and 35 imply Relation 33. The convegence of the tems of the fist line in 35 towad the coesponding tems of 33 as n is tivial. As fo the gadient tem we have, owing to Schwaz inequality in R d, Relation 34, the boundedness of the coefficients k i,j on [, T ] and the definition of the nom 4, the sequence of estimates dx vx, τ p n x, τ, kx, τ u 1 ϕx, τ R d dx vx, τ p n x, τ kx, τ u 1 ϕx, τ

21 c v p n C 1,t c n d i, dx k i,j x, τu 1 ϕ,x j x, τ u 1 ϕ.,. 1,,T ; 36 a.s. as n. In a simila way we have dx vx, τ p n x, τ gu 1 ϕx, τ c 1 u 1 n ϕ.,. 1,,T ; 37 a.s. as n since g is Lipschitz continuous. It emains to investigate the convegence of the stochastic integals in 35. Moe specifically, we wish to show that lim l λ 1 j v., τ p nl., τ, hu 1 ϕ., τe j B j = 38 a.s. fo evey t [, T ] along a suitable subsequence of polynomials p nl l N. In ode to achieve this it is sufficient to pove that lim E n λ 1 j v., τ p n., τ, hu 1 ϕ., τe j B j = 39 fo evey t [, T ]. Using successively the isomety popety of Itô s integal, the definition of the nom 3, Schwaz inequality and the fact that h is Lipschitz continuous we obtain E λ 1 j v., τ p n., τ, hu 1 ϕ., τe j B j λ j E c v p n C 1,t v., τ p n., τ, hu 1 ϕ., τe j c n T C 1 E T T λ j E dx1 u 1 ϕx, τ u 1 ϕ., τ 1

22 as n, because of 34, 5 and the fact that e j = 1 fo each j N. This poves 39 and hence 38, so that the above emaks along with Relations 36, 37, and 38 pove Relation 33. The above consideations now lead to the following. Poof of Theoem 1. Let u 1 ϕ., t t [,T ] be a vaiational solution of the fist kind, fix t > and let v H 1, t; since the base of the cylinde, t is smooth, thee exists a sequence v n n N C 1 [, t] such that the estimate v v n 1,,t 1 4 n holds fo evey n N see, fo instance, [38]. Futhemoe, we have dxv n x, tu 1 ϕx, t = dxv n x, ϕx dxv n,τ x, τu 1 ϕx, τ dx v n x, τ, kx, τ u 1 ϕx, τ R d dxv n x, τgu 1 ϕx, τ λ 1 j vn., τ, hu 1 ϕ., τe j B j 41 a.s. fo evey n N and evey t [, T ], by the statement of Poposition active@pefix active@cha. We can now ensue the convegence of each tem of the fist line in 41 towad the coesponding tem in 33 by means of standad Sobolev tace-inequalities, while we can handle the thid and fouth tem on the ight-hand side of 41 exactly as we did in the poof of Poposition. Regading the convegence of the stochastic integals, we have to ague slightly diffeently than we did to establish Relation 38 in ode to etieve the appopiate nom; in fact, it is sufficient to poceed exactly as we did to establish Relation 8; this gives E λ 1 j v., τ v n., τ, hu 1 ϕ., τe j B j c v v n 1,,t λ j e j 1 sup E u 1 ϕ., t t [,T ]

23 c n λ j e j 1 sup E u 1 ϕ., t t [,T ] as n by vitue of 4, 6 and 4. This poves that an appopiate subsequence of the stochastic integals in 41 conveges to the stochastic integal in 33 a.s. fo each t [, T ], theeby completing the poof of Relation 33 fo v H 1, t; fom this and the standad existence and uniqueness esults fo vaiational solutions of the fist kind [3], [4], we can conclude that thee exists a unique vaiational solution of the second kind to 3 such that u 1 ϕ., t = u ϕ., t a.s. as equalities in L fo evey t [, T ]. We now tun to the poof of Theoem, which will equie one pepaatoy esult. Let q: [, T ] H 1 H 1 R be the symmetic quadatic fom defined by q t; v, ˆv = dxkx, t vx, ˆvx R d and set q t; v = q t; v, v. Fom this definition and Hypothesis K, we infe in paticula that the Hölde continuity estimate qs; v qt; v c s t β qt; v 4 holds fo all s, t [, T ] and evey v H 1, whee β 1, 1]. Fom 4, the unifom ellipticity of 3 and the geneal theoy of linea paabolic equations see, fo instance, [1], [6], [34], [48], we conclude that thee exists a two-paamete family of evolution opeatos Ut; s s t T in L associated with the pincipal pat of 3 given by Ut, sv = { v if s = t dyg., t; y, svy if s < t 43 whee G is the Geen s function that entes Relation 1. We also infe fom the fist epesentation theoem fo foms [7], o fom the geneal consideations of [34], that thee exists a self-adjoint, positive ealization At = divk., t of the elliptic patial diffeential opeato with conomal bounday conditions in the pincipal pat of 3; this opeato geneates the family Ut; s s t T, and its self-adjointness domain in L is At = { v H 1 : Atv L, Atv, ˆv = qt; v, ˆv } 44 3

24 fo evey ˆv H 1. We note that the self-adjointness of At implies the self-adjointness of each one of the Ut, s see, fo instance, [1], [48], which in tun implies that the Geen s function G is symmetic in its space vaiables, a fact we shall use fequently in the sequel. The pepaatoy esult we alluded to above is cental to the poof of Theoem ; it shows that we can cancel out two tems in Relation 1 povided we choose an appopiate class of test functions thee, which we constuct fom the opeatos Ut; s s t T. We wite C fo the space of all eal-valued, twice continuously diffeentiable functions with compact suppot in. Lemma 3. Assume that the same hypotheses as in Theoem hold and let u ϕ., t t [,T ] be a vaiational solution of the second kind to Poblem 3. Fo evey v C, define v t., s = Ut, sv fo all s, t [, T ] such that s t. Then v t H 1, t fo evey t, T ] and the elation dxvxu ϕx, t = holds a.s. fo evey t [, T ]. dxv t x, ϕx λ 1 j v t., τ, hu ϕ., τe j dxv t x, τgu ϕx, τ B j 45 Poof. The symmety of G and Relation 43 imply that { vx if s = t v t x, s = dygy, t; x, svy if s < t 46 fo evey x. Moeove, fo s < t the function G is twice continuously diffeentiable in x, once continuously diffeentiable in s and is a classical solution to the bounday-value poblem G s y, t; x, s = divkx, s x Gy, t; x, s, x, s, T ], Gy, t; x, s nk =, x, s, T ] 47 see, fo instance, [17] o [19]. Fom these consideations, the fact that G satisfies the heat-kenel estimates 11 and fom Gauss divegence theoem, we easily infe that v t x, s = dy x Gy, t; x, svy 48 4

25 along with vsx, t s = dyg s y, t; x, svy = dygy, t; x, s divky, s vy 49 and we have v t H 1, t; we may then choose v t as a test function in 1, which shows that 45 holds if, and only if, the elation = dxv t τ x, τu ϕx, τ dx v t x, τ, kx, τ u ϕx, τ R d 5 holds a.s. fo evey t [, T ]. In ode to pove 5 we assume t >, choose ɛ > sufficiently small and fist show that we have ɛ = ɛ dxv t τ x, τu ϕx, τ dx v t x, τ, kx, τ u ϕx, τ R d 51 a.s.. Fom Relation 11 and fo a fixed τ [, t ɛ] we fist have x, y G τ y, t; x, τvyu ϕx, τ L 1 a.s. as a consequence of the integability popeties of v and u ϕ., τ. Theefoe, by invoking successively the fist equality in 49, the fist equation in 47 and Fubini s theoem, we obtain ɛ dxvτ t x, τu ϕx, τ ɛ = dy dx divkx, τ x Gy, t; x, τu ϕx, τ vy 5 a.s.. Futhemoe, Relation 11 also implies that d k i,j., t G xj., t; y, s H 1 fo evey i {1,..., d} since the k i,j s and the k i,j,xl s ae bounded fom above on, T because of Hypothesis K. This popety along with the fact that u ϕ., τ H 1 a.s. allows us to tansfom the integal between the paentheses of 5 by using Gauss divegence theoem, the second equation in 47 along with the fact that kx, τ is a 5

26 symmetic matix; we get ɛ dxvτ t x, τu ϕx, τ = = ɛ ɛ dy dx x Gy, t; x, τvy, kx, τ u ϕx, τ R d dx v t x, τ, kx, τ u ϕx, τ R d a.s. as a consequence of 48 and Fubini s theoem, which is the desied assetion. It emains to investigate the limit ɛ in 51. Regading the convegence of the left-hand side of that expession we have successively ɛ E dxvτ t x, τu ϕx, τ dxvτ t x, τu ϕx, τ c t E t ɛ u ϕ., τ 1 c t E 1 u ϕ., τ t ɛ c t ɛe 1 u ϕ., τ sup τ [,T ] < by vitue of the boundedness of vτ t in, t; as fo the coesponding estimate of the ight-hand side of 51 we get E dx v t x, τ, kx, τ u ϕx, τ R d c t ɛ d T t ɛ dx vx t j x, τ 1 E 1 u ϕ., τ 1, as a consequence of Schwaz inequality and 5, so that the peceding expession goes to zeo as ɛ by vitue of the absolute continuity of the Lebesgue integal in the second to last facto. Theefoe, thee exists a sequence ɛ n n N R conveging to zeo such that the two elations ɛn lim dxvτ t x, τu ϕx, τ = n 6 dxv t τ x, τu ϕx, τ 53

27 and ɛn lim n = dx v t x, τ, kx, τ u ϕx, τ R d dx v t x, τ, kx, τ u ϕx, τ R d 54 hold a.s.. It is now plain that 51, 53 and 54 imply 5. The peceding consideations now lead to the following. Poof of Theoem. By substituting 46 into 45, by applying the deteministic and stochastic vesions of Fubini s theoem to the esulting expession and by egouping tems we get dxvx u ϕx, t dygx, t; y, ϕy = dxvx dxvx dygx, t; y, τgu ϕy, τ dygx, t; y, τhu ϕy, τw y, a.s. fo evey v C and evey t [, T ]; fom this, we infe that u ϕ., t dyg., t; y, ϕy dyg., t; y, τgu ϕy, τ dyg., t; y, τhu ϕy, τw y, is othogonal to C a.s. fo evey t [, T ], so that u ϕ., t t [,T ] is a mild solution to 3 since C is dense in L. Convesely, let u 3 ϕ., t t [,T ] be a mild solution to 3; then, both u ϕ., t t [,T ] and u 3 ϕ., t t [,T ] satisfy 1, so that we get E u ϕx, t u 3 ϕx, t ce dygx, t; y, τ gu ϕy, τ gu 3 ϕy, τ ce dygx, t; y, τ hu ϕy, τ hu 3 ϕy, τ W y, c dy Gx, t; y, τ E u ϕy, τ u 3 ϕy, τ 7

28 by using techniques simila to the ones above. By integating the peceding inequality with espect to x, by applying Fubini s theoem and by invoking the Gaussian popety fo G, we obtain E u ϕ., t u 3 ϕ., t c dye u ϕ y, τ u 3 ϕy, τ = c E u ϕ., τ u 3 ϕ., τ 55 fo evey t [, T ]. We now notice that τ E u ϕ., τ u 3 ϕ., τ L 1, t by vitue of 6, so that fom 55 and Gonwall s inequality we can conclude that u ϕ., t = u 3 ϕ., t a.s. in L fo evey t [, T ] since u ϕ, u 3 ϕ L Ω; C[, T ] ; L. Theefoe, evey mild solution to 3 is a vaiational solution of the second kind, and thee exists a unique such mild solution to 3. We now tun to the poof of Theoem 3, which will equie seveal pepaatoy esults as it is not a pioi evident that the above andom fields should also satisfy 14 along with joint Hölde egulaity popeties in x, t; in fact, we will need quite a few additional aguments to show that thee exists a vesion of u 1 ϕ., t t [,T ] with these popeties. In ou next esult we pove the existence of a pogessively measuable, eal-valued pocess u ϕ x, t x,t [,T ] that satisfies 1 along with 14, though a suitable fixed point agument. Fom now on we take [, without esticting the geneality, and define B as the eal Banach space consisting of all eal-valued equivalence classes of pocesses u indexed by x, t [, T ], pogessively measuable on [, T ] Ω, endowed with the usual pointwise opeations and the nom u 1 sup E ux, t x,t [,T ] <. 56 Let M ϕ : B B be the map induced by 1, that is M ϕ ux, t = dygx, t; y, ϕy dygx, t; y, τguy, τ a.s.. Regading M ϕ we have the following esult. dygx, t; y, τhuy, τw y, 57 8

29 Poposition 3. Assume that the same hypotheses as in Theoem 3 hold; then M ϕ possesses a unique fixed point u ϕ in B fo evey [,. Poof. We begin by showing that M ϕ is indeed well defined on B. Owing to the boundedness of ϕ and the Gaussian popety fo G we fist get E M ϕ ux, t c 1 E E dygx, t; y, τguy, τ dygx, t; y, τhuy, τw y,. 58 Futhemoe, as a consequence of Hypothesis L, the Gaussian popety and Hölde s inequality elative to the finite measue dy Gx, t; y, τ on, t, we can estimate the fist expectation in 58 as E dygx, t; y, τguy, τ c 1 E dy Gx, t; y, τ uy, τ c 1 c dy Gx, t; y, τ E uy, τ 1 sup E uy, τ < 59 y,τ [,T ] since u B. In ode to obtain a simila estimate fo the second expectation in 58, we invoke successively the definition of the stochastic integal, Bukholde s inequality and Hölde s inequality elative to the measue on the inteval, t; we get E dygx, t; y, τhuy, τw y, ce ce λ j dygx, t; y, τhuy, τe j y λ j dygx, t; y, τhuy, τe j y 9

30 ce c 1 c 1 dy Gx, t; y, τ uy, τ dy Gx, t; y, τ E uy, τ 1 sup E uy, τ < 6 y,τ [,T ] whee we have also used Hypotheses L and C along with Hölde s inequality elative to the finite measue dy Gx, t; y, τ on. Fom 58, 59 and 6 we infe that sup x,t [,T ] E M ϕ ux, t <, so that M ϕ u B. Now let u, u B ; then, fom 57 we have M ϕ ux, t M ϕ u x, t = dygx, t; y, τ guy, τ gu y, τ dygx, t; y, τ huy, τ hu y, τ W y, a.s., so that the Lipschitz popeties of g and h along with aguments entiely simila to those leading to 59 and 6 give E M ϕ ux, t M ϕ u x, t c dy Gx, t; y, τ E uy, τ u y, τ c sup E uy, τ u y, τ y fo evey x, t [, T ]. The peceding elation along with standad consideations now show that the N th iteate M ϕ N of M ϕ is a contaction in B fo N sufficiently lage. It is woth stessing the fact that the peceding constuction does not imply u ϕ should exhibit any Sobolev egulaity in x o any continuity in t as u 1 ϕ., t t [,T ] does, so that the peceding esult does not yet pove that u ϕ is a solution to 3; in fact, thus fa the vaiables x, t [, T ] meely index u ϕ but we shall show below that u 1 ϕ., t t [,T ] and u ϕ x, t x,t [,T ] ae actually indistinguishable; fo the time being we pove a seies of esults that will lead us to the existence of a jointly Hölde continuous vesion of u ϕ. Fo this we also use Relation 1, each tem of which we investigate sepaately; we begin with the following poposition, 3

31 which is an immediate consequence of the theoy developed in [17] as the fist tem of 1 is a classical solution to 3 when g = h =. Poposition 4. Assume that Hypotheses K and I hold; then we have x, t dygx, t; y, ϕy Cα, α [, T ]. We next tun to the analysis of the second tem in 1 fo which we have the following esult. Poposition 5. Assume that the same hypotheses as in Theoem 3 hold and let u ϕ x, t x,t [,T ] be the andom field of Poposition 3; then we have E dy Gx 1, t; y, τ Gx, t; y, τgu ϕ y, τ c x 1 x 61 fo all x 1, x unifomly in t [, T ], along with 1 E dygx, t 1 ; y, τgu ϕ y, τ dygx, t ; y, τgu ϕ y, τ c t 1 t γ 6 fo all t 1, t [, T ] unifomly in x, fo evey γ, 1 and evey [,. Poof. Since is convex, we fist invoke the mean-value theoem fo G along with estimate 11 ] and Hypothesis L; since the measue dyt τ d1 exp [ c x y t τ is finite on, t by vitue of the Gaussian popety, we obtain E dy Gx 1, t; y, τ Gx, t; y, τgu ϕ y, τ [ ] ce dyt τ d1 exp c x y 1 u ϕ y, τ x 1 x t τ [ ] ce 1 dyt τ d1 exp c x y u ϕ y, τ x 1 x t τ 31

32 whee x belongs to the segment connecting x 1 and x ; consequently, by applying Hölde s inequality fo this measue to the last integal and by taking 56 into account, we get Relation 61. Without esticting the geneality we now choose σ > sufficiently small and set t 1 = t σ and t = t in ode to pove 6; fo the left-hand side of 6 we fist get the uppe bounds E dy Gx, t σ; y, τ Gx, t; y, τgu ϕ y, τ σ E dygx, t σ; y, τgu ϕ y, τ t E dy Gx, t σ; y, τ Gx, t; y, τgu ϕ y, τ c 1 sup E u ϕ y, τ σ 63 y,τ [,T ] whee we have used the Gaussian popety fo G, Hypothesis L along with 56 to obtain the last inequality. It emains to estimate the integal involving the time-incement of G; the fist pat of the agument is essentially simila to what we just did and we obtain E dy Gx, t σ; y, τ Gx, t; y, τgu ϕ y, τ c 1 sup E u ϕ y, τ y,τ [,T ] dy Gx, t σ; y, τ Gx, t; y, τ. 64 But we now have to poceed diffeently than we did to establish 61 in ode d to contol the singulaity on the time-diagonal of G. Let γ d ;, 1 then, by invoking successively the mean-value theoem, 11 and the Gaussian popety, we can estimate the last integal in 64 as dy Gx, t σ; y, τ Gx, t; y, τ c dy Gx, t σ; y, τ Gx, t; y, τ 1 γ G t x, t ; y, τ γ σ γ c t τ d 1 γ t τ d γ d 3

33 c c dyt τ d exp [ c ] x y t σ γ τ t τ d 1 γ t τ d γ d t τ γ σ γ cσ γ σ γ since σ >, t t, t σ, d γ d < and d d 1 γ γ d = γ. The substitution of the peceding estimate into 64 along with 63 lead to d Relation 6 fo evey γ d,, 1 and a fotioi fo evey γ, 1. Finally, egading the thid tem in 1 we have the following esult in which the dimension d does impose a estiction on one of the estimates fo the fist time. Poposition 6. Assume that the same hypotheses as in Theoem 3 hold and let u ϕ x, t x,t [,T ] be the andom field of Poposition 3; then we have E dy Gx 1, t; y, τ Gx, t; y, τhu ϕ y, τw y, c x 1 x γ 65 fo all x 1, x unifomly in t [, T ], fo evey γ, 1 and evey [, ; moeove, we have 1 E dygx, t 1 ; y, τhu ϕ y, τw y, dygx, t ; y, τhu ϕ y, τw y, c t 1 t γ 66 fo all t 1, t [, T ] unifomly in x, fo evey γ, 1 d and evey [,. The poof of this poposition is much moe elaboate than that of Poposition 5 and elies on two lemmas; it is based on an extension of the so-called factoization method, which was oiginally intoduced in [15] to deal with egulaity questions concening autonomous, linea stochastic patial diffeential equations; the method povides a way to expess the stochastic 33

34 integal in 1 by means of an auxiliay andom field whose moments ae unifomly bounded in some sense. Fo evey δ, 1, we define Yδ by Y δ x, t = t τ δ dygx, t; y, τhu ϕ y, τw y, 67 fo evey x and evey t [, T ]; as befoe, we can easily pove that this expession is well-defined and finite a.s.; in fact, this is a tivial consequence of the boundedness popety we just alluded to, which we descibe in the following esult. Lemma 4. Assume that the same hypotheses as in Theoem 3 hold; then we have sup x,t [,T ] E Y δ x, t < 68 fo evey [,. Poof. We show that 68 is a diect consequence of 56 fo u ϕ. Owing to Bukholde s inequality and to Hypotheses C and L, we fist get the estimates E Y δ x, t c λ j e j E t τ δ ce t τ δ dy Gx, t; y, τhu ϕ y, τ dy Gx, t; y, τ 1 u ϕ y, τ. Futhemoe, the measue t τ δ is finite on, t since we have δ, 1 ; by applying Hölde s inequality fo this measue to the last integal and by invoking the Gaussian popety of G, we obtain E Y δ x, t ce ce c t τ δ dy Gx, t; y, τ 1 u ϕ y, τ t τ 1 δ dy Gx, t; y, τ u ϕ y, τ 1 sup E u ϕ y, τ < y,τ [,T ] 34

35 unifomly in x, t [, T ], whee we used Hölde s inequality elative to the finite measue dy Gx, t; y, τ on along with Relation 56 fo u ϕ. Popety 68 along with the following elation between the stochastic integal in 1 and Y δ will be cucial to ou poof of Poposition 6. Lemma 5. Assume that the same hypotheses as in Theoem 3 hold; then the elation dygx, t; y, τhu ϕ y, τw y, = sinδπ π t τ δ 1 dygx, t; y, τy δ y, τ 69 holds a.s. fo evey δ, 1 and evey x, t [, T ]. Poof. Let σ, τ, t [, T ] such that σ < τ < t; then the evolution opeatos 43 satisfy the fundamental popety Ut, τuτ, σ = Ut, σ; equivalently, we have Gx, t; z, σ = dygx, t; y, τgy, τ; z, σ 7 fo the coesponding Geen s function, fo all x, z. Relation 69 then follows fom the substitution of 67 into 69, the deteministic and stochastic vesions of Fubini s theoem, Relation 7 and the identity σ t τ δ 1 τ σ δ = π sinδπ. The peceding esults now lead to the following. Poof of Poposition 6. In ode to pove 65, we have to ague diffeently than we did to pove 61 because of the singula facto t τ δ 1 in 69; owing to the Gaussian popety fo G, we fist notice that the measue dy t τ δ 1 Gx 1, t; y, τ Gx, t; y, τ is finite on, t; then, by using successively 69, Hölde s inequality elative to this measue along with 68 we get E dy Gx 1, t; y, τ Gx, t; y, τhu ϕ y, τw y, c sup E Y δ y, τ y,τ [,T ] t τ δ 1 dy Gx 1, t; y, τ Gx, t; y, τ

36 Let γ, 1; by using the Gaussian popety we can estimate the last integal in the peceding expession as c t τ δ 1 t τ δ 1 dy Gx 1, t; y, τ Gx, t; y, τ dy Gx 1, t; y, τ Gx, t; y, τ 1 γ Gx 1, t; y, τ Gx, t; y, τ γ t τ δ 1 d 1 γ dy Gx 1, t; y, τ Gx, t; y, τ γ. 7 In ode to contol the space-incement of G in the last line of 7 we now choose δ γ, 1, and then use successively the mean-value theoem along with 11 and the Gaussian popety; witing again x fo a point on the segment between x 1 and x we obtain fom 7 the estimates t τ δ 1 dy Gx 1, t; y, τ Gx, t; y, τ c t τ δ 1 d 1 γ [ ] dyt τ d1 γ exp c x y x 1 x γ t τ [ = c t τ δ 1 γ dyt τ d exp c t τ δ 1 γ x 1 x γ c x 1 x γ ] c x y t τ x 1 x γ since δ γ >. The substitution of the peceding estimate into 71 poves 65. We now show that 66 holds by choosing again σ > sufficiently small, t 1 = t σ and t = t; owing to 69 we can fist bound the left-hand side of 66 fom above by E t σ τ δ 1 dygx, t σ; y, τy δ y, τ t τ δ 1 dygx, t; y, τy δ y, τ 36

37 E σ t σ τ δ 1 t dygx, t σ; y, τy δ y, τ 73 and we poceed by investigating each tem of 73 sepaately. Regading the second tem, we notice by aguing as befoe that the measue dyt σ τ δ 1 Gx, t σ; y, τ is finite on t, tσ; then, letting γ, 1, δ γ, 1 and using Hölde s inequality elative to this measue along with 68 and the Gaussian popety, we obtain E σ t t σ τ δ 1 sup E Y δ y, τ y,τ [,T ] σ t σ τ δ 1 c t σ t dygx, t σ; y, τy δ y, τ dy Gx, t σ; y, τ t σ τ δ 1 cσ δ cσ γ. 74 The analysis of the fist tem in 73 is moe complicated; the fist pat of the agument is simila to what we just did to deive the fist inequality in 74; this emak and the Gaussian popety lead to E t σ τ δ 1 dygx, t σ; y, τy δ y, τ t τ δ 1 dygx, t; y, τy δ y, τ sup E Y δ y, τ y,τ [,T ] dy t σ τ δ 1 Gx, t σ; y, τ t τ δ 1 Gx, t; y, τ c t σ τ δ 1 t τ δ 1 c t σ τ δ 1 dy Gx, t σ; y, τ Gx, t; y, τ. 75 On the one hand, in ode to contol the incement of the line befoe last in 75, let us choose γ, 1 and δ γ, 1 again; fom the mean-value 37