Journal of Mathematical Analysis and Applications. Stochastic representation of partial differential inclusions

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1 J. Math. Anal. Appl Content lit available at ScienceDiect Jounal of Mathematical Analyi and Application Stochatic epeentation of patial diffeential incluion Michał Kiielewicz Faculty of Mathematic, Compute Science and Econometic, Univeity of Zielona Góa, Podgóna 5, Zielona Góa, Poland aticle info abtact Aticle hitoy: Received 2 Octobe 27 Available online 24 Decembe 28 Submitted by H. Fanowa Keywod: Patial diffeential incluion Diffuion pocee Exitence and epeentation theoem Patial diffeential incluion of the fom u t L FGu + cu and ψ L FG u + cu, wheel FG i a unifomly paabolic econd ode et-valued opeato, ae conideed. In paticula, baing on diffuion popetie of wea olution to tochatic diffeential incluion, ome exitence and epeentation theoem fo olution of uch type patial diffeential incluion ae given. 28 Elevie Inc. All ight eeved. 1. Intoduction The peent pape deal with ome patial diffeential incluion that ae invetigated by tochatic method connected with diffuion popetie of wea olution to tochatic diffeential incluion conideed in the autho pape 9. In paticula, the pape contain futhe much moe pecie idea dealing with connection between wea olution of tochatic diffeential incluion and olution to ome patial diffeential incluion. In the lat yea ome popetie of patial diffeential incluion have been invetigated by G. Batuzel and A. Fyzowi ee 1 3 a application of ome geneal method of abtact diffeential incluion. Patial diffeential incluion conideed by G. Batuzel and A. Fyzowi have the fom Du F u, whee D i a patial diffeential opeato and F i a given lowe emicontinuou l..c. multifunction. In the peent pape we conide patial diffeential incluion of the fom u t t, x L FGut, x + ct, xut, x and ψt, x L FG ut, x + ct, xut, x, whee c and ψ ae given continuou function, u denote an unnown function and L FG i a et-valued patial diffeential opeato geneated by given l..c. et-valued mapping F and G. Patial diffeential incluion, conideed in the pape ae invetigated togethe with ome initial and bounday condition. Solution to uch initial and bounday valued poblem ae chaacteied by wea olution to tochatic diffeential incluion. Thi appoach lead to natual method of olving ome optimal contol poblem fo ytem decibed by patial diffeential equation depending on contol paamete. Thee method ae conequence of the wea compactne with epect to ditibution of the et of ditibution of all wea olution to tochatic diffeential incluion ee 8 and 9. In what follow we hall denote by C 1,2 Rn+1 and C 2 Rn the pace of all function h C 1,2 R n+1, R and h C 2 R n, R, epectively with compact uppot. By D and U we denote the boundaie of given domain D R n and U R n+1, epectively. By P F we hall denote a complete filteed pobability pace Ω, F, F, P with a filtation F = F t t< atifying the uual condition, i.e., uch that F contain all A F uch that PA = and F t = ε> F t+ε fo t,. We hall deal with et-valued mapping F :, R n ClR n and G :, R n ClR n m, whee ClR n and ClR n m denote the pace of all nonempty cloed ubet of R n and R n m,epectively. Given the above et-valued mapping F and G we hall denote by CF and CG et of all continuou electo f :, R n R n and g :, R n R n m,off and G, epectively. Fo a given n-dimenional continuou tochatic adde: M.Kiielewicz@wmie.uz.zgoa.pl X/$ ee font matte 28 Elevie Inc. All ight eeved. doi:1.116/j.jmaa

2 M. Kiielewicz / J. Math. Anal. Appl poce X,x = X,x t t< on P F atifying X,x = x a.. fo, x, R n, we hall denote by Y,x the n + 1- dimenional tochatic poce defined on P F by Y,x = Y,x t t<, whee Y,x t = + t, X,x + t fo t <. Fo given above tochatic pocee, T > and a domain D R n we can define the fit exit time τ D and τ U of X,x and Y,x fom D and U =, T D, epectively, i.e., τ D = inf > : X,x / D} and τ U = inft > : Y,xt / U}. Itcanbeveified ee 11, p. 226 that τ U = τ D. In what follow we hall need ome continuou election theoem. We ecall it ee 9, Theoem 3 in the geneal fom. Let X, ρ, Y, and Z, be Polih and Banach pace, epectively. Similaly a above by ClY we denote the pace of all nonempty cloed ubet of Y. Theoem 1. See 9, Theoem 3. Let λ : X Y Z and u : X Z be continuou and H : X ClY be l..c. uch that ux λx, Hx fo x X. Aume λx, i affine and H tae convex value fo evey x X. Then fo evey ε > thee i a continuou function f ε : X Yuchthatf ε x Hx and λx, f ε x ux ε fo x X. The pape i oganized a follow. In Section 2 we ecall the baic notion dealing with tochatic diffeential incluion and intoduce et-valued patial diffeential opeato geneated by uch incluion. Some popetie of the fit exit time of continuou tochatic pocee ae given in Section 3, wheea Section 4 contain ome eult dealing with tochatic epeentation of olution to patial diffeential equation by wea olution to non-autonomou tochatic diffeential equation. Section 5 i devoted to exitence and epeentation theoem fo olution of initial and bounday value poblem fo patial diffeential incluion. It eult ae baed on the eult of two peviou ection. In paticula, the eult of Section 3 ae eential in the poof of the exitence and epeentation theoem fo bounday value poblem, wheea the exitence and epeentation theoem fo initial value poblem ae connected with eult of Section Stochatic diffeential incluion and et-valued patial diffeential opeato Given et-valued meauable and bounded mapping F :, R n ClR n and G :, R n ClR n m by a tochatic diffeential incluion SDIF, G we mean a elation x t x cl L 2 t F τ, x τ + Gτ, x τ db τ that ha to be atified fo evey t < by a ytem P F, X, B coniting of a complete filteed pobability pace P F, the n-dimenional F-nonanticipative tochatic poce X = Xt t< and an m-dimenional F-Bownian motion B = B t t<, whee cl L 2 denote the cloue in the L 2 Ω, R n -nom topology. In the paticula cae, when F and G tae convex value, then the above SDIF, G tae the fom x t x F τ, x τ + Gτ, x τ db τ. If a ytem P F, X, B atifie 1 then the poce X poee a continuou modification. It can be teated a a andom vaiable X : Ω, F C,βC, whee C = C,, R n and βc denote the Boel σ -algeba on C. We call the above ytem P F, X, B aweaolutiontosdif, G. AweaolutionP F, X, B to SDIF, G i aid to be unique in law if fo evey othe it wea olution P F, X, B one ha P = PX 1, whee P and PX 1 denote the ditibution of X and X, epectively defined by P A = P A and PX 1 A = P X 1 A fo A βc. In what follow we hall conide tochatic diffeential incluion 1 togethe with the initial condition X = x, a.., fo fixed, x, R n. Let u obeve that on the family of all wea olution to 1 atifying the above initial condition, we can define an equivalence elation by etting P F, X, B P F, F, P if and only if P = P X 1 fo evey wea olution P F, X, B and P F, F, P atifying X = X = x a.. In what follow we hall denote by X,x F, G the family of all -equivalence clae defined by all uch wea olution to 1. If P F, X, B X,x F, G we hall identify the cla P F, X, B with an -equivalence cla X of all n-dimenional continuou pocee Z uch that P = PZ 1. It i clea that we can aociate with evey X X,x F, G exactly one pobability meaue P. The pace of all uch pobability meaue coeponding to all clae belonging to the family X,x F, G will be denoted by X,x P F, G. We hall conide SDIF, G with F and G atifying ome with the following condition. Condition A. i F :, R n ClR n and G :, R n ClR n m ae bounded and tae convex value. ii F and G ae l..c. iii G i diagonally convex-valued, i.e., fo evey t, x, R n the et lgt, x =: g g T : g Gt, x} i convex, whee g T denote the tanpoition of g Gt, x. 1

3 594 M. Kiielewicz / J. Math. Anal. Appl iv G i uch that all continuou electo of the multifunction D = lg ae unifomly poitive defined and fo evey σ CD thee i g CG uch that σ = g g T. v F and G ae continuou. Obeve that lcg CD and iv implie that CD lcg. Then iv can be fomulate that G i uch that evey continuou electo σ of a multifunction D = lg i unifomly poitive defined and CD = lcg. Let u note that the above condition i iv could be witten in a weae fom by equiing: F and G atify condition i and ii, ae uch that L FG i unifomly paabolic, i.e. that fo evey σ lg thee i a poitive numbe M uch that Σσ ij t, xξ i ξ j M ξ 2 fo all t, x R + R n and ξ R n, and G i diagonally convex and uch that CD = cl C lcg, whee cl C denote the cloue in a topology of the unifom convegence of continuou function on R + R n. Some ufficient condition implying the hypothei iv ae given in 9, Lemma 11. Similaly a in 9, by the above condition we obtain the following eult. Theoem 2. See 9, Theoem 1. Aume condition i, iii v of A ae atified. Then fo evey, x, R n the family X,x F, G i nonempty and X,x P F, G i wealy compact with epect to the convegence in ditibution. Theoem 3. Aume condition i iv of A ae atified. Fo evey f, g CF CG and, x, R n thee i a unique cla X,x X,x F, G uch that the poce Y,x = Y,xt t< with Y,xt = + t, X,x + t fo t < i an Itô diffuion uch that Y,x =, x a.. Poof. Let f, g CF CG and, x, R n be fixed. By vitue of 6, Theoem IV.6.1 and Stooc and Vaadhan uniquene theoem ee 12 thee i a unique in law wea olution P F, X,x, B to the tochatic diffeential equation x t = x + f τ, x τ + gτ, x τ db τ.leta f = 1, f T T and b g =, g 1,...,g n T with, g i R 1+m, whee =,..., and g i denote the ith ow of g fo i = 1,...,n. Similaly a in 9, p. 144 we can veify that the poce Y,x defined above i a unique in law wea olution to the tochatic diffeential equation y t =, x + a f y τ + b gy τ db τ. Theefoe, imilaly a in 9, we can ee that Y,x i an Itô diffuion on P F atifying Y,x =, x a.. Immediately fom 11, Theoem it follow that by the aumption of Theoem 3 fo evey f, g CF CG and h C 1,2 Rn+1 the limit E,x hy,xt h, x lim t t 2 exit fo evey, x, R n, whee E,x denote the mean value opeato with epect to a pobability meaue Q,x uch that Q,x Y,xt 1 E 1,...,Y,xt E =PY,xt 1 E 1,...,Y,xt E fo evey t i < and E i βr n+1 with 1 i fo evey 1. Then the et D R n+1 of all h : R + R n R uch that the limit 2 exit i nonempty. In what follow the above limit will be denoted by L C h, x. The opeato L C defined by 2 on D R n+1 i called the infiniteimal geneato of Y,x. Similaly a in 11, Theoem we can veify that fo evey h C 1,2 Rn+1 and, x, R n one ha L C h, x = h t, x + n f i, x h xi, x i=1 n i, j=1 g g T, x h ij xi x j, x. Hence fo h C 2 Rn and, x, R n we get L C h, x = n i=1 f i, xh x i x + 1 n 2 i, j=1 g g T ij, xh x i x j x. Similaly a in 11, Theoem we obtain the following Dynin fomula E,x h Y,xτ = h, x + E,x τ L C h Y,xt dt fo h 1,2 C Rn+1,, x, R n and a topping time τ uch that E,x τ <. In paticula, fo h C 2 Rn one obtain E,x hx τ,x =hx + E,x τ LC ht, X,x t dt fo x Rn. Given the above et-valued mapping F and G we can define on C 1,2 Rn+1 a et-valued patial diffeential opeato L FG by etting L FG h, x = Luv h, x: u F, x, v G, x }, 3

4 M. Kiielewicz / J. Math. Anal. Appl whee n L uv h, x = u i h xi, x i=1 n v v T ij h x i x j, x 3 i, j=1 fo, x, R n. Immediately fom Theoem 1 the following eult follow. Theoem 4. Aume F and G atify condition i iv of A,letR + =, and u, v : R + R + R n R be continuou uch that ut, C 1,2 Rn+1 and vt,, x L FG ut,, x fo, x, R n and t,. Then fo evey ε > thee i f ε, g ε CF CG uch that g ε gε T i unifomly poitive defined and vt,, x L fε gε ut,, x ε fo evey, x, Rn and t,. Poof. Let X = R + R + R n, Y = R n R n n, λt,, x, z, σ = n i=1 z iu x i t,, x + 1 n 2 i, σ j=1 iju x i x j t,, x, F t,, x = F πt,, x and Gt,, x = Gπt,, x fo t,, x X, whee π denote the othogonal pojection of R R R n onto R R n. By the popetie of v one ha vt,, x λt,, x, Ht,, x fo t,, x X, whee Ht,, x = F t,, x l Gt,, x. By vitue of Theoem 1 fo evey ε > thee i a continuou electo hε of H uch that vt,, x λt,, x, hε t,, x ε fo evey t,, x X. By the definition of H and iv thee ae continuou electo fε and g ε of F and G, epectively uch that hε = fε, g ε g ε T. Hence it follow that thee ae continuou function f ε and g ε, electo of F and G, uch that f ε = f ε π and g ε = g ε π. Immediately fom iv it follow that g ε gε T i unifomly poitive defined. 3. Some popetie of the fit exit time Let D be a domain in R d and, x R + D. Aume X = X t t and X n = Xt n t ae continuou tochatic pocee on a tochatic bae P F = Ω, F, F, P uch that X = X n = x fo n = 1, 2,... and up t X n t Xt a..a n. We hall how that τ n τ a.. a n, wheneve τ n < a.. fo n 1, whee τ = inf > : X / D} and τ n = inf > : X n / D} fo n = 1, 2,... Similaly a in 4, Theoem VI.5.1 it can be veified that if X n ae fo evey n 1 wea olution to SDIF, G atifying initial condition X n = x a.., then fo appopiate taen F and G we have τ n < a.. fo n 1. Let u obeve that fo the above pocee X n and X we alo have τ < a.., wheneve τ n < a.. fo n = 1, 2,... Futhemoe, if T : Ω R + i uch that T maxτ n, τ a.. fo n 1thenτ n = inf, T : X n / D} and τ = inf, T : X / D} a.. fo n 1. Fo a given equence A n n=1 of ubet of a metic pace X, ρ by Lim um A n and Lim inf A n we hall denote it limit upeio and inteio, epectively. Similaly, by Li A n and L A n we denote it Kuatowi extemal limit. Recall, that by the definition x Li A n and z L A n if and only if thee i a equence x n n=1 of Rd uch that lim n x n = x and x n A n fo n 1 and thee i a ubequence A n =1 of A n n=1 and a equence x =1 of Rd uch that lim x = z and x A n fo 1, epectively. The popetie of the above limit can be find among othe in W. Hildenband 5, K. Kuatowi 1 and M. Kiielewicz 7. In paticula, it can be poved ee 5, Theoem B.1 that if X, ρ i a compact metic pace then a equence A n n=1 of nonempty ubet of X, ρ convege to A X in the Haudoff ditance if and only if Li A n = A = L A n. Lemma 5. Let D be a domain in R d and, x R + D. Aume X = X t t and X n = Xt n t ae d-dimenional continuou tochatic pocee on a tochatic bae P F = Ω, F, F, P uch that X = X n = x a.. fo n = 1, 2,... and up t X n t Xt a.. a n.thenli Xn 1D = D = L Xn 1D a.., whee D = R d \ D. Poof. Fo implicity aume that X ω and X n ω ae continuou and lim n up t X n tω Xtω = fo evey ω Ω and n = 1, 2,... Fo evey ω Ω and ε > thee exit N ε ω 1 uch that X n tω Xtω + εb and Xtω X n tω + εb fo t and n N ε ω, whee B i the cloed unit ball of R d. Then Xn 1X nt} Xn 1Xtω +εb} and X 1 Xt} X n t +εb} a.. fo n N ε ω. Let u obeve that fo evey A R + and C R d one ha A Xn 1X na, A XA, X n C C +εb and X C C fo n = 1, 2,... Taing in paticula in the above incluion, A = D and C = D we obtain D Xn 1X n D Xn 1XX 1 D + εb Xn 1D + εb a.. fo n N ε ω. Similaly, taing A = Xn 1D and C = D we obtain Xn 1D D + εb a.. fo n N ε ω. Hence it follow D = n=1 = = Nε 1 +Nε D + εb n=1 = +n D + εb = Lim inf Xn 1 +n D + εb D + εb = +Nε D + εb n=nε+1 = +n D + εb

5 596 M. Kiielewicz / J. Math. Anal. Appl a.. fo evey ε >, which implie D ε> Lim inf X 1 n D + εb = Lim inf Xn 1D a.. Hence it follow D Li Xn 1D a.. In a imila way we get = X 1 +Nε D + εb D + εb a.. Then Nε 1 n=1 = +n D = +Nε D n=nε+1 = a.. fo evey ε >. Hence, by vitue of 1, IV.8 it follow that L Xn 1 D = n=1 = +n D D + εb +n D D + εb a.. fo evey ε >. Then L Xn 1D ε> X 1 D + εb = D a.. Fom the above incluion we obtain D Li Xn 1D L Xn 1D D a.. Then L Xn 1D D Li Xn 1D, which by 1, V.1 implie that Li Xn 1D = L Xn 1D = D a.. In a imila way we can pove the following lemma. Lemma 6. Let D be a domain in R d and, x R + D. Aume X = X t t and X n = Xt n t ae d-dimenional continuou tochatic pocee on a tochatic bae P F = Ω, F, F, P uch that X = X n = x a.. fo n = 1, 2,... and up t X n t Xt a.. a n. If thee i a mapping T : Ω R + uch that maxτ, τ n T a.. fo n = 1, 2,..., whee τ = inf > : X / D} and τ n = inf > : X n / D} then D, T =LiXn 1D, T = LXn 1D, T a.. Lemma 7. Let D be a domain in R d and, x R + D. Aume X = X t t and X n = X n t t ae d-dimenional continuou tochatic pocee on a tochatic bae P F = Ω, F, F, P uch that X = X n = x a.. fo n = 1, 2,... and up t X n t Xt a.. a n.ifmaxτ n, τ < a.. fo n = 1, 2,...,whee τ n = inf n D and τ = inf D, then thee i a mapping T : Ω R + uch that maxτ n, τ T a.. fo n 1,wheeD = R d \ D. Poof. By vitue of Lemma 5 we have τ = infli Xn 1D a.. Then fo evey ε > thee i t ε Li Xn 1D uch that t ε < τ + ε a.., which implie that fo evey n 1theeit n Xn 1D uch that t n t ε a n. Theefoe, lim up τ n t ε < τ + ε a.. fo evey ε >, which implie that lim up τ n τ a.. Then fo a.e. ω Ω thee exit a poitive intege Nω 1 uch that τ n ω<τω fo n Nω. TaingT ω = maxτ 1 ω, τ 2 ω,...,τ Nω ω, τ ω} fo a.e. ω Ω, we can define a mapping T : Ω R + uch that maxτ n, τ T a.. fo n 1. Now we can pove the following convegence theoem. Theoem 8. Let D be a domain in R d and, x R + D. Aume X = X t t and X n = X n t t ae d-dimenional continuou tochatic pocee on a tochatic bae P F = Ω, F, F, P uch that X = X n = x a.. fo n = 1, 2,... and up t X n t Xt a.. a n.ifτ n = inf > : X n / D} < a.. fo n = 1, 2,... then lim n τ n = τ,whee τ = inf > : X / D} a.. Poof. It i eay to ee that τ < a.. Then maxτ n, τ < a.. fo n = 1, 2,... Theefoe, by vitue of Lemma 7 thee i a mapping T : Ω R + uch that maxτ n, τ T a.. fo n = 1, 2,... Then τ n = infxn 1D, T and τ = inf D, T a.. fo n = 1, 2,... Theefoe, by vitue of Lemma 6 and 5, Theoem B.1 we get lim n hxn 1D, T, D, T = a.., whee h i the Haudoff metic on Cl, T ω fo evey fixed ω Ω. Let ε > and t ε D, T be uch that t ε < τ +ε a.. By the above popety of a equence Xn 1D, T n=1 and the definition of the Haudoff metic h we have ditt ε, Xn 1D, T a..foeveyε > an. Theefoe, thee exit a equence t n ε n=1 uch that tn ε X 1 n D, T fo n 1 and t n ε t ε an. Hence it follow that τ n t n ε tn ε t ε +t ε < t n ε t ε +τ + ε fo ε > and n 1. Theefoe, lim up n τ n τ, a.. Similaly, fo evey ε > and n 1wecanelectt n ε X 1 n D, T and tn ε D, T uch that t n ε τ n + ε and tn ε t n ε an. Hence it follow τ tn ε tn ε t n ε +tn ε tn ε t n ε +τ n + ε fo ε > and n 1. Theefoe, τ lim inf n τ n a.. Then lim up n τ n τ lim inf n τ n a.., which implie that lim n τ n = τ a.. Lemma 9. Let D R n be a bounded domain, Xt t T and Xt t T be continuou n-dimenional tochatic pocee on Ω, F, P and Ω, F, P, epectively uch that X = X = xa..andp = P X 1. Fo evey bounded continuou function ψ :, T R n R one ha Eψσ D, Xσ D =Ẽψ σ D, X σd and E σ D ψt, Xt dt=ẽ σ D ψt, Xt dt, wheeσd = inft, T : Xt / D} Tand σ D = inft, T : Xt / D} T.

6 M. Kiielewicz / J. Math. Anal. Appl Poof. It i eay to ee that fo evey meauable mapping Φ : C T,β Y, G, whee Y, G i a meauable pace and β i a Boel σ -algeba on C T =: C, T, R n, one ha PΦ = PΦ. Taing in paticula, Φx = ηx, Ψ ηx, x fo x C T, whee ηx = inft, T : xt / D} T and Ψt, x = xt, weobtainpσ D, X σ D 1 = P σ D, X σd 1.Hence, by the definition of function η and Φ the above eult follow. 4. Initial and bounday value poblem fo patial diffeential equation We conide hee ome initial and bounday valued poblem fo patial diffeential equation geneated by patial diffeential opeato L C fo f, g CF CG with F and G atifying Condition A. To begin with let u ecall that if Condition A ae atified, then fo a given T >,, x, T R n and a wea olution X,x to the tochatic diffeential equation dx t = f t, x t dt + gt, x t db t with initial condition x = x, we can define the diffuion poce Y,x = + t, X,x + t t T with the infiniteimal geneato L C and uch that Y,x =, x. Similaly a in 11, Theoem we obtain the following theoem. Theoem 1. Aume condition i iv of A ae atified, T > and let c C, T R n be bounded. Then fo evey f, g CF CG and, x, T R n theeiauniqueinlawolutionx,x to SDE f, g atifying X,x = x a.., and uch that the function v defined by vt,, x = E exp,x c Y,xτ h Y,xt fo h C 1,2 R n+1,, x, T R n and t, T atifie v t t,, x = L C vt,, x c, xvt,, x fo, x, T Rn and t, T, v,, x = h, x fo, x, T R n. 4 Poof. The exitence and law uniquene of X,x follow immediately fom 6, Theoem IV.6.1 and the uniquene theoem of Stooc and Vaadhan ee 12. Fix, T and let Ut = hy,xt and V t = exp cy,xτ fo t, T. Then vt,, x = E,x Ut V t, dv t = V t cy,xt dt and U t = h, x + L C h, x d + n g g T i,, x j db. i, j=1 Hence it follow dut V t = U t dv t + V t du t,incedu t dv t =. Then U t V t t T i an Itô poce and by Itô fomula t E,x Ut V t = h, x + E,x Vτ L C h Y,xτ t E,x Uτ c Y,xτ Vτ fo t, T. Hence it follow that v,, x i diffeentiable fo fixed, x, T R n. Similaly a in the poof of 11, Theoem we obtain 1 E,x v t, Y,x vt,, x But = 1 E,x E Y,x V t h Y,xt E,x V t h Y,xt } = 1 E,x = 1 E,x E,x h Y,xt + V t+ h Y,xt + exp c Y,xτ + c Y,xτ F E,x V t h Y,xt } F V t h Y,xt = 1 E,x V t+ h Y,xt + V t h Y,xt + 1 E,x V t+ h Y,xt + exp c Y,xτ } 1.

7 598 M. Kiielewicz / J. Math. Anal. Appl and lim lim 1 1 V t+ h Y,xt + V t h Y,xt = lim V t+ h Y,xt + exp = E,x h Y,xt V t lim = E,x h Y,xt V t lim 1 exp 1 c Y,xτ 1 c Y,x exp 1 vt +,, x vt,, x = v t t,, x c Y,xτ } 1 c Y,xτ } = vt,, x c, x. Theefoe L C vt,, x = v t t,, x + vt,, x c, x and v,, x = E,x hy,x = h, x. Fo given T >, f CF, g CG and, x, T R n, by X,x olution to tochatic diffeential equation and X,x we hall denote unique in law wea x t = x + f τ, x τ + gτ, x τ db τ 5 and x t = x + f τ, x τ + gτ, x τ db τ, 6 epectively. Similaly a in 11, p. 18 we can veify that the poce Y,x = t, X,x t t T i time homogenou. Lemma 11. Aume condition i iv of A ae atified, T > and let f, g CF CG. Fo evey, x, T R n thee ae unique in law olution X,x and X,x to 5 and 6, epectively uch that the pocee Y,x = Y,xt t T and Y,x = Y,x t t T with Y,xt = + t, X,x + t and Y,x t = t, X,x t have the ame ditibution. Poof. The exitence of unique in law olution X,x and X,x fo evey, x, T Rn follow immediately fom 6, Theoem IV.6.1 and the uniquene theoem of Stooc and Vaadhan ee 12. Similaly a in 11, p. 18 we can veify that Y,x and Y,x have the ame ditibution fo evey, x, T R n. Coollay 1. If condition i iv of A ae atified then fo evey T >, f, g CF CG and h C 2 Rn the infiniteimal geneato of the diffuion poce Y,x = Y,x t t T i defined by L C h t, x = n f i t, xh x i x i=1 fo evey t, x, T R n. n i, j=1 g g T t, xh ij x i x j x Indeed, let h 1,2 C Rn+1 be defined by h = h π, whee π denote the othogonal pojection of R n+1 onto R n.by vitue of Lemma 11 we have E,x hy,x t=e,x hy,x t=e,x hy,xt. Hence, by 9, Theoem one obtain E,x hy,x t h, x E,x hy,xt h, x lim = lim = t t t t fo evey t, x, T R n. A a pecial cae of Theoem 1 we obtain the following theoem. n f i t, xh x i x i=1 n i, j=1 g g T t, xh ij x i x j x

8 M. Kiielewicz / J. Math. Anal. Appl Theoem 12. Aume condition i iv of A ae atified, T > and let c C, T R n, R be bounded. Then fo evey f, g CF CG and x R n theeiauniqueinlawolutionx,x to SDE f, g atifying X,x = x a.., and uch that the function v defined by vt, x = E exp,x c Y,x τ h π Y,x t fo h C 2 Rn and t, x, T R n atifie v t t, x = LC vt, t, x ct, xvt, x fo t, x, T Rn, v, x = hx fo x R n. Similaly a in 11, Theoem we can alo pove the following theoem. Theoem 13. Aume condition i iv of A ae atified, T > and let c C, T R n, R and v C 1,1,2, T, T R n be bounded and uch that v t t,, x = L C vt,, x c, xvt,, x fo, x, T Rn and t, T, v,, x = h, x fo, x, T R n fo f, g CF CG and h 1,2 C Rn+1. Then fo evey, x, T R n thee i a unique in law olution X,x to the tochatic diffeential equation 5 uch that vt,, x = E exp,x +t fo evey, x, T R n and t, T. c τ, X,xτ h + t, X,x + t Rema 1. If the aumption of Theoem 1 ae atified, w C 1,2, T R n, R and u C, T R n, R ae bounded and uch that the function v defined by v, x = wt, x fo < T,atifie v, x + L C v, x = u, x fo, x, T Rn, v, x = ht, x fo x R n then fo evey, x, T R n thee i a unique in law wea olution X,x to the tochatic diffeential equation 5 uch that wt, x = E,x h T, X,xT + E,x fo evey, x, T R n. T u τ, X,xτ We hall conide now ome genealized Diichlet Poion poblem with patial diffeential opeato geneated by f, g CF CG with F and G atifying Condition A. Similaly a in 11, Theoem we obtain the following eult. Theoem 14. Aume condition i iv of A ae atified, T > and let D be a bounded domain in R n.letu C, T R n, R and Φ C, T D, R be bounded and f, g CF CG. Ifv C 1,2 Rn+1 i bounded and uch that ut, x = L C vt, x fo t, x, T D, 7 lim D x y vt, x = Φt, y fo t, y, T D then fo evey, x, T D thee i a unique in law wea olution X,x to thetochaticdiffeentialequation 5 uch that v, x = E,x Φ τ D, X,xτ D τd E,x u t, X,xt dt fo evey, x, T D, whee τ D = inf, T : X,x / D} T.

9 6 M. Kiielewicz / J. Math. Anal. Appl Poof. By vitue of 6, Theoem IV.6.1 and the uniquene theoem of Stooc and Vaadhan ee 12 fo evey, x, T D thee i a unique in law wea olution X,x to the tochatic diffeential equation 5. Similaly a in 9 we can veify that the poce Y,x = + t, X,x + t t T i an Itô diffuion with the infiniteimal geneato L C defined above. Let U =, T D and τu = inf, T : Y,xt / U } T, whee D i an inceaing equence =1 of open et D uch that D D and D = =1. It can be veified ee 11, p. 226 that τ U = τ, whee τ = inf, T : X,x / D } T.ByDynin fomulafoevey = 1, 2,... we get E,x v Y,x τ U = v, x + E,x τ U L C v Y,xt dt fo evey, x, T D. By7wehaveu + t, X,x + t = L C vy,xt. Hence, by the definition of Y,x and the equality τu = τ, foevey = 1, 2,... we obtain v, x = E,x v τ, X,x τ E,x τ u t, X,xt dt On the othe hand by 7 and the boundedne of the function u we get lim E,x v τ, X,x τ = E,x Φ τ D, X,xτ D and lim E,x τ u t, X,xt dt fo, x, T D. Then τd = E,x u t, X,xt dt. v, x = E,x Φ τ D, X,xτ D τd E,x u t, X,xt dt fo evey, x, T D. Theoem 15. Aume condition i iv of A ae atified, T >, D i a bounded domain in R n and let c, u C, T R n, R and Φ C, T D, R be bounded. If v C 1,2 Rn+1 i bounded and uch that ut, x = L C vt, x ct, xvt, x lim D x y vt, x = Φt, y fo t, x, T D, fo t, y, T D then fo evey f, g CF CG and, x, T D thee i a unique in law wea olution X,x to the tochatic diffeential equation 5 uch that v, x = E,x Φ τ D, X,xτ D τ D c t, X,xt τd dt E,x u, X,x + c, X,xt } dt d fo evey, x, T D, whee τ D = inf, T : X,x / D} T. Poof. Similaly a in the poof of Theoem 14 we can veify that fo evey f, g CF CG and, x, T D thee i a unique in law wea olution X,x + t, X,x + t t T i an Itô diffuion with the infiniteimal geneato L C τu = inf, T : Y to the tochatic diffeential equation 5 uch that the poce Y,x = 8 defined above. Let U =, T D and,xt / U} T. It can be veified that τu = τ D. Fix, x, z, T R R n and define Zt = z + cy,xτ and Ht = Y,xt, Zt. It can be veified that H t t T i an Itô diffuion with the infiniteimal geneato L H ψ, x, z = L C ψ, x, z + ψ 1,2 z, x, zc, x fo ψ C, T Rn+1.HencebyDynin fomulait follow

10 E,x,z ψ τu τ R Hτ U τ R = ψ, x, z + E,x,z L H ψ H d, whee τ R = inft, T : Ht R}. Taingψ, x, z = e z v, x we get and E,x,z ψ Hτ U τ R = E,x,z L H ψ H = Hence and by 8 it follow e z v, x = E exp,x,z M. Kiielewicz / J. Math. Anal. Appl τu τ R c Y,x d v Y,x τ U τ R c Y,xτ L C v Y,x c Y,x v Y,x. τu τ R c Y,x d v Y,x τ U τ R τu τ R E,x,z c Y,xτ u Y,x d. Taing z = and paing to the limit when R one obtain τ U v, x = E exp,x c Y,xτ Φ Y τ U,x τ U E,x c Y,xτ u Y,x d becaue lim R vy,xτ U τ R = ΦY,xτ U. Hence by the definition of Y,xt = + t, X,x + t it follow +τ U v, x = E exp,x +τ U E,x c τ, X,xτ + Theefoe v, x = E,x Φ τ D, X,xτ D τ D τd E,x u, X,x + fo evey, x, T D, becaue + τ U = τ D. Φ + τu, X,x + τ U c τ, X,xτ u, X,x d. c t, X,xt dt c, X,xt dt Coollay 2. If the aumption of Theoem 15 ae atified and v C 1,2 Rn+1 i bounded and uch that v t t, x = L vt, x ct, xvt, x fo t, x, T D, d } lim D x y vt, x = Φt, y fo t, y, T D then fo evey, x, T D thee i a unique in law wea olution X,x to thetochaticdiffeentialequation 5 uch that v, x = E,x Φ τ D, X,xτ D τ D c t, X,xt dt fo evey, x, T D.

11 62 M. Kiielewicz / J. Math. Anal. Appl Initial and bounday valued poblem fo patial diffeential incluion Let F and G atify condition i iv of A and L FG be a et-valued patial diffeential opeato on C 1,2 Rn+1 defined by 3 above. Futhemoe, by L C FG we hall denote a family LC FG =LC : f, g CF CG}. Foevey f, g CF CG and h D FG := D R n+1 : f, g CF CG} one ha L C ht, x L C FG ht, x fo t, x, T R n. Immediately fom the above exitence and epeentation theoem the following eult follow. Theoem 16. Aume condition i iv of A ae atified, T >, h C 1,2 Rn+1 and let c C, T R n, R be bounded. Fo evey, x, T R n thee i a wea olution X,x to SDIF, G with the initial condition x = x a.., defined on a pobability pace Ω, F, P uch that the function +t vt,, x = E exp,x c τ, X,x τ h + t, X,x + t atifie v t t,, x LC FGvt,, x c, xvt,, x fo v,, x = h, x fo, x, T R n., x, T Rn and t, T, Poof. By vitue of Michael continuou election theoem ee 7 thee ae f CF and g CG. Then, by 6, Theoem IV.6.1 and the uniquene theoem thee i a unique in law wea olution X,x to the tochatic diffeential equation dx t = f t, x t dt + gt, x t db t with initial condition x = x a.. uch that the poce Y,x = + t, X,x + t t T i an Itô diffuion with infiniteimal geneato L C LC FG. By vitue of Theoem 1 the function vt,, x = E exp,x c Y,xτ h Y,xt defined fo h C 1,2 Rn+1,, x, T R n and t, T atifie condition 4. But L C vt,, x L C FG vt,, x fo, x, T Rn and t, T. Theefoe condition 9 ae alo atified. It i eay to ee that +t vt,, x = E exp,x c τ, X,x τ h + t, X,x + t. Theoem 17. Aume condition i, iii v of A ae atified, T > and let h C 1,2 Rn+1. Suppoe c C, T R n, R and v C 1,1,2, T, T R n, R ae bounded and uch that 9 v t t,, x v t,, x L FGvt,, x c, xvt,, x fo, x, T R n and t, T, v,, x = h, x fo, x, T R n. 1 Then fo evey, x, T R n thee exit X,x X,x F, G uch that +t vt,, x = Ẽ c τ, X,x τ h + t, X,x + t fo evey, x, T R n and t, T. Poof. By vitue of Theoem 4, fo evey 1 thee ae f CF and g CG uch that v t t,, x v t,, x L f g vt,, x c, xvt,, x 1/ 11 fo evey, x, T R n and t, T, whee L f g i defined by 3. But L C vt,, x = v f g t,, x + L f g vt,, x. Then an inequality 11 can be witten in the fom v t t,, x L C f g vt,, x c, xvt,, x 1/ 12 fo, x, T R n and t, T. By vitue of 6, Theoem IV.6.1 and the uniquene theoem of Stooc and Vaadhan ee 12 fo evey, x, T R n and = 1, 2,... thee i a unique in law wea olution X,x to the

12 M. Kiielewicz / J. Math. Anal. Appl tochatic diffeential equation dx t = f t, x t dt + g t, x t db t with initial condition x = x, a.., uch that a poce Y,x = + t, X,x + t t T i an Itô diffuion with the infiniteimal geneato L C L C f g FG.WehaveX,x X,xF, G fo 1. By the wea compactne of the et X,x P F, G and 6, TheoemI.2.1 thee i an inceaing ubequence n =1 of =1, a pobability pace Ω, F, P and continuou pocee n Xn,x, X,x on Ω, F, P uch that PX,x 1 = P Xn,x 1 fo = 1, 2,... and up t T Xn,xt X,x t, P -a.. a. By the Stooc and Vaadhan uniquene theoem it follow that Xn,x i a wea olution to a tochatic diffeential equation dx t = f n t, x t dt + g n t, x t db t with an initial condition x = x a.., becaue P n X n,x = x} = P Xn,x = x} fo = 1, 2,..., whee P n F n = Ω n, F n, F n, P n i a filteed pobability pace uch that P n F n, X n,x, B n i a wea olution to the above tochatic diffeential equation. Then Xn,x X,x F, G which by the wea compactne of X,x P F, G implie that X,x X,x F, G, too. Denote Y,x = + t, X,x + t t T, Ỹ,x = + t, X,x + t t T and Ỹ,x = + t, X,x + t t T. Letσ = inft, T : Y,x t / K } T, σ = inft, T : Ỹ,x t / K } T and σ = inft, T : Ỹ,x t / K } T, whee K =y R n : y < }. Similaly a in 9, p. 151 we can ee that lim σ = T a.. Hence in paticula, it follow that lim σ t} vt σ, Ỹ,x σ d P = fo, T and t, T. By vitue of Lemma 9 and the popetie of Y n,x and Ỹ n,x we have E,x n exp t σ = Ẽ c Y n,xτ t σ c Ỹ n,xτ v t t σ, Y n,x t σ v t t σ, Ỹ n,x t σ fo fixed t, T, t < t and, T. ByItô fomulaweobtain E,x n exp = E,x t σ t σ c Y n,xτ τ v t t σ, Y n,x t σ vt,, x c Y n,xτ L C f n g n vt τ, Y n,xτ v t t τ, Y n,xτ c Y n,xτ v t τ, Y n,xτ. Similaly a above, by the popetie of Y n,x and Ỹ n,x and Lemma 9 we alo have E,x n Theefoe t σ = Ẽ τ t σ Ẽ = Ẽ t σ τ t σ c Y n,xτ L C f n g n v t τ, Y n,xτ v t t τ, Y n,xτ c Y n,xτ v t τ, Y n,xτ c Ỹ n,xτ L C f n g n v t τ, Ỹ n,xτ v t t τ, Ỹ n,xτ c Ỹ n,xτ v t τ, Ỹ n,xτ. c Ỹ n,xτ τ v t t σ, Ỹ n,x t σ vt,, x c Ỹ n,xτ L C f n g n v t τ, Ỹ n,xτ v t t τ, Ỹ n,xτ c Ỹ n,xτ v t τ, Ỹ n,xτ fo = 1, 2,...,, T and t < t. Hence and by 12 it follow

13 64 M. Kiielewicz / J. Math. Anal. Appl Ẽ up x R n Ẽ t σ t σ c Ỹ n,xτ τ v t t σ, Ỹ n,x t σ vt,, x c + τ, x L C f n g n vt τ, + t, x v t t τ, + t, x c + t, xvt τ, + t, x 1 T +τ cu, x du T n n fo = 1, 2,... and, T. Theefoe lim Ẽ t σ c Ỹ n,xτ v t t σ, Ỹ n,x t σ = vt,, x unifomly with epect to t,, x. Hence, by vitue of Theoem 8 and the popetie of a equence Ỹ n,x =1 and σ =1, it follow that Ẽ t σ c Ỹ,x τ v t t σ, Ỹ,x t σ = vt,, x fo, x, T R n, t, t and = 1, 2,... which can be witten in the fom vt,, x = σ t} + σ >t} σ c Ỹ,x τ v t σ, Ỹ,x σ d P c Ỹ,x τ v t t, Ỹ,x t d P. Hence, imilaly a in the poof of 9, Theoem 14, it follow that Ẽ c Ỹ,x τ v t t, Ỹ,x t = vt,, x fo, x, T R n and t, t. Paing to the limit in the lat equality when t t and then taing t = t we obtain vt,, x = Ẽ c Ỹ,x τ v, Ỹ,x t which by 1 can be witten in the fom +t vt,, x = Ẽ c τ, X,x τ h + t, X,x + t fo evey, x, T R n and t, T. Theoem 18. Aume condition i, iii v of A ae atified, T >, D i a bounded domain in R n and let Φ C, T D, R and u C, T D, R be bounded. If v C 1,2 Rn+1 i bounded and uch that ut, x v t t, x L FG vt, x fo t, x, T D, lim D x y vt, x = Φt, y fo t, y, T D 13 then fo evey, x, T D thee exit X,x X,x F, G uch that

14 v, x = Ẽ Φ τ D, X,x τ D Ẽ M. Kiielewicz / J. Math. Anal. Appl τ D u t, X,x t dt fo evey, x, T D, whee τ D = inf, T : X,x / D} T. Poof. By vitue of Theoem 4 fo evey 1 thee ae f CF and g CG uch that ut, x v t t, x L f g vt, x 1/ fo t, x, T D. Similaly a in the poof of Theoem 17 we can veify that fo evey, x, T D and 1 thee i a unique in law wea olution X,x to the tochatic diffeential equation dx t = f n t, x t dt + g n t, x t db t with initial condition x = x a.., uch that the poce Y,x = + t, X,x + t t T i an Itô diffuion with infiniteimal geneato L C uch that L C vt, x = v f g f g t t, x + L f g vt, x fo v C 1,2 Rn+1.Then ut, x L C vt, x 1/ fo t, x, T D. By the wea compactne of f g X,x P F, G and 6, Theoem I.2.1 thee i an inceaing ubequence n =1 of =1, a pobability pace Ω, F, P and continuou pocee Xn,x, X,x on Ω, F, P uch that PX n,x 1 = P Xn,x 1 fo = 1, 2,..., and up t T Xn,xt X,x t, P -a.. a. By the uniquene Stooc and Vaadhan theoem it follow that Xn,x i a wea olution to the above tochatic diffeential equation, becaue P n X n n,x = x} = P Xn,x = x} fo = 1, 2,..., whee P n F n = Ω n, F n, F n, P n i a filteed pobability pace uch that P n F n, X n,x, B n i a wea olution to the above tochatic diffeential equation. Then Xn,x X,x F, G which, by the wea compactne of X,x P F, G implie that alo X,x X,x F, G. LetτU = inf, T : Y n,xt /, T D} T, τ = inf, T : X n,xt / D} T, τ = inf, T : Xn,xt / D} T and τ D = inf, T : X,x / D} T. WehaveEn,x τ < and τu = τ fo = 1, 2,... Let u t, x = L C f n g n vt, x fo t, x, T D and = 1, 2,... Similaly a in the poof of Theoem 14 by Dynin fomula, fo evey = 1, 2,..., we obtain En,x f n g n v Y,x τ τ U = v, x + E,x L C f n g n v Y f n g n,x t dt which, by the definition of Y,x and the equality τu = τ, can be witten in the fom v, x = En,x v τ, X f n g n,x τ τ E,x L C f n g n v t, X f n g n,x t dt. By the popetie of the pocee X f n g n,x v, x = τ f n g n Ẽ v, X,x τ Ẽ f n g n f n g n But ut, X,x t u t, X,x v, x Ẽ τ f n g n v, X,x Ẽ τ f n g n u t, X,x τ + Ẽ f n g n and X,x and Lemma 9 hence it follow τ L C f n g n f n g n v t, X,x t dt. t 1/n P -a.. Theefoe, τ u t, X f n g n,x t dt t u t, X f n g n,x t dt 1 n. Then lim Ẽ v, x τ f n g n v, X,x τ τ + f n g n u t, X,x t dt = unifomly with epect to, x, T D. Hence, by Theoem 8 and 13, it follow that v, x = Ẽ Φ τ D, X,x τ D τ D Ẽ u t, X,x t dt fo evey, x, T D. Quite imilaly we obtain the following eult.

15 66 M. Kiielewicz / J. Math. Anal. Appl Theoem 19. Aume condition i, iii v of A ae atified, T >, D i a bounded domain in R n and let Φ C, T D, R, c C, T D, R and u C, T D, R be bounded. If v C 1,2 Rn+1 i bounded and uch that ut, x v t t, x L FG vt, x ct, xvt, x fo t, x, T D, lim D x y vt, x = Φt, y fo, y, T D then fo evey, x, T D thee exit X,x X,x F, G uch that v, x = Ẽ Φ τ D, X,x τ D τ D c t, X,x t τ D dt Ẽ u t, X,x t +t c z, X,x z } dz dt fo evey, x, T D, whee τ D = inf, T : X,x / D} T. Rema 2. TheeultofthepapeaealotuefoT =, becaue by the boundedne of D we have τ D < a.., which implie that lim T τ D T = τ D a.. The cae T < i much moe compatible with pactical application. Rema 3. The eult of the pape ae alo tue if intead of ClG = lcg in iv we tae ClG = cl C lcg. Acnowledgment The autho i deeply indebted to the anonymou efeee of the pape fo vey valuable uggetion and ema geatly impoving the text. Refeence 1 G. Batuzel, A. Fyzowi, Abtact diffeential incluion G with ome application to patial diffeential one, Ann. Polon. Math. LIII G. Batuzel, A. Fyzowi, Stability of pincipal eigenvalue of Schödinge type poblem fo diffeential incluion, Topol. Method Nonlinea Anal G. Batuzel, A. Fyzowi, A cla of etact in L p with ome application to diffeential incluion, Dicu. Math. Diffe. Incl. Contol Optim A. Fiedman, Stochatic Diffeential Equation and Application, Acad. Pe, New Yo, W. Hildenband, Coe and Equilibia of a Lage Economy, Pinceton Univ. Pe, New Jeey, N. Ieda, S. Watanabe, Stochatic Diffeential Equation and Diffuion Pocee, Noth Holland, Amtedam, M. Kiielewicz, Diffeential Incluion and Optimal Contol, Kluwe Academic Publihe, New Yo, M. Kiielewicz, Wea compactne of olution et to tochatic diffeential incluion with convex ight-hand ide, Topol. Method Nonlinea Anal M. Kiielewicz, Stochatic diffeential incluion and diffuion pocee, J. Math. Anal. Appl C. Kuatowi, Topologie, vol. I, Polie Wydawnictwo Nauowe, Wazawa, B. Øendal, Stochatic Diffeential Equation, Spinge-Velag, Belin Heidelbeg, D.W. Stooc, S.R.S. Vaadhan, Diffuion poce with continuou coefficient, I, II, Comm. Pue Appl. Math ,