Abstract
We consider some stochastic difference partial differential equations of the form du(x, t, c) = L(x, t, D)u(x, t, c)dt + M(x, t, D)u(x, t − a, c)dw(t), where L(x, t, D) is a linear uniformly elliptic partial differential operator of the second order, M(x, t, D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. Some properties are also studied. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications.
Highlights
Consider the stochastic linear system nk du(t,c) = Au(t,c)dt +bij Biu t − cj, c dwij (t), i=1 j=1 (1.1)where A is a linear closed operator generating the strongly continuous semigroup Q(t) on a separable Hilbert space H, and wij are mutually independent Wiener processes on a separable Hilbert space K with covariance operators Wij, positive nuclear operators in the space L(K,K) of continuous linear mapping of K into itself.It is assumed that A is defined on S1 ⊂ H into H and S1 is dense in H
A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications
It is assumed that B1, . . . , Bn are linear closed operators defined on S2 ⊃ S1, S2 ⊂ H, and with values in H. bij(·) are elements of L(H, L(K, H))
Summary
We consider some stochastic difference partial differential equations of the form du(x,t,c) = L(x,t,D)u(x,t,c)dt + M(x,t,D)u(x,t − a,c)dw(t), where L(x,t,D) is a linear uniformly elliptic partial differential operator of the second order, M(x,t,D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness of the solution of suitable mixed problems are studied for the considered equation. A more general stochastic problem is considered in a Hilbert space and the results concerning stochastic partial differential equations are obtained as applications
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