Abstract stochastic problems with generators of regularized semigroups

Abstract

We study the stochastic Cauchy problem dX(t) = AX(t)dt + BdW(t), t ∈ [0, T ], X(0) = ξ, where A is the generator of a regularized semigroup in a separable Hilbert space H, B is a bounded linear operator and W is an H-valuedWiener process on a probability space (Ω, F, P). We construct regularized solutions to this problem in L2(Ω;H) and in spaces of abstract stochastic distributions. We also study the semi-linear problem dX(t) = [AX(t) + F(t,X)]dt + B(t,X)dW(t), X(0) = ξ, where F and B satisfy some appropriate growth and Lipschitz conditions.