Introduction and Preliminaries

Abstract

We are here concerned with a stochastic differential equation in a separable Hilbert space H,

$$\left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {dX(t,x) = (AX(t,x) + F(X(t,x)))dt + B dW(t),} & {t > 0, x \in H,} \\ \end{array} } \hfill \\ {\begin{array}{*{20}{c}} {X(0,x) = x,} & {x \in H.} \\ \end{array} } \hfill \\ \end{array} } \right.$$

(1)

Here A: D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup e^tA in H, B is a bounded operator from another Hilbert space U and H, F: D(F) ⊂ H → H is a nonlinear mapping and W(t), t ≥ 0, is a cylindrical Wiener process in U defined in some probability space (Ω, ℱ, ℙ), see Chapter 2 for a precise definition.