EXISTENCE, UNIQUENESS AND INVARIANT-MEASURES FOR STOCHASTIC SEMILINEAR EQUATIONS ON HILBERT-SPACES

A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroup R(t) are obtained. We show that R(t) is a compact C-o-semigroup in all Sobolev spaces W-n,W-p which are built on its invariant measure mu. Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations in L(P)(mu) spaces and spaces W-1,W-p. As a consequence we prove the uniqueness of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to mu. It is shown also that the density of this measure with respect to mu is in L(p)(mu) for, all p greater than or equal to 1.