Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces

Abstract

A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupRt are obtained. We show thatRt is a compactC0-semigroup in all Sobolev spacesWn,p which are built on its invariant measure μ. Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inLp(μ) spaces and spacesW1,p. As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to μ. It is shown also that the density of this measure with respect to μ is inLp(μ) for allp≧1.