Invariant measures for stochastic Cauchy problems with asymptotically unstable drift semigroup

Abstract

We investigate existence and permanence properties of invariant measures for abstract stochastic Cauchy problems of the form $$dU(t) = (AU(t)+f)\,dt + B\,dW_H(t), \ \ t\ge 0,$$ governed by the generator $A$ of an asymptotically unstable $C_0$-semigroup on a Banach space $E$. Here $f \in E$ is fixed, $W_H$ is a cylindrical Brownian motion over a separable real Hilbert space $H$, and $B$ is a bounded operator from $H$ to $E$. We show that if $E$ does not contain a copy of $c_0$, such invariant measures fail to exist generically but may exist for a dense set of operators $B$. It turns out that many results on invariant measures which hold under the assumption of uniform exponential stability of $S$ break down without this assumption.