Asymptotic Behavior of Infinite Dimensional Stochastic Differential Equations by Anticipative Variation of Constants Formula

Abstract

We consider the long time behavior of an infinite dimensional stochastic evolution equation with respect to a cylindrical Wiener process. New estimates on the disturbance operator related to the problem are proved using a ``variation of constants''-type formula. Such estimates, under the natural assumption of mean-square stability for the linear part of the equation, lead directly to sufficient conditions for the exponential stability of the problem. In the last part of the paper we prove that, under suitable conditions, the equation admits a unique invariant measure that is strongly mixing. To complete the paper, we present examples of interesting situations where our construction applies.