Long term behavior of dichotomous stochastic differential equations in Hilbert spaces

Abstract

Abstract: We study existence of invariant measures for semilinear stochastic differential equations in Hilbert spaces. The noise is infinite dimensional, white in time, and colored in space. We show that if the equation is exponentially dichotomous in the sense that the semigroup generated by the linear part is hyperbolic and the Lipschitz constants of the nonlinearities are not too large, then existence of a solution with bounded mean squares implies existence of an invariant measure. Moreover, we show that every bounded solution satisfies a certain "Cauchy condition", which implies that its distributions converge weakly to a limit distribution.

Keywords: Coupling; dichotomy; Gronwall's lemma; hyperbolic semigroup; invariant measure; stochastic delay differential equation; uniform tightness
AMSC numbers: 34K50, 60H20, 93E15