Dissipative stochastic equations in Hilbert space with time dependent coefficients

Abstract

We prove existence and, under an additional assumption, uniqueness of an evolution system of measures $({\nu }_t{)}_{t\in \mathbb{R}}$ for a stochastic differential equation with time dependent dissipative coefficients. We prove that if $P_{s,t}$ denotes the corresponding transition evolution operator, then $P_{s,t}\phi$ behaves asymptotically as $t\to +\infty$ like a limit curve (which is independent of $s$) for any continuous and bounded ``observable'' $\phi$.