  The limiting stability of invariant probability measures of time homogeneous
transition semigroups for autonomous stochastic systems has been extensively
discussed in the literature. In this paper we initially initiate a program to
study the asymptotic stability of evolution systems of probability measures of
time inhomogeneous transition operators for nonautonomous stochastic systems. A
general theoretical result on this topic is established in a Polish space by
establishing some sufficient conditions which can be verified in applications.
Our abstract results are applied to a stochastic lattice reaction-diffusion
equation driven by a time-dependent nonlinear noise. A time-average method and
a mild condition on the time-dependent diffusion function are used to prove
that the limit of every evolution system of probability measures must be an
evolution system of probability measures of the limiting equation. The
theoretical results are expected to be applied to various stochastic lattice
systems/ODEs/PDEs in the future.
