Evolution systems of probability measures for nonautonomous Klein-Gordon Itô equations on ZN

Invariant probability measures of time-homogeneous transition semigroups for autonomous stochastic ODEs/PDEs/lattice systems have been widely discussed in the literature. In this work we investigate evolution systems of probability measures of time inhomogeneous transition operators for a class of nonautonomous Klein-Gordon Itô equation defined on an integer set ZN. The existence, periodicity, pointwise ergodicity and tightness of evolution systems of probability measures are established in an infinite-dimensional phase space by using a generalized Krylov-Bogolyubov method developed by Da Prato and Röckner [21]. The asymptotic stability of every periodic evolution system of probability measures with respect to the noise intensity is discussed by using a recent result on this topic established by Wang, Caraballo and Tuan [49]. The uniqueness, global exponential mixing, forward strong mixing and backward strong mixing of every evolution system of probability measures are also discussed under some globally monotone conditions. The idea of uniform tail-estimate due to Wang [41] is used to overcome several difficulties caused by the lack of compactness in infinite lattices. It seems that this is the first time to study evolution systems of probability measures for nonautonomous hyperbolic stochastic equations.