Non-autonomous stochastic lattice systems with Markovian switching

TL;DR

This paper investigates the long-term behavior of non-autonomous stochastic lattice systems with Markovian switching, establishing existence, stability, and convergence properties of their measures as noise diminishes.

Contribution

It introduces new results on the existence and stability of evolution systems of measures for stochastic lattice systems with Markovian switching, including zero-noise limits.

Findings

Existence of an evolution system of measures for the stochastic system.

Pullback and forward asymptotic stability in distribution.

Convergence of measures as noise intensity approaches zero.

Abstract

The aim of this paper is to study the dynamical behavior of non-autonomous stochastic lattice systems with Markovian switching. We first show existence of an evolution system of measures of the stochastic system. We then study the pullback (or forward) asymptotic stability in distribution of the evolution system of measures. We finally prove that any limit point of a tight sequence of an evolution system of measures of the stochastic lattice systems must be an evolution system of measures of the corresponding limiting system as the intensity of noise converges zero. In particular, when the coefficients are periodic with respect to time, we show every limit point of a sequence of periodic measures of the stochastic system must be a periodic measure of the limiting system as the noise intensity goes to zero.