On Limiting Behavior of Stationary Measures for Stochastic Evolution Systems with Small Noise Intensity

Abstract

The limiting behavior of stochastic evolution processes with small noise intensity $\epsilon$ is investigated in distribution-based approach. Let $\mu^{\epsilon}$ be stationary measure for stochastic process $X^{\epsilon}$ with small $\epsilon$ and $X^{0}$ be a semiflow on a Polish space. Assume that $\{\mu^{\epsilon}: 0<\epsilon\leq\epsilon_0\}$ is tight. Then all their limits in weak sense are $X^0-$invariant and their supports are contained in Birkhoff center of $X^0$. Applications are made to various stochastic evolution systems, including stochastic ordinary differential equations, stochastic partial differential equations, stochastic functional differential equations driven by Brownian motion or L\'{e}vy process.