On Limiting Behavior of Stationary Measures for Stochastic Evolution Systems with Small Noise Intensity\@textsuperscript\safe@setrefT1thanksT1\@nil,\safe@setrefT1thanks\@nil\@@T1,\safe@setrefT1thanks

The limiting behavior of stochastic evolution processes with small noise intensity $ϵ$ is investigated in distribution-based approach. Let ${\mu }^{ϵ}$ be stationary measure for stochastic process $X^{ϵ}$ with small $ϵ$ and $X^0$ be a semiflow on a Polish space. Assume that $\{{\mu }^{ϵ}:0<ϵ\le {ϵ}_0\}$ 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 Lévy process.