國立清華大學 數學系

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\'evy process. This is a joint work with Dr. Chen Lifeng and Profs. Dong Zhao and Zhai Jianliang.