This talk focuses on the connection between deterministic dynamical systems and their small stochastic perturbations. Let µǫ be stationary measure for stochastic process Xǫ with small ǫ and X0 be a semiflow on a Polish space. Assume that {µǫ : 0 < ǫ ≤ ǫ0} is tight. Then all their limits in weak sense are X0−invariant and their supports are contained in Birkhoff center of X0. Applications are made to various stochastic evolution systems, including stochastic ordinary differential equations, stochastic partial differential equations, and stochastic functional differential equations driven by Brownian motion or L´evy process. Peculiar examples are presented to show that saddle points are stochastically stable with respect to non-degenerate noise.

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