  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.
