Averaging and mixing for stochastic perturbations of linear conservative systems

Abstract

We study stochastic perturbations of linear systems of the form $$dv(t)+Av(t)dt=\epsilon P(v(t))dt+\sqrt{\epsilon }B(v(t))dW(t),v\in {\mathbb{R}}^D,(\ast )$$ where $A$ is a linear operator with non-zero imaginary spectrum. It is assumed that the vector field $P(v)$ and the matrix-function $B(v)$ are locally Lipschitz with at most a polynomial growth at infinity, that the equation is well posed and first few moments of norms of solutions $v(t)$ are bounded uniformly in $\epsilon$. We use the Khasminski approach to stochastic averaging to show that as $\epsilon \to 0$, a solution $v(t)$, written in the interaction representation in terms of operator $A$, for $0\le t\le Const{\epsilon }^{-1}$ converges in distribution to a solution of an effective equation. The latter is obtained from (*) by means of certain averaging. Assuming that eq.(*) and/or the effective equation are mixing, we examine this convergence further.