Averaging and mixing for stochastic perturbations of linear conservative systems

Abstract

We study stochastic perturbations of linear systems of the form \begin{equation} dv(t)+Av(t) dt =\varepsilon P(v(t)) dt+\sqrt{\varepsilon} \mathcal{B}(v(t)) dW (t), \qquad v\in\mathbb{R}^D, \tag{*} \end{equation} where $A$ is a linear operator with non-zero imaginary spectrum. It is assumed that the vector field $P(v)$ and the matrix function $\mathcal{B}(v)$ are locally Lipschitz with at most polynomial growth at infinity, that the equation is well-posed and a few of first moments of the norms of solutions $v(t)$ are bounded uniformly in $\varepsilon$. We use Khasminski's approach to stochastic averaging to show that, as $\varepsilon\to0$, a solution $v(t)$, written in the interaction representation in terms of the operator $A$, for $0\leqslant t\leqslantConst\cdot\varepsilon^{-1}$ converges in distribution to a solution of an effective equation. The latter is obtained from $(*)$ by means of certain averaging. Assuming that equation $(*)$ and/or the effective equation are mixing, we examine this convergence further. Bibliography: 27 titles.