Averaging and mixing for stochastic perturbations of linear conservative systems

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 $0\leqslant t\leqslant\text{Const}\cdot\varepsilon^{-1}$ 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.
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