Averaging and mixing for stochastic perturbations of linear conservative systems

We study stochastic perturbations of linear systems of the form (Formula present) 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 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 ε. We use Khasminski’s approach to stochastic averaging to show that, as ε → 0, a solution v(t), written in the interaction representation in terms of the operator A, for 0 ⩽ t ⩽ Const · ε⁻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.