AVERAGING AND MIXING FOR STOCHASTIC PERTURBATIONS OF LINEAR CONSERVATIVE SYSTEMS

We study stochastic perturbations of linear systems of the form dv(t) + Av(t) dt = εP (v(t))dt + √ ε B(v(t))dW (t), v ∈ R D , (*) 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 ε. We use the Khasminski approach to stochastic averaging to show that as ε → 0, a solution v(t), written in the interaction representation in terms of 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 eq. (*) and/or the effective equation are mixing, we examine this convergence further