Averaging and mixing for stochastic perturbations of linear conservative systems

TL;DR

This paper investigates stochastic perturbations of linear conservative systems with oscillatory behavior, demonstrating convergence to an effective averaged equation using stochastic averaging techniques, under certain mixing conditions.

Contribution

It introduces a stochastic averaging approach for linear systems with oscillatory spectra, establishing convergence to an effective equation as perturbation parameter tends to zero.

Findings

Solutions converge in distribution to an effective averaged equation

The approach applies to systems with mixing properties

Uniform bounds on moments of solutions are established

Abstract

We study stochastic perturbations of linear systems of the form $$ dv(t)+Av(t)dt = \epsilon P(v(t))dt+\sqrt{\epsilon}B(v(t)) dW (t), v\in\mathbb{R}^{D}, (*) $$ 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 $\epsilon\to0$, 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.