Averaging for stochastic perturbations of integrable systems

TL;DR

This paper proves averaging theorems for small stochastic perturbations of integrable systems, establishing convergence of solutions to averaged equations and constructing effective stochastic models, especially under mixing conditions.

Contribution

It introduces new averaging results for stochastic perturbations of integrable systems and constructs effective equations with uniform convergence under mixing assumptions.

Findings

Solutions of perturbed systems converge to averaged equations as perturbation vanishes.

Effective stochastic equations accurately describe the long-term behavior of perturbed systems.

Uniform convergence occurs when the averaged system exhibits mixing properties.

Abstract

We are concerned with averaging theorems for $\epsilon$-small stochastic perturbations of integrable equations in $\mathbb{R}^d \times \mathbb{T}^n =\{(I,\varphi)\}$ $$ \dot I(t) =0,\quad \dot \varphi(t) = \theta(I), \qquad (1)$$ and in $\mathbb{R}^{2n} = \{v=(\mathbf{v}_1, \dots, \mathbf{v}_n), \; \mathbf{v}_j \in \mathbb{R}^2\}$, $$ \dot{\mathbf{v}}_k(t) =W_k(I) \mathbf{v}_k^\bot, \quad k=1, \dots, n, \qquad (2) $$ where $I=(I_1, \dots, I_n)$ is the vector of actions, $I_j = \frac12 \| \mathbf{v}_j\|^2$. The vector-functions $\theta$ and $W$ are locally Lipschitz and non-degenerate. Perturbations of these equations are assumed to be locally Lipschitz and such that some few first moments of the norms of their solutions are bounded uniformly in $\epsilon$, for $0\le t\le \epsilon^{-1} T$. For $I$-components of solutions for perturbations of (1) we establish their convergence in law to solutions of the corresponding averaged $I$-equations, when $0\le \tau := \epsilon t\le T$ and $\epsilon\to0$. Then we show that if the system of averaged $I$-equations is mixing, then the convergence is uniform in the slow time $\tau=\epsilon t\ge0$. Next using these results, for $\epsilon$-perturbed equations of (2) we construct well posed {\it effective stochastic equations} for $v(\tau)\in \mathbb{R}^{2n}$ (independent from $\epsilon$) such that when $\epsilon\to0$, actions of solutions of the perturbed equations of (2) with $t:= \tau/\epsilon$ converge in distribution to actions of solutions for the effective equations. Again, if the effective system is mixing, this convergence is uniform in the slow time $\tau \ge0$. We provide easy sufficient conditions on the perturbed equations which ensure that our results apply to their solutions.