Averaging for stochastic perturbations of integrable systems

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.