Averaging for stochastic perturbations of integrable systems

We are concerned with averaging theorems for \(\varepsilon \) $\epsilon$ -small stochastic perturbations of integrable equations in \({\mathbb {R}}^d \times {\mathbb {T}}^n =\{(I,\varphi )\}\) ${\mathbb{R}}^d\times {\mathbb{T}}^n=\{(I,\phi )\}$ 1 \(\begin{aligned} \dot{I}(t) =0, \quad {\dot{\varphi }}(t) = \theta (I), \end{aligned}\) $$\begin{array}{c}\hfill \dot{I}(t)=0,\dot{\phi }(t)=\theta (I),\end{array}$$ and in \({\mathbb {R}}^{2n} = \{ v=(\textbf{v}_1, \dots , \textbf{v}_n), \ \textbf{v}_j \in {\mathbb {R}}^2\}\) ${\mathbb{R}}^{2n}=\{v=({\mathbf{v}}_1,\cdots ,{\mathbf{v}}_n),{\mathbf{v}}_j\in {\mathbb{R}}^2\}$ , 2 \(\begin{aligned} {\dot{\textbf{v}}}_k(t) =W_k(I) \textbf{v}_k^\perp , \quad k=1, \dots , n, \end{aligned}\) $$\begin{array}{c}\hfill {\dot{\mathbf{v}}}_k(t)=W_k(I){\mathbf{v}}_k^{\perp },k=1,\cdots ,n,\end{array}$$ where \(I=(I_1, \dots , I_n)\) $I=(I_1,\cdots ,I_n)$ is the vector of actions, \(I_j = \frac{1}{2} \Vert \textbf{v}_j\Vert ^2\) $I_j=\frac{1}{2}{\Vert {\mathbf{v}}_j\Vert }^2$ . The vector-functions \(\theta \) $\theta$ 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 \(\varepsilon \) $\epsilon$ , for \(0\le t\le \varepsilon ^{-1} T\) $0\le t\le {\epsilon }^{-1}T$ . For I-components of solutions for perturbations of (1) we establish their convergence in law to solutions of the corresponding averaged I-equations, when \(0\le \tau := \varepsilon t\le T\) $0\le \tau :=\epsilon t\le T$ and \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ . Then we show that if the system of averaged I-equations is mixing, then the convergence is uniform in the slow time \(\tau =\varepsilon t\ge 0\) $\tau =\epsilon t\ge 0$ . Next using these results, for \(\varepsilon \) $\epsilon$ -perturbed equations (2) we construct well posed effective stochastic equations for \(v(\tau )\in {\mathbb {R}}^{2n}\) $v(\tau )\in {\mathbb{R}}^{2n}$ (independent of \(\varepsilon \) $\epsilon$ ) such that when \(\varepsilon \rightarrow 0\) $\epsilon \to 0$ , the actions of solutions for the perturbed equations with \(t:= \tau /\varepsilon \) $t:=\tau /\epsilon$ 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 \(\tau \ge 0\) $\tau \ge 0$ . We provide easy sufficient conditions on the perturbed equations which ensure that our results apply to their solutions.