Averaging for stochastic perturbations of integrable systems

We are concerned with averaging theorems for ε-small stochastic perturbations of integrable equations in Rd×Tn={(I,φ)} (Formula presented.) and in R2n={v=(v1,⋯,vn),vj∈R2}, (Formula presented.) where I=(I1,⋯,In) is the vector of actions, Ij=12‖vj‖2. The vector-functions θ 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 ε, for 0≤t≤ε-1T. 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≤τ:=εt≤T and ε→0. Then we show that if the system of averaged I-equations is mixing, then the convergence is uniform in the slow time τ=εt≥0. Next using these results, for ε-perturbed equations (2) we construct well posed effective stochastic equations for v(τ)∈R2n (independent of ε) such that when ε→0, the actions of solutions for the perturbed equations with t:=τ/ε 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 τ≥0. We provide easy sufficient conditions on the perturbed equations which ensure that our results apply to their solutions.