I will discuss ε-small stochastic perturbations of an integrable Hamiltonian system in R2n. Firstly I will write the perturbed equation using the action-angle variables of the integrable system, and formally average the obtained fast-slow system. The averaged equation for actions which we get in this way indeed describes the dynamics of the original equation for t ≾ε-1, but only under some serious restrictions, which I will explain. A better way to study the long time dynamics of actions is inspired by the Krylov–Bogolyubov averaging: motivated by the latter, we guess in R2n a regular auxiliary equation, obtained by some averaging of the original one. Then we prove that under much weaker restrictions the actions of its solutions approximate those for solutions of the original equation for t ≾ε-1. Moreover, imposing some more restrictions on the equation we prove that this approximation holds uniformly in time.

The talk is based on joint works with Andrey Piatnitski, Huang Guan and Guo Jing.

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