Abstract:  I will discuss stochastic epsilon-perturbations of an integrable Hamiltonian system in R^{2n}. I will show that, firstly, on time intervals of order 1/epsilon the actions of solutions for perturbed equations are close to those of solutions for specially constructed effective stochastic equations, independent from epsilon. Secondly, if the effective equation is mixing, then the approximation of actions of solutions for perturbed equations, provided by this equation, is uniform in time. All imposed restrictions admit easy sufficient conditions. I will discuss applications of the obtained results to perturbations of chains of nonlinear oscillators, related to the problem of describing the heat conduct in crystals. 

Bio Sketch: Sergei Kuksin is a professor of mathematics at the Université Paris Cité and Sorbonne Université and at Steklov Mathematical Institute of RAS, a head of laboratory in the RUDN University (Moscow) and a professor at the Institute for Financial Studies, Shandong University. His research is the broad area of analysis and probability, including KAM theory, PDEs with randomness, turbulence and statistical hydrodynamics, and elliptic PDEs for functions between compact manifolds. In 1992 he was a plenary speaker with talk "KAM theory for partial differential equations" at the European Congress of European Mathematicians in Paris. In 1998 he was an invited speaker at International Congress of Mathematicians in Berlin. In 2016 he received the Lyapunov Prize of the Russian Academy of Sciences.