Random Perturbations of Dynamical Systems: Large Deviations and Averaging

Abstract

A rewiew of the theory of random perturbations of dynamical systems is presented in this paper.Limit theorems for large deviations is an important tool in problems concerning the long time behavior of the perturbed system. But for some important classes of dynamical systellls ,for example,for Halniltonian systems,such an approach docs not works. A new approach based on a developement of the averaging principle has been suggested.It turns out that for the white noise type perturbations the slow component of the perturbed motion converges,under some assumptions,to a diffusion process on a graph corresponding to the first integral of the nonperturbed system. Perturbations of the Hamiltonian systems in the plane and of area-preserving systems on a torus are considered. The slow component of the perturbed system converges to a jumping process on the graph in the case of impuls-like perturbations.