Averaging Principle for Stochastic Perturbations of Multifrequency Systems

Abstract

We consider the averaging principle for deterministic and stochastic perturbations of multidimensional dynamical systems for which coordinates can be introduced in such a way that the "fast" coordinates change in a torus (for Hamiltonian systems, "action-angle coordinates"). Stochastic perturbations of the white-noise type are considered. Our main assumption is that the set of action values for which the frequencies of the motion on corresponding tori are rationally dependent (and so the motion reduces to a torus of smaller dimension) has Lebesgue measure zero. Our results about stochastic perturbations imply some new results for averaging of purely deterministic perturbations.