A generalization of the Freidlin–Wentcell theorem on averaging of Hamiltonian systems

Abstract

In this paper, we generalize the classical Freidlin-Wentzell’s theorem for random perturbations of Hamiltonian systems. In (Probability Theory and Related Fields 128 (2004) 441–466), M.Freidlin and M.Weber generalized the original result in the sense that the coefficient for the noise term is no longer the identity matrix but a state-dependent matrix and taking the drift term into consideration. In this paper, We generalize the result by adding a state-dependent matrix that converges uniformly to 0 on any compact sets as ϵ tends to 0 to a state-dependent noise and considering the drift term which contains two parts, the state-dependent mapping and a state-dependent mapping that converges uniformly to 0 on any compact sets as ϵ tends to 0. In the proof, we adapt a new way to prove the weak convergence inside the edge by constructing an auxiliary process and modify the proof in (Probability Theory and Related Fields 128 (2004) 441–466) when proving gluing condition.