Random perturbations of nonlinear oscillators

Abstract

Degenerate white noise perturbations of Hamiltonian systems in R2 are studied. In particular, perturbations of a nonlinear oscillator with 1 degree of freedom are considered. If the oscillator has more than one stable equilibrium, the long time behavior of the system is denned by a diffusion process on a graph. Inside the edges the process is denned by a standard averaging procedure, but to define the process for all t > 0, one should add gluing conditions at the vertices. Calculation of the gluing conditions is based on delicate Hörmander-type estimates.