RANDOM PERTURBATIONS OF NONLINEAR OSCILLATORS

Degenerate white noise perturbations of Hamiltonian systems in R-2 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 defined b y a diffusion process on a graph. Inside the edges the process is defi ned by a standard averaging procedure, but to define the process for a ll t > 0, one should add gluing conditions at the vertices. Calculatio n of the gluing conditions is based on delicate Hormander-type estimat es.