A four-thirds law for phase randomization of stochastically perturbed oscillators and related phenomena

LetI be a set of invariants and be a set of angle variables for a system of differential equations with anO() vector field. When time dependent stochastic perturbations, also ofO(), are added to the system, we have shown that under suitable conditionsI becomes a stochastic adiabatic invariant satisfying a diffusion equation on time scales of order 1/², in the limit as »0. Here we show that the angle variables converge weakly to a Gaussian Markov process on an O(^-4/3) time scale, and thus the phase becomes randomized at these times. Application to nearly integrable Hamiltonian systems is considered.Supported by NSF grant DMR-8704348