A 4/3 LAW FOR PHASE RANDOMIZATION OF STOCHASTICALLY PERTURBED OSCILLATORS AND RELATED PHENOMENA

Let I be a set of invariants and theta be a set of angle variables for a system of differential equations with an O(epsilon) vector field. When time dependent stochastic perturbations, also of O(epsilon), are added to the system, we have shown that under suitable conditions I becomes a stochastic adiabatic invariant satisfying a diffusion equation on time scales of order 1/epsilon2, in the limit as epsilon --> 0. Here we show that the angle variables converge weakly to a Gaussian Markov process on an O(epsilon-4/3) time scale, and thus the phase becomes randomized at these times. Application to nearly integrable Hamiltonian systems is considered.