A STOCHASTIC-THEORY OF ADIABATIC INVARIANCE

Let I be a set of invariants for a system of differential equations with an order o(epsilon) vector field. When order epsilon-perturbations of zero mean are added to the system we show that, under suitable regularity and ergodicity conditions, I becomes an adiabatic invariant with maximal variations of order one on time scales of order 1/epsilon-2. In the stochastically perturbed case, I behaves asymptotically (for small epsilon) like a diffusion process on 1/epsilon-2 time scales. The results also apply to an interesting class of deterministic perturbations. This study extends the results of Khas'minskii on stochastically averaged systems, as well as some of the deterministic methods of averaging, to such invariants.