Geometric singular perturbation theory for stochastic differential equations

Abstract

We consider slow-fast systems of differential equations, in which both the slow and fast variables are perturbed by noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold, which we construct explicitly. Depending on the dynamics of the reduced system, the results cover time spans which can be exponentially long in the noise intensity squared (that is, up to Kramers' time). We obtain exponentially small upper and lower bounds on the probability of exceptional paths. If the slow manifold contains bifurcation points, we show similar concentration properties for the fast variables corresponding to non-bifurcating modes. We also give conditions under which the system can be approximated by a lower-dimensional one, in which the fast variables contain only bifurcating modes.