Geometric singular perturbation theory for stochastic differential equations

Nils Berglund and Barbara Gentz
J. Differential Equations 191:1-54 (2003)

We consider slow-fast systems of differential equations, in which both the slow and fast variables are perturbed by additive 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 give 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.

2000 Mathematics Subject Classification: 37H20, 34E15 (primary), 60H10 (secondary).

Keywords and phrases: Singular perturbations, slow-fast systems, invariant manifolds, dynamic bifurcations, stochastic differential equations, first-exit times, concentration of measure.

 

Journal Homepage Published article: 10.1016/S0022-0396(03)00020-2 MR1973280 Zbl1053.34048

PDF file (576 Kb) hal-00003006 arXiv/math.PR/0204008