Nils Berglund and Barbara Gentz
Geometric singular perturbation theory 
for stochastic differential equations
(556K, Postscript)

ABSTRACT.  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.