  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.
