Nils Berglund and Barbara Gentz
Pathwise description of dynamic pitchfork bifurcations
with additive noise
(673K, Postscript)
ABSTRACT. The slow drift (with speed $\eps$) of a parameter
through a pitchfork bifurcation point, known as the dynamic
pitchfork bifurcation, is characterized by a significant delay
of the transition from the unstable to the stable state. We
describe the effect of an additive noise, of intensity $\sigma$,
by giving precise estimates on the behaviour of the individual
paths. We show that until time $\sqrt\eps$ after the bifurcation,
the paths are concentrated in a region of size $\sigma/\eps^{1/4}$
around the bifurcating equilibrium. With high probability, they
leave a neighbourhood of this equilibrium during a time interval
$[\sqrt\eps,c\sqrt{\eps\abs{\log\sigma}}]$, after which they are
likely to stay close to the corresponding deterministic solution.
We derive exponentially small upper bounds for the probability of
the sets of exceptional paths, with explicit values for the exponents.