Pathwise description of dynamic pitchfork bifurcations with additive noise

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.