Pathwise description of dynamic pitchfork bifurcations with additive noise

The slow drift (with speed epsilon) 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 rootepsilon after the bifurcation, the paths are concentrated in a region of size sigma/epsilon(1/4) around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval [rootepsilon, crootepsilon\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.