Pathwise description of dynamic pitchfork bifurcations with additive noise
- Berglund, Nils
- Gentz, Barbara
2010 Mathematics Subject Classification
- 37H20 60H10 34E15 93E03
- Dynamic bifurcation, pitchfork bifurcation, additive noise, bifurcation delay, singular perturbations, stochastic differential equations, random dynamical systems, pathwise description, concentration of measure
The slow drift (with speed e) 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 s, by giving precise estimates on the behaviour of the individual paths. We show that until time e1/2 after the bifurcation, the paths are concentrated in a region of size s/e1/4 around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval [e1/2, c (e log s )1/2], 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.
- Probab. Theory Related Fields 122 (2002) 3, 341-388