Pathwise description of dynamic pitchfork bifurcations with additive noise

Nils Berglund and Barbara Gentz
Probab. Theory Related Fields 122:341-388 (2002)

The slow drift (with speed ε) 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 σ, by giving precise estimates on the behaviour of the individual paths. We show that until time ε^1/2 after the bifurcation, the paths are concentrated in a region of size σ/ε^1/4 around the bifurcating equilibrium. With high probability, they leave a neighbourhood of this equilibrium during a time interval [ε^1/2, c(ε|log σ|)^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.

2000 Mathematics Subject Classification: 37H20, 60H10 (primary), 34E15, 93E03 (secondary).

Keywords and phrases: Dynamic bifurcation, pitchfork bifurcation, additive noise, bifurcation delay, singular perturbations, stochastic differential equations, random dynamical systems, pathwise description, concentration of measure.

 

Journal Homepage Published article: 10.1007/s004400100174 MR1892851 Zbl1008.37031

PDF file (608 Kb) hal-00130541 arXiv/math.PR/0008208