WIAS Preprint No. 688, (2001)

Beyond the Fokker-Planck equation: Pathwise control of noisy bistable systems


  • Berglund, Nils
  • Gentz, Barbara

2010 Mathematics Subject Classification

  • 37H20 60H10 34E15 82C31

2008 Physics and Astronomy Classification Scheme

  • 02.50.-r, 05.10.Gg, 75.60.-d, 92.40.Cy


  • Langevin equation, Fokker-Planck equation, double-well potential, first-exit time, scaling laws, stochastic resonance, dynamicalhysteresis, bifurcation delay, white noise, coloured noise


We introduce a new method, allowing to describe slowly time-dependent Langevin equations through the behaviour of individual paths. This approach yields considerably more information than the computation of the probability density. The main idea is to show that for sufficiently small noise intensity and slow time dependence, the vast majority of paths remain in small space-time sets, typically in the neighbourhood of potential wells. The size of these sets often has a power-law dependence on the small parameters, with universal exponents. The overall probability of exceptional paths is exponentially small, with an exponent also showing power-law behaviour. The results cover time spans up to the maximal Kramers time of the system. We apply our method to three phenomena characteristic for bistable systems: stochastic resonance, dynamical hysteresis and bifurcation delay, where it yields precise bounds on transition probabilities, and the distribution of hysteresis areas and first-exit times. We also discuss the effect of coloured noise.

Appeared in

  • J. Phys. A: Math. Gen. 35 (2002), 2057-2091

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