WIAS Preprint No. 627, (2001)

A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential


  • Berglund, Nils
  • Gentz, Barbara

2010 Mathematics Subject Classification

  • 37H99 60H10 34F05 34E15


  • Stochastic resonance, noise-induced synchronization, double-well potential, additive noise, random dynamical systems, non-autonomous stochastic differential equations, singular perturbations, pathwise description, concentration of measure


Additive white noise may significantly increase the response of bistable systems to a periodic driving signal. We consider two classes of double-well potentials, symmetric and asymmetric, modulated periodically in time with period 1/ε where ε is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity σ crosses a threshold value. Below the threshold, paths are concentrated near one potential well, and have an exponentially small probability to jump to the other well. Above the threshold, transitions between the wells occur with probability exponentially close to 1/2 in the symmetric case, and exponentially close to 1 in the asymmetric case. The transition zones are localised in time near the points of minimal barrier height. We give a mathematically rigorous description of the behaviour of individual paths, which allows us, in particular, to determine the power-law dependence of the critical noise intensity on ε and on the minimal barrier height, as well as the asymptotics of the transition and non-transition probabilities.

Appeared in

  • Ann. Appl. Probab. 12, 1419-1470 (2002)

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