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November 2002 A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential
Nils Berglund, Barbara Gentz
Ann. Appl. Probab. 12(4): 1419-1470 (November 2002). DOI: 10.1214/aoap/1037125869

Abstract

Additive white noise may significantly increase the response of bistable systems to a periodic driving signal. We consider the overdamped motion of a Brownian particle in two classes of double-well potentials, symmetric and asymmetric ones. These potentials are modulated periodically in time with period $1/\eps$, where $\eps$ is a moderately (not exponentially) small parameter. We show that the response of the system changes drastically when the noise intensity $\sigma$ crosses a threshold value. Below the threshold, paths are concentrated in 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 localized in time near the instants of minimal barrier height. We give a mathematically rigorous description of the behavior of individual paths, which allows us, in particular, to determine the power-law dependence of the critical noise intensity on $\eps$ and on the minimal barrier height, as well as the asymptotics of the transition and nontransition probabilities.

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Nils Berglund. Barbara Gentz. "A sample-paths approach to noise-induced synchronization: Stochastic resonance in a double-well potential." Ann. Appl. Probab. 12 (4) 1419 - 1470, November 2002. https://doi.org/10.1214/aoap/1037125869

Information

Published: November 2002
First available in Project Euclid: 12 November 2002

zbMATH: 1023.60052
MathSciNet: MR1936599
Digital Object Identifier: 10.1214/aoap/1037125869

Subjects:
Primary: 37H99 , 60H10
Secondary: 34E15 , 34F05

Keywords: Additive Noise , concentration of measure , double-well potential , noise-induced synchronization , nonautonomous stochastic differential equations , pathwise description , Random dynamical systems , singular perturbations , Stochastic resonance

Rights: Copyright © 2002 Institute of Mathematical Statistics

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Vol.12 • No. 4 • November 2002
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