The Annals of Applied Probability

Transition times and stochastic resonance for multidimensional diffusions with time periodic drift: A large deviations approach

Samuel Herrmann, Peter Imkeller, and Dierk Peithmann

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Abstract

We consider potential type dynamical systems in finite dimensions with two meta-stable states. They are subject to two sources of perturbation: a slow external periodic perturbation of period T and a small Gaussian random perturbation of intensity ɛ, and, therefore, are mathematically described as weakly time inhomogeneous diffusion processes. A system is in stochastic resonance, provided the small noisy perturbation is tuned in such a way that its random trajectories follow the exterior periodic motion in an optimal fashion, that is, for some optimal intensity ɛ(T). The physicists’ favorite, measures of quality of periodic tuning—and thus stochastic resonance—such as spectral power amplification or signal-to-noise ratio, have proven to be defective. They are not robust w.r.t. effective model reduction, that is, for the passage to a simplified finite state Markov chain model reducing the dynamics to a pure jumping between the meta-stable states of the original system. An entirely probabilistic notion of stochastic resonance based on the transition dynamics between the domains of attraction of the meta-stable states—and thus failing to suffer from this robustness defect—was proposed before in the context of one-dimensional diffusions. It is investigated for higher-dimensional systems here, by using extensions and refinements of the Freidlin–Wentzell theory of large deviations for time homogeneous diffusions. Large deviations principles developed for weakly time inhomogeneous diffusions prove to be key tools for a treatment of the problem of diffusion exit from a domain and thus for the approach of stochastic resonance via transition probabilities between meta-stable sets.

Article information

Source
Ann. Appl. Probab., Volume 16, Number 4 (2006), 1851-1892.

Dates
First available in Project Euclid: 17 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1169065210

Digital Object Identifier
doi:10.1214/105051606000000385

Mathematical Reviews number (MathSciNet)
MR2288707

Zentralblatt MATH identifier
1155.60022

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J60: Diffusion processes [See also 58J65] 60F10: Large deviations
Secondary: 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx] 86A10: Meteorology and atmospheric physics [See also 76Bxx, 76E20, 76N15, 76Q05, 76Rxx, 76U05] 34D45: Attractors [See also 37C70, 37D45]

Keywords
Stochastic resonance optimal tuning large deviations diffusion periodic potential double well potential noise induced transition exit time distribution perturbed dynamical system

Citation

Herrmann, Samuel; Imkeller, Peter; Peithmann, Dierk. Transition times and stochastic resonance for multidimensional diffusions with time periodic drift: A large deviations approach. Ann. Appl. Probab. 16 (2006), no. 4, 1851--1892. doi:10.1214/105051606000000385. https://projecteuclid.org/euclid.aoap/1169065210


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