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Workshop

Large deviations for diffusions and stochastic resonance

  • Samuel Herrmann (Université Nancy, Nancy, France)
G3 10 (Lecture hall)

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 $\eps$, and therefore 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, i.e. for some optimal intensity $\eps(T)$.

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, i.e. 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 -- is investigated by using extensions and refinements of the Freidlin-Wentzell theory of large deviations for time homogeneous diffusions. Large deviation 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.

Katja Bieling

Max Planck Institute for Mathematics in the Sciences, Leipzig Contact via Mail

Peter Imkeller

Humboldt Universität zu Berlin

Stefan Müller

Max Planck Institute for Mathematics in the Sciences, Leipzig