We consider potential type dynamical systems in finite dimensions with twometa-stable states. They are subject to two sources of perturbation: a slowexternal periodic perturbation of period $T$ and a small Gaussian randomperturbation of intensity $\epsilon$, and, therefore, are mathematicallydescribed as weakly time inhomogeneous diffusion processes. A system is instochastic resonance, provided the small noisy perturbation is tuned in such away that its random trajectories follow the exterior periodic motion in anoptimal fashion, that is, for some optimal intensity $\epsilon (T)$. Thephysicists' favorite, measures of quality of periodic tuning--and thusstochastic resonance--such as spectral power amplification or signal-to-noiseratio, have proven to be defective. They are not robust w.r.t. effective modelreduction, that is, for the passage to a simplified finite state Markov chainmodel reducing the dynamics to a pure jumping between the meta-stable states ofthe original system. An entirely probabilistic notion of stochastic resonancebased on the transition dynamics between the domains of attraction of themeta-stable states--and thus failing to suffer from this robustness defect--wasproposed before in the context of one-dimensional diffusions. It isinvestigated for higher-dimensional systems here, by using extensions andrefinements of the Freidlin--Wentzell theory of large deviations for timehomogeneous diffusions. Large deviations principles developed for weakly timeinhomogeneous diffusions prove to be key tools for a treatment of the problemof diffusion exit from a domain and thus for the approach of stochasticresonance via transition probabilities between meta-stable sets.
展开▼
