Large deviations principles of Non-Freidlin-Wentzell type.

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 s, 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 s(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 notrobust 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 su er 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 di usions. Large deviation principles developed forweakly time inhomogeneous di usions 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.