Heuristics of noise induced transitions Transitions for time homogeneous dynamical systems with small noise Semiclassical theory of stochastic resonance in dimension 1 Large deviations and transitions between meta-stable states of dynamical systems with small noise and weak inhomogeneity Supplementary tools Laplace's method Bibliography Index

We investigate exit times from domains of attraction for the motion of a self-stabilized particle traveling in a geometric (potential type) landscape and perturbed by Brownian noise of small amplitude. Self-stabilization is the effect of including an ensemble-average attraction in addition to the usual state-dependent drift, where the particle is supposed to be suspended in a large population of identical ones. A Kramers' type law for the particle's exit from the potential's domains of attraction and a large deviations principle for the self-stabilizing diffusion are proved. It turns out that the exit law for the self-stabilizing diffusion coincides with the exit law of a potential diffusion without self-stabilization and a drift component perturbed by average attraction. We show that self-stabilization may substantially delay the exit from domains of attraction, and that the exit location may be completely different.

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.

We consider potential type dynamical systems in flnite 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 ", and therefore mathematically described as weakly time inhomogeneous difiusion 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 "(T). The physicists' favorite measures of quality of periodic tuning { and thus stochastic resonance { such as spectral power ampliflcation or signal-to-noise ratio have proven to be defective. They are not robust w.r.t. efiective model reduction, i.e. for the passage to a simplifled flnite 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 sufier from this robustness defect { was proposed before in the context of one-dimensional difiusions. It is investigated for higher dimensional systems here, by using extensions and reflnements of the Freidlin-Wentzell theory of large deviations for time homogeneous difiusions. Large deviation principles developed for weakly time inhomogeneous difiusions prove to be key tools for a treatment of the problem of difiusion exit from a domain and thus for the approach of stochastic resonance via transition probabilities between meta-stable sets.