A mathematical approach of stochastic resonance

We consider a dynamical system describing the diffusive motion of a particle in a double well potential with a periodic perturbation of very small frequency, and an additive stochastic perturbation of small amplitude. It is in stochastic resonance if the solution trajectories amplify the small periodic perturbation in a 'best possible way'. Systems of this type first appeared in simple energy balance models designed for a qualitative explanation of global glacial cycles. Large deviations theory provides a lower bound for the proportion of the amplitude and the logarithm of the period above which quasi-deterministic periodic behavior can be observed. To obtain optimality, one has to measure periodicity with a measure of quality of tuning. Notions of quality of tuning widely used in physics such as the spectral power amplification or the signal-to-noise ratio depend on the spectral properties of the averaged trajectories of the diffusion. These notions pose serious mathematical problems if the underlying system is reduced to simpler Markov chain models on the finite state space composed of the meta-stable states of the potential landscape in the limit of small noise. As a way out of this dilemma we propose to measure the quality of periodic tuning by the probability that transitions between the domains of attraction of the potential wells happen during a parametrized time window maximized in the window parameter. This notion can be investigated by means of uniform large deviations estimates and turns out to be robust for the passage to dimension reduced Markov chains.