Two Mathematical Approaches to Stochastic Resonance

Summary

We consider a random dynamical system describing the diffusion of a small-noise Brownian particle in a double-well potential with a periodic perturbation of very large period. According to the physics literature, the system is in stochastic resonance if its random trajectories are tuned in an optimal way to the deterministic periodic forcing. The quality of periodic tuning is measured mostly by the amplitudes of the spectral components of the random trajectories corresponding to the forcing frequency. Reduction of the diffusion dynamics in the small noise limit to a Markov chain jumping between its meta-stable states plays an important role.

We study two different measures of tuning quality for stochastic resonance, with special emphasis on their robustness properties when passing to the reduced dynamics of the Markov chains in the small noise limit. The first one is the physicists favourite, spectral power amplification. It is analyzed by means of the spectral properties of the diffusion’s infinitesimal generator in a framework where the system switches every half period between two spatially antisymmetric potential states. Surprisingly, resonance properties of diffusion and Markov chain differ due to the crucial significance of small intra-well fluctuations for spectral concepts. To avoid this defect, we design a second measure of tuning quality which is based on the pure transition mechanism between the meta-stable states. It is investigated by refined large deviation methods in the more general framework of smooth periodically varying potentials, and proves to be robust for the passage to the reduced dynamics.