Two mathematical approaches of stochastic resonance, Interacting Stochastic Systems

Summary. We consider a random dynamical system describing the diusion 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 diusion dynamics in the small noise limit to a Markov chain jumping between its meta-stable states plays an important role. We study two dierent measures of tuning quality for stochastic resonance, with special emphasis on their robustness properties when passing to the reduced dy-namics of the Markov chains in the small noise limit. The rst one is the physi-cists favourite, spectral power amplication. It is analyzed by means of the spectral properties of the diusion's innitesimal generator in a framework where the sys-tem switches every half period between two spatially antisymmetric potential states. Surprisingly, resonance properties of diusion and Markov chain dier due to the crucial signicance of small intra-well uctuations 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 re-ned 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.