The Reduction of Potential Diffusions to Finite State Markov Chains and Stochastic Resonance

Abstract

Consider a dynamical system describing the motion of a particle in a double well potential with a periodic perturbation of very small frequency, and a white noise perturbation of intensity e. If its trajectories amplify the small periodic perturbation in a ‘best possible way’, it is said to be in stochastic resonance. A lower bound for the ratio of amplitude and logarithm of the period above which quasi-deterministic periodic behavior can be observed is obtained via large deviations theory. However, to obtain optimality, periodicity of trajectories has to be studied by means of a measure of quality of tuning such as spectral power amplification. In the particular setting where the potential alternates every half period between two spatially antisymmetric double well states we encounter a surprise. The stochastic resonance pattern is not correctly described by the reduced dynamics associated with a two state Markov chain whose periodic hopping rates between the potential minima mimic the large (spatial) scale motion of the diffusion. Only if small scale fluctuations inside the potential wells where the diffusion spends most of its time are carefully eliminated, the reduced dynamics is robust.