The exit problem for diffusions with time-periodic drift and stochastic resonance

TL;DR

This paper introduces a new, robust notion of stochastic resonance for one-dimensional diffusions in periodically changing double-well potentials, based on exact transition rates and effective dynamics, improving upon previous spectral measures.

Contribution

It develops a refined concept of stochastic resonance using exponential transition rates that remain robust under effective Markov chain approximations.

Findings

New notion of stochastic resonance based on transition probabilities

Robustness of the measure under effective Markov chain dynamics

Identification of stochastic resonance points through transition probability maximization

Abstract

Physical notions of stochastic resonance for potential diffusions in periodically changing double-well potentials such as the spectral power amplification have proved to be defective. They are not robust for the passage to their effective dynamics: continuous-time finite-state Markov chains describing the rough features of transitions between different domains of attraction of metastable points. In the framework of one-dimensional diffusions moving in periodically changing double-well potentials we design a new notion of stochastic resonance which refines Freidlin's concept of quasi-periodic motion. It is based on exact exponential rates for the transition probabilities between the domains of attraction which are robust with respect to the reduced Markov chains. The quality of periodic tuning is measured by the probability for transition during fixed time windows depending on a time scale parameter. Maximizing it in this parameter produces the stochastic resonance points.