Stochastic resonance: non-robust and robust tuning notions

We consider a dynamical system describing the diusive motion of a particle in a double well potential with a periodic perturbation of very small frequency, and an additive stochastic perturbation of amplitude ". It is in stochastic res-onance if the solution trajectories amplify the small periodic perturbation in a `best possible way'. Systems of this type rst appeared in simple energy balance models designed for a qualitative explanation of global glacial cycles. Large de-viations 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 amplication or the signal-to-noise ratio depend on the spectral properties of the averaged trajectories of the diusion. We show that these notions pose serious mathematical problems if the underlying system is reduced to simpler Markov chain models on the nite state space compose