Stochastic resonance in stochastic PDEs

TL;DR

This paper investigates how noise intensity influences the likelihood of state transitions in stochastic PDEs with periodic forcing, revealing a critical threshold that determines exponential transition probabilities.

Contribution

It extends the understanding of stochastic resonance from 1D SDEs to infinite-dimensional SPDEs with periodic drift and noise, establishing a critical noise level for transition probabilities.

Findings

Existence of a critical noise intensity for transitions.

Transition probabilities are exponentially small below the threshold.

Transition probabilities are nearly certain above the threshold.

Abstract

We consider stochastic partial differential equations (SPDEs) on the one-dimensional torus, driven by space-time white noise, and with a time-periodic drift term, which vanishes on two stable and one unstable equilibrium branches. Each of the stable branches approaches the unstable one once per period. We prove that there exists a critical noise intensity, depending on the forcing period and on the minimal distance between equilibrium branches, such that the probability that solutions of the SPDE make transitions between stable equilibria is exponentially small for subcritical noise intensity, while they happen with probability exponentially close to $1$ for supercritical noise intensity. Concentration estimates of solutions are given in the $H^s$ Sobolev norm for any $s<\frac12$. The results generalise to an infinite-dimensional setting those obtained for $1$-dimensional SDEs in [Nils Berglund and Barbara Gentz. A sample-paths approach to noise-induced synchronization: stochastic resonance in a double-well potential. Ann. Appl. Probab., 12(4):1419-1470, 2002].