Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond

TL;DR

This paper establishes precise estimates for the expected transition times in certain one-dimensional parabolic SPDEs with bistable potentials, extending Kramers' law to cases involving bifurcations and vanishing determinants.

Contribution

It rigorously proves a Kramers-type law for metastable transition times in parabolic SPDEs, including bifurcation scenarios where traditional determinants vanish.

Findings

Transition times depend exponentially on energy barriers.

Explicit prefactors involve functional determinants.

Results extend to bifurcation cases with vanishing determinants.

Abstract

We prove a Kramers-type law for metastable transition times for a class of one-dimensional parabolic stochastic partial differential equations (SPDEs) with bistable potential. The expected transition time between local minima of the potential energy depends exponentially on the energy barrier to overcome, with an explicit prefactor related to functional determinants. Our results cover situations where the functional determinants vanish owing to a bifurcation, thereby rigorously proving the results of formal computations announced in [Berglund and Gentz, J. Phys. A 42:052001 (2009)]. The proofs rely on a spectral Galerkin approximation of the SPDE by a finite-dimensional system, and on a potential-theoretic approach to the computation of transition times in finite dimension.