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

Nils Berglund (1) , Barbara Gentz (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 Fakultät für Mathematik = Faculty of Mathematics [Bielefeld]

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.

Mots clés: Stochastic partial differential equations; parabolic equations; reaction-diffusion equations; metastability; Kramers' law; exit problem; transition time; large deviations; Wentzell-Freidlin theory; potential theory; capacities; Galerkin approximation; subexponential asymptotics; pitchfork bifurcation.; pitchfork bifurcation

Domaines: Probabilités [math.PR]

Nils Berglund, Barbara Gentz. Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond. 2012. ⟨hal-00666605v1⟩