Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond
Résumé
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]Origine | Fichiers produits par l'(les) auteur(s) |
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