×

Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers law and beyond. (English) Zbl 1285.60060

Summary: 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 a previous work. 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.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35K57 Reaction-diffusion equations
60J45 Probabilistic potential theory
37H20 Bifurcation theory for random and stochastic dynamical systems
PDF BibTeX XML Cite
Full Text: DOI arXiv