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

We prove a Kramers-type law for metastable transition times for a class of onedimensional 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 [BG09]. 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. 5 February 2012. 2010 Mathematical Subject Classification. 60H15, 35K57 (primary), 60J45, 37H20 (secondary)