  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.
