Transition metastable times for a class of stochastic partial differential equations

In this talk, we study a class of scalar, parabolic, semi-linear stochastic partial differential equations perturbed by a space-time white noise on a bounded real intervalin the small noise asymptotic.

Due to the stochastic term, metastable transitions occur between different stable equilibriums of the deterministic dynamical system and different behaviors on different timescales happen. We compute the expectation of the transition times for some models (so-called Eyring-Kramers Formula).

The proof use a finite difference approximation and a coupling and apply finite dimensional estimates to the approximation (using potential theory and capacities). We prove the uniformity of the estimates in the dimension and then we take the limit to recover the infinite dimensional equation.