Abstract

We consider a small perturbation by space-time Gaussian white noise of the Allen-Cahn equation. The latter is a nonlinear PDE, which can be seen as a gradient flow with respect to a double-well potential. The perturbed stochastic evolution is then a paradigmatic model exhibiting metastable dynamics: before exploring the full state space and reaching equilibrium, the system remains localized at the bottom of one well for a very long time. In the talk I will present a general approach to get metastability estimates in this infinite-dimensional setting. The focus is on sharp estimates that go beyond rough large deviation asymptotics and that are crucial for deriving coarse-grained effective dynamics. A key ingredient of the method is the systematic use of log-Sobolev inequalities in order to lift tunnelling calculations to infinite dimensions. As a main application we show how to compute the leading asymptotic behavior of the exponentially small spectral gap. We obtain an explicit formula expressed in terms of a certain Fredhom determinant as prefactor. This result shows that the gap behaves like the inverse of the average tunnelling time between wells and provides an alternative, spectral-theoretic way to prove the Eyring-Kramers formula.

Based on joint work with Morris Brooks (IST Austria)

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Seminars in Probability and Finance