We would like to invite you to the following Probability seminar
that will take place on January 15 at 14.30 by the zoom platform.
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Speaker: Giacomo Di Gesù (Università di Pisa)
Title: Metastability for an SPDE via functional inequalities
15 JANUARY (Friday) - 14:30 zoom link: TBA
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).