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An Eyring-Kramers formula for the spectral gap of the stochastic Allen-Cahn equation

Date: Tuesday, January 7, 2020 17:30 - 18:30
Speaker: Giacomo Di Gesù (TU Wien)
Location: Heinzel Seminar Room / Office Bldg West (I21.EG.101)
Series: Mathematics and CS Seminar
Host: M. Beiglböck, N. Berestycki, L. Erdös, J. Maas

We consider the Allen-Cahn equation on the one-dimensional torus, perturbed by a small spacetime white noise. The deterministic equation is a nonlinear PDE, which can be seen as a gradient flow with respect to a double-well energy. If a small noise is added, the typical picture of a metastable dynamics emerges: the system quickly reaches a local equilibrium in one of the two wells; this state endures for an exponentially long time, until a sufficiently large stochastic fluctuation enables the system to overcome the energetic barrier separating the two wells. This behavior produces a slowdown in the relaxation to the equilibrium measure, reflected e.g. by an exponentially small spectral gap. In th alk I will present a technique which provides a formula for the precise asymptotic behavior of the spectral gap, showing that the prefactor is given by a suitable Fredholm determinant. The formula shows that the gap behaves like twice the inverse of the metastable transition time from one well to the other.
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