An Eyring-Kramers law for the stochastic Allen-Cahn equation in dimension two

TL;DR

This paper establishes an Eyring-Kramers law for the stochastic Allen-Cahn equation in two dimensions, incorporating Wick renormalisation and providing uniform transition time bounds as noise diminishes.

Contribution

It introduces a Wick renormalisation for the 2D stochastic Allen-Cahn equation and derives a modified Eyring-Kramers formula with renormalised determinants.

Findings

Sharp bounds on transition times between stable states

Uniform estimates suggesting a limiting Eyring-Kramers law

Modification of the prefactor due to infinite renormalisation

Abstract

We study spectral Galerkin approximations of an Allen--Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon}$. We introduce a Wick renormalisation of the equation in order to have a system that is well-defined as the regularisation is removed. We show sharp upper and lower bounds on the transition times from a neighbourhood of the stable configuration $-1$ to the stable configuration $1$ in the asymptotic regime $\varepsilon \to 0$. These estimates are uniform in the discretisation parameter $N$, suggesting an Eyring-Kramers formula for the limiting renormalised stochastic PDE. The effect of the "infinite renormalisation" is to modify the prefactor and to replace the ratio of determinants in the finite-dimensional Eyring-Kramers law by a renormalised Carleman-Fredholm determinant.