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

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 $\sqrt{\epsilon }$. 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 $\epsilon \to 0$. 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.

28 pages, 1 Figure. Revised version with expanded discussion

References in corpus (2)

• Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond
• Metastability in Interacting Nonlinear Stochastic Differential Equations I: From Weak Coupling to Synchronisation

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