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

Nils Berglund, Giacomo Di Gesù and Hendrik Weber
Electronic J. Probability 22 (41):1-27 (2017)

We study spectral Galerkin approximations of an Allen-Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength ε^1/2. 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 ε → 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.

Mathematical Subject Classification: 60H15, 60K35 (primary), 81S20, 82C28 (secondary).

Keywords and phrases: Stochastic partial differential equations, metastability, Kramers' law, renormalisation, potential theory, capacities, spectral Galerkin approximation, Wick calculus.

 

Journal Homepage Published article: 10.1214/17-EJP60 MR3646067 Zbl1362.60059

PDF file (992 Kb) hal-01304559 arXiv/1604.05742