2016/04/30 by Nils Berglund, Giacomo Di Gesù, Hendrik Weber · 1 citation
Mathematics · Physics and Astronomy · #math.PR #math-ph #math.MP #msc:60H15 #msc:35K57 #msc:81S20 #msc:82C28
paper · pdf · doi:10.1214/17-ejp60
published as Electron. J. Probab. 22 (2017), no. 41, 27 pp · 28 pages, 1 Figure. Revised version with expanded discussion
arxiv created 2017/01/21 · arxiv updated 2017/05/01
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). 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.