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

作者：Berglund Nils; Di Gesu Giacomo; Weber Hendrik

来源：Electronic Journal of Probability, 2017, 22(0): 41.

DOI：10.1214/17-EJP60

• Stochastic partial differential equations
• metastability
• Kramers' law
• renormalisation
• potential theory
• capacities
• spectral Galerkin approximation
• Wick calculus

We study spectral Galerkin approximations of an Allen-Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength root 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 -> 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.

• 出版日期2017