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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 √ 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.
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https://hal.archives-ouvertes.fr/hal-01304559
Contributor : Nils Berglund <>
Submitted on : Monday, January 23, 2017 - 7:06:21 AM
Last modification on : Thursday, February 7, 2019 - 4:57:00 PM
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Nils Berglund, Giacomo Di Gesù, Hendrik Weber. An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two. Electronic Journal of Probability, Institute of Mathematical Statistics (IMS), 2017, 22 (41), pp.1-27. ⟨10.1214/17-EJP60⟩. ⟨hal-01304559v3⟩

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