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Article Dans Une Revue Electronic Journal of Probability Année : 2017

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

Résumé

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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Dates et versions

hal-01304559 , version 1 (19-04-2016)
hal-01304559 , version 2 (27-04-2016)
hal-01304559 , version 3 (23-01-2017)

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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, 2017, 22 (41), pp.1-27. ⟨10.1214/17-EJP60⟩. ⟨hal-01304559v3⟩
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