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2017 An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two
Nils Berglund, Giacomo Di Gesù, Hendrik Weber
Electron. J. Probab. 22: 1-27 (2017). DOI: 10.1214/17-EJP60


We study spectral Galerkin approximations of an Allen–Cahn equation over the two-dimensional torus perturbed by weak space-time white noise of strength $\sqrt{\varepsilon } $. 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 $\varepsilon \to 0$. 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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Nils Berglund. Giacomo Di Gesù. Hendrik Weber. "An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two." Electron. J. Probab. 22 1 - 27, 2017.


Received: 27 April 2016; Accepted: 23 April 2017; Published: 2017
First available in Project Euclid: 28 April 2017

zbMATH: 1362.60059
MathSciNet: MR3646067
Digital Object Identifier: 10.1214/17-EJP60

Primary: 35K57 , 60H15
Secondary: 81S20 , 82C28

Keywords: Capacities , Kramers’ law , metastability , potential theory , renormalisation , spectral Galerkin approximation , Stochastic partial differential equations , Wick calculus

Vol.22 • 2017
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