Electronic Journal of Probability

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

Nils Berglund, Giacomo Di Gesù, and Hendrik Weber

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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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 41, 27 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35K57: Reaction-diffusion equations
Secondary: 81S20: Stochastic quantization 82C28: Dynamic renormalization group methods [See also 81T17]

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

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Berglund, Nils; Di Gesù, Giacomo; Weber, Hendrik. An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two. Electron. J. Probab. 22 (2017), paper no. 41, 27 pp. doi:10.1214/17-EJP60. https://projecteuclid.org/euclid.ejp/1493345028

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