We study spectral Galerkin approximations of an Allen-Cahn equation over the two-√ε 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.

An Eyring-Kramers law for the stochastic allen-cahn equation in dimension two

Di Gesu' G.;
2017-01-01

Abstract

We study spectral Galerkin approximations of an Allen-Cahn equation over the two-√ε 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.
2017
Berglund, N.; Di Gesu', G.; Weber, H.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11568/1081561
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