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An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two
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Berglund, Nils, Di Gesù, Giacomo and Weber, Hendrik (2017) An Eyring–Kramers law for the stochastic Allen–Cahn equation in dimension two. Electronic Journal of Probability, 22 . 41. doi:10.1214/17-EJP60
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Official URL: http://dx.doi.org/10.1214/17-EJP60
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√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 −1−1 to the stable configuration 11 in the asymptotic regime ε→0ε→0. These estimates are uniform in the discretisation parameter NN, 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.
| Item Type: | Journal Article | ||||||
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| Subjects: | Q Science > QA Mathematics Q Science > QC Physics |
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| Divisions: | Faculty of Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Stochastic partial differential equations, Potential theory (Physics), Galerkin methods | ||||||
| Journal or Publication Title: | Electronic Journal of Probability | ||||||
| Publisher: | University of Washington. Dept. of Mathematics | ||||||
| ISSN: | 1083-6489 | ||||||
| Official Date: | 28 April 2017 | ||||||
| Dates: |
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| Volume: | 22 | ||||||
| Article Number: | 41 | ||||||
| DOI: | 10.1214/17-EJP60 | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Access rights to Published version: | Open Access | ||||||
| Funder: | European Research Council (ERC), Seventh Framework Programme (European Commission) (FP7), Engineering and Physical Sciences Research Council (EPSRC), Royal Society (Great Britain). University Research Fellowship | ||||||
| Grant number: | (FP/2007-2013) / ERC Grant Agreement number 614492 (ERC) (FP7) |
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