[Zurück]

G. Di Gesu, N Berglund, H. Weber:
"An Eyring-Kramers law for the stochastic Allen-Cahn equation in dimension two";
Electronic Journal of Probability, 22 (2017), 41; S. 1 - 27.

We study spectral Galerkin approximations of an Allen-Cahn equation over the twodimensional 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.

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

http://dx.doi.org/10.1214/17-EJP60

http://publik.tuwien.ac.at/files/publik_268287.pdf