Join the Mathematics Department Colloquia for a lecture with Professeur Nils Berglund from the Institut Denis Poisson, Universite d'Orleans.

In this talk, we will consider parabolic stochastic partial differential equations (SPDEs) of Allen-Cahn type, which are used as models for domain evolution in ferromagnetic materials, and of phase separation in alloys. These equations show a metastable behavior, meaning that for weak noise, or low temperature, there are two or more states in which the system spends exponentially long times. We obtain sharp asymptotics for the expectation of transition times between such states in two situations. The first situation occurs for a general class of parabolic SPDEs on the one-dimensional torus, subjected to space-time white noise. In that case, we find that the mean transition time is proportional to a Fredholm determinant. The second situation occurs for the Allen-Cahn SPDE on the two-dimensional torus, again with space-time white noise. This equation requires a renormalization procedure to be well-posed, and we show that the mean transition time is proportional to a renormalized Carleman-Frehholm determinant. The talk is based on joint works with Barbara Gentz, Giacomo Di Gesù, and Hendrik Weber.