An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures

Abstract

These notes have been prepared for a series of lectures given at the Sarajevo Stochastic Analysis Winter School, from January 28 to February 1, 2019. There already exist several excellent lecture notes and reviews on the subject, such as (Hairer 2009) on (non-singular) stochastic PDEs, and (Hairer 2015, Chandra and Weber 2017) on singular stochastic PDEs and regularity structures. The present notes have two main specificities. The first one is that they focus on a particular example, the Allen-Cahn equation, which allows to introduce several of the difficulties of the theory in a gradual way, by increasing the space dimension step by step. The hope is that while this limits the generality of the theory presented, this limitation is more that made up by a gain in clarity. The second specific aspect of these notes is that they go beyond existence and uniqueness of solutions, by covering a few recent results on convergence to equilibrium and metastability in these system.