The Allen–Cahn action functional in higher dimensions

Abstract

The Allen--Cahn action functional is related to the probability of rare events in the stochastically perturbed Allen--Cahn equation. Formal calculations suggest a \emph{reduced action functional} in the sharp interface limit. We prove the corresponding lower bound in two and three space dimensions. One difficulty is that diffuse interfaces may collapse in the limit. We therefore consider the limit of diffuse surface area measures and introduce a generalized velocity and generalized reduced action functional in a class of evolving measures.