The Allen–Cahn action functional in higher dimensions

The Allen-Cahn action functional is related to the probability of rare eventsin the stochastically perturbed Allen-Cahn equation. Formal calculationssuggest a reduced action functional in the sharp interface limit. We prove intwo and three space dimensions the corresponding lower bound. One difficulty isthat diffuse interfaces may collapse in the limit. We therefore consider thelimit of diffuse surface area measures and introduce a generalized velocity andgeneralized reduced action functional in a class of evolving measures. As acorollary we obtain the Gamma convergence of the action functional in a classof regularly evolving hypersurfaces.