Stochastic Allen-Cahn approximation of the mean curvature flow: large deviations upper bound

TL;DR

This paper investigates the large deviations of a stochastic Allen-Cahn equation in the sharp interface limit, linking the probabilistic behavior to mean curvature flow and nucleation phenomena.

Contribution

It establishes a large deviation upper bound for the stochastic Allen-Cahn equation, connecting the rate function to interface evolution and nucleation events in the sharp interface limit.

Findings

Large deviation upper bound proven for stochastic Allen-Cahn

Rate function characterized by interface velocity deviation and nucleation

Zero level set corresponds to Brakke's mean curvature evolution

Abstract

Consider the Allen-Cahn equation on the $d$-dimensional torus, $d=2,3$, in the sharp interface limit. As it is well known, the limiting dynamics is described by the motion by mean curvature of the interface between the two stable phases. Here, we analyze a stochastic perturbation of the Allen-Cahn equation and describe its large deviation asymptotics in a joint sharp interface and small noise limit. Relying on previous results on the variational convergence of the action functional, we prove the large deviation upper bound. The corresponding rate function is finite only when there exists a time evolving interface of codimension one between the two stable phases. The zero level set of this rate function is given by the evolution by mean curvature in the sense of Brakke. Finally, the rate function can be written in terms of the sum of two non-negative quantities: the first measures how much the velocity of the interface deviates from its mean curvature, while the second is due to the possible occurrence of nucleation events.