Low noise limit for the invariant measure of a multi-dimensional stochastic Allen-Cahn equation

TL;DR

This paper investigates how the invariant measure of a discretized multi-dimensional stochastic Allen-Cahn equation behaves in the low noise limit, showing exponential concentration around free energy minimizers under certain domain growth conditions.

Contribution

It provides a rigorous analysis of the low noise limit for the invariant measure of a discretized stochastic Allen-Cahn equation in multiple dimensions, considering domain growth and boundary conditions.

Findings

Invariant measures concentrate exponentially fast around free energy minimizers.

Concentration holds if the domain growth is sufficiently slow.

Results extend understanding of stochastic PDEs in low noise regimes.

Abstract

We study the invariant measure of a discretized stochastic Allen-Cahn equation in d+1 dimensions in the low noise limit. We consider a cuboidal domain and impose the two stable phases as boundary conditions at two opposite faces. We then take a joint limit where the temperature and the mesh of the discretization go to zero while the size of the domain grows. Our main result is that the invariant measures concentrate exponentially fast around the minimizers of the free energy functional if the domain does not grow too fast.