Sharp interface limit for invariant measures of a stochastic Allen-Cahn equation

TL;DR

This paper investigates the invariant measure of a stochastic Allen-Cahn equation in one dimension, analyzing its behavior in the sharp interface limit and revealing concentration on single-jump configurations with uniform distribution.

Contribution

It provides a rigorous analysis of the invariant measure's sharp interface limit, connecting stochastic PDEs to Gibbs measures and interface concentration phenomena.

Findings

Invariant measure is absolutely continuous with respect to a Brownian bridge.

In the sharp interface limit, the measure concentrates on configurations with one jump.

The jump location is uniformly distributed across the domain.

Abstract

The invariant measure of a one-dimensional Allen-Cahn equation with an additive space-time white noise is studied. This measure is absolutely continuous with respect to a Brownian bridge with a density which can be interpreted as a potential energy term. We consider the sharp interface limit in this setup. In the right scaling this corresponds to a Gibbs type measure on a growing interval with decreasing temperature. Our main result is that in the limit we still see exponential convergence towards a curve of minimizers of the energy if the interval does not grow too fast. In the original scaling the limit measure is concentrated on configurations with precisely one jump. This jump is distributed uniformly.