Tightness for a stochastic Allen--Cahn equation

TL;DR

This paper investigates a stochastic Allen-Cahn equation with multiplicative noise, establishing uniform energy bounds, tightness of solutions, and convergence to phase indicators, thus connecting stochastic PDEs to mean curvature flow.

Contribution

It provides the first rigorous analysis of the tight interface limit for a stochastic Allen-Cahn equation with multiplicative noise, linking it to stochastic mean curvature flow.

Findings

Established uniform energy bounds for solutions.

Proved tightness and convergence to phase-indicator functions.

Connected stochastic Allen-Cahn dynamics to stochastic mean curvature flow.

Abstract

We study an Allen-Cahn equation perturbed by a multiplicative stochastic noise which is white in time and correlated in space. Formally this equation approximates a stochastically forced mean curvature flow. We derive uniform energy bounds and prove tightness of of solutions in the sharp interface limit, and show convergence to phase-indicator functions.