On the short time asymptotic of the stochastic Allen-Cahn equation

TL;DR

This paper analyzes the short time behavior of solutions to the stochastic Allen-Cahn equation, showing that in the sharp interface limit, solutions evolve according to mean curvature motion with stochastic forcing, extending prior results to higher dimensions.

Contribution

It extends Funaki's result from 2D to arbitrary dimensions, describing the stochastic motion by mean curvature in the short time regime.

Findings

Solutions follow mean curvature motion with stochastic forcing

Extension of previous 2D results to higher dimensions

Provides a detailed description of short time behavior

Abstract

A description of the short time behavior of solutions of the Allen-Cahn equation with a smoothened additive noise is presented. The key result is that in the sharp interface limit solutions move according to motion by mean curvature with an additional stochastic forcing. This extends a similar result of Funaki in spatial dimension $n=2$ to arbitrary dimensions.