Generation and motion of interfaces in one-dimensional stochastic Allen-Cahn equation

TL;DR

This paper investigates the formation and evolution of interfaces in a one-dimensional stochastic Allen-Cahn equation with space-time white noise, establishing the time scale for interface generation and linking it to interface motion dynamics.

Contribution

It provides a rigorous analysis of interface generation in stochastic Allen-Cahn equations with general initial conditions and connects this to known interface motion results.

Findings

Interfaces are generated in logarithmic time scale.

The study extends interface motion analysis to more general initial conditions.

The results bridge interface generation and subsequent motion in stochastic settings.

Abstract

In this paper we study a sharp interface limit for a stochastic reaction-diffusion equation. We consider the case that the noise is a space-time white noise multiplied by a small parameter and a smooth function which has a compact support. We show a generation of interfaces for one-dimensional stochastic Allen-Cahn equation with general initial values. We prove that interfaces are generated in a time of logarithmic order. After the generation of interfaces, we connect it to the motion of interfaces which was investigated by Funaki for special initial values.