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

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.