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 which is parameterized by a sufficiently small parameter $\varepsilon >0$. We consider the case that the noise is a space–time white noise multiplied by $\varepsilon ^\gamma a(x)$ where the function a(x) is a smooth function which has compact support. First, we show a generation of interfaces for a one-dimensional stochastic Allen–Cahn equation with general initial values. We prove that interfaces are generated in time of order $O(\varepsilon |\log \varepsilon |)$. After the generation of interfaces, we connect it to the motion of interfaces which was investigated by Funaki (Probab Theory Relat Fields 102(2):221–288, 1995) for special initial values. Funaki (Probab Theory Relat Fields 102(2):221–288, 1995) proved that the interface moved in a proper time scale obeying a certain stochastic differential equation (SDE) if the interface formed at the initial time. We take the time scale of order $O(\varepsilon ^{-2\gamma - \frac{1}{2}})$. This time scale is the same as that of Funaki (Probab Theory Relat Fields 102(2):221–288, 1995) and interface moves in this time scale obeying some SDE with high probability.