Sharp Interface Limits for a Stochastic Allen-Cahn Equation

Abstract

This chapter discusses the problem of the sharp interface limits for a stochastic Allen-Cahn equation as mentioned in Sect. 3.1 The original motivation comes from statistical physics in studying the dynamic phase transition (Kawasaki and Ohta, Prog Theor Phys 67:147–163, 1982; Kawasaki, Non-equilibrium and phase transition–statistical physics in mesoscopic scale, Asakura 2000). The proper scaling in time is different and changes according to the types of noises and the spatial dimension d. We start with explaining the results in a deterministic case and then formulate the results in a stochastic case. A brief survey is given on the motions by mean curvature with or without noises, which arise in the limit, and the sharp interface limit.