Generation of interfaces for multi-dimensional stochastic Allen–Cahn equation with a noise smooth in space

Abstract

In this paper, we study the generation of interfaces for a stochastic Allen–Cahn equation with general initial value in the multi-dimensional case that external noise is given by Q-Brownian motion. We prove that interfaces, for d-dimensional stochastic Allen–Cahn equation with scaling parameter , are generated at the time of order . Especially, in one-dimensional case, we give more detailed estimate and shape of the generated interface than that obtained in our previous result. Assuming that the Q-Brownian motion is smooth in space variable, we extend a comparison theorem for PDE to SPDE’s in order to prove the generation. Moreover, we connect the generated interface to the motion of interface in one-dimension. In this case, we consider the white noise only in time multiplied by as the noise term, where a is a smooth function which has a compact support. This is the special case of Q-Brownian motion. We take the time scale of order for studying the motion of interface.