Stochastic Perturbations of the Allen-Cahn Equation

Abstract

Consider the Allen-Cahn equation with small diffusion ∊2 perturbed by a space time white noise of intensity σ. In the limit, σ/∊2 → 0, solutions converge to the noise free problem in the L2 norm. Under these conditions, asymptotic results for the evolution of phase boundaries in the deterministic setting are extended, to describe the behaviour of the stochastic Allen-Cahn PDE by a system of stochastic differential equations. Computations are described, which support the asymptotic derivation.