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.