Large deviation principle for a stochastic Allen-Cahn equation

Abstract

In this paper we consider the Allen–Cahn equation perturbed by a stochastic flux term and prove a large deviation principle. Using an associated stochastic flow of diffeomorphisms the equation can be transformed to a parabolic partial differential equation with random coefficients. We use this structure and first provide a large deviation principle for stochastic flows in function spaces with Hölder-continuity in time. Second, we use a continuity argument and deduce a large deviation principle for the stochastic Allen–Cahn equation.