Large deviation principle for a stochastic Allen-Cahn equation

TL;DR

This paper establishes a large deviation principle for a stochastic Allen-Cahn equation with a flux term, using stochastic flow transformations and continuity arguments to handle the randomness in the PDE.

Contribution

It introduces a novel approach by transforming the stochastic PDE into a PDE with random coefficients via stochastic flows, enabling the derivation of large deviation principles.

Findings

Large deviation principle proved for the stochastic Allen-Cahn equation.

Transformation to PDE with random coefficients facilitates analysis.

Method applicable to other stochastic PDEs with similar structures.

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 H\"older-continuity in time. Second, we use a continuity argument and deduce a large deviation principle for the stochastic Allen-Cahn equation.