Front Motion in the One-Dimensional Stochastic Cahn--Hilliard Equation

Abstract

In this paper, we consider the one-dimensional Cahn--Hilliard equation perturbed by additive noise and study the dynamics of interfaces for the stochastic model. The noise is smooth in space and defined as a Fourier series with independent Brownian motions in time. Motivated by the work of Bates and Xun on slow manifolds for the integrated Cahn--Hilliard equation, our analysis reveals the significant difficulties and differences in comparison to the deterministic problem. New higher order terms that we estimate appear due to Itô calculus and stochastic integration and dominate the exponentially slow deterministic dynamics. Using a local coordinate system and defining the admissible interface positions as a multidimensional diffusion process, we derive a first order linear system of stochastic ordinary differential equations approximating the equations of front motion. Furthermore, we prove stochastic stability of the approximate slow manifold of solutions over a very long time scale and evaluate the noise effect.