Front fluctuations for the stochastic Cahn-Hilliard equation

Abstract

We consider the Cahn-Hilliard equation in one space dimension, perturbed by the derivative of a space and time white noise of intensity ${\epsilon }^{\frac{1}{2}}$, and we investigate the effect of the noise, as $\epsilon \to 0$, on the solutions when the initial condition is a front that separates the two stable phases. We prove that, given $\gamma <\frac{2}{3}$, with probability going to one as $\epsilon \to 0$, the solution remains close to a front for times of the order of ${\epsilon }^{-\gamma }$, and we study the fluctuations of the front in this time scaling. They are given by a one dimensional continuous process, self similar of order $\frac{1}{4}$ and non Markovian, related to a fractional Brownian motion and for which a couple of representations are given.

33 pages