Front fluctuations for the stochastic Cahn–Hilliard equation

We consider the Cahn–Hilliard equation in one space dimension, perturbed by the derivative of a space and time white noise of small intensity, and we investigate the effect of the noise on the solutions when the initial condition is a front that separates the two stable phases. We prove that, given with probability going to one as the noise intensity vanishes, the solution remains close to a front for long times, 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 one and non-Markovian, related to a fractional Brownian motion and for which a couple of representations are given.