Exponential loss of memory for the 2-dimensional Allen\u2013Cahn equation with small noise

Abstract

We prove an asymptotic coupling theorem for the 2-dimensional Allen–Cahn equation perturbed by a small space-time white noise. We show that with overwhelming probability two profiles that start close to the minimisers of the potential of the deterministic system contract exponentially fast in a suitable topology. In the 1-dimensional case a similar result was shown in Martinelli et al. (Commun Math Phys 120(1):25–69, 1988; J Stat Phys 55(3–4):477–504, 1989). It is well-known that in two or more dimensions solutions of this equation are distribution-valued, and the equation has to be interpreted in a renormalised sense. Formally, this renormalisation corresponds to moving the minima of the potential infinitely far apart and making them infinitely deep. We show that despite this renormalisation, solutions behave like perturbations of the deterministic system without renormalisation: they spend large stretches of time close to the minimisers of the (un-renormalised) potential and the exponential contraction rate of different profiles is given by the second derivative of the potential in these points. As an application we prove an Eyring–Kramers law for the transition times between the stable solutions of the deterministic system for fixed initial conditions.