Sharp interface limit for the stochastic Allen-Cahn equation

Abstract

The behavior of the Allen-Cahn equation<br /> ∂ _t u ^ε(x,t)= Δ u ^ε(x,t) - ε ^-2 F'(u ^ε (x,t))+ ξ _t^ε<br /> with an additional stochastic term ξ _t^ε is studied for small values of ε. This equation is a reaction diffusion equation with a particular shape of the reaction term -F' which is the negative derivative of a double-well potential with two wells of equal depth.<br /> In the first part the invariant measure for this equation is studied in the case, where x ∈ [-1,1] takes values in a compact one-dimensional domain, and where ξ _t^ε denotes a space-time white noise. This measure is absolutely continuous with respect to a Brownian bridge with appropriate boundary conditions. A scaled version of this measure is transformed to a Gibbs-type measure on a growing interval. Then it is shown that these transformed measures concentrate around a one-dimensional curve of minimizers in the infinite-dimensional space of possible configurations. This implies that in the original scaling the measures concentrate on the set of step functions with precisely one jump.<br /> In the second part the dynamical system is studied in higher dimensions. Here the noise term ξ _t^ε is constant in space, and smoothened in time, with a correlation length that goes to zero at a precise rate as ε↓ 0. In the limit one obtains almost surely configurations that are concentrated on { ± 1 }. The development of the phase boundaries is driven by its mean curvature with an additional stochastic forcing term.