Interface Fluctuations for the D = 1 Stochastic Ginzburg–Landau Equation with Nonsymmetric Reaction Term

Abstract

We consider a Ginzburg–Landau equation in the interval [−ε⁻¹, ε⁻¹], ε>0, with Neumann boundary conditions, perturbed by an additive white noise of strength $\sqrt {\varepsilon }$ and reaction term being the derivative of a function which has two equal–depth wells at ±1, but is not symmetric. When ε=0, the equation has equilibrium solutions that are increasing, and connect −1 with +1. We call them instantons, and we study the evolution of the solutions of the perturbed equation in the limit ε→0⁺, when the initial datum is close to an instanton. We prove that, for times that may be of the order of ε⁻¹, the solution stays close to some instanton whose center, suitably normalized, converges to a Brownian motion plus a drift. This drift is known to be zero in the symmetric case, and, using a perturbative analysis, we show that if the nonsymmetric part of the reaction term is sufficiently small, it determines the sign of the drift.