Interface fluctuations for the D=1 stochastic Ginzburg-Landau equation with nonsymmetric reaction term

We consider a Ginzburg-Landau equation in the interval [-epsilon(-1), epsilon(-1)], epsilon > 0, with Neumann boundary conditions, perturbed by an additive white noise of strength root epsilon, and reaction term being the derivative of a function which has two equal-depth wells at +/- 1; but is not symmetric. When epsilon = 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 epsilon --> 0(+), when the initial datum is close to an instanton. We prove that, for limes that may be of the order of epsilon(-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.