98-46 S. Brassesco, P. Butta'

Interface fluctuations for the d=1 stochastic Ginzburg-Landau equation with non-symmetric reaction term (306K, PostScript) Feb 4, 98

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Abstract. We consider a Ginzburg-Landau equation in the interval $[-\eps^{-1},\eps^{-1}]$, $\eps>0$, with Neumann boundary conditions, perturbed by an additive white noise of strength $\sqrt \eps$, and reaction term being the derivative of a function which has two equal depth wells at $\pm1$, but is not symmetric. When $\eps=0$, 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 $\eps\to 0^+$ , when the initial datum is close to an instanton. We prove that, for times that may be of the order of $\eps^{-1}$, 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 non-symmetric part of the reaction term is sufficiently small, it determines the sign of the drift.

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