INTERFACE FLUCTUATIONS AND COUPLINGS IN THE D=1 GINZBURG-LANDAU EQUATION WITH NOISE

We consider a Ginzburg-Landau equation in the interval [-epsilon(-kapp a), epsilon(-kappa)], epsilon > 0, kappa greater than or equal to 1, w ith Neumann boundary conditions, perturbed by an additive white noise of strength root epsilon. We prove that if the initial datum is close to an ''instanton'' then, in the limit epsilon --> 0(+), the solution stays close to some instanton for times that may grow as fast as any i nverse power of epsilon, as long as ''the center of the instanton is f ar from the endpoints of the interval''. We prove that the center of t he instanton, suitably normalized, converges to a Brownian motion. Mor eover, given any two initial data, each one close to an instanton, we construct a coupling of the corresponding processes so that in the lim it epsilon --> 0(+) the time of success of the coupling (suitably norm alized) converges in law to the first encounter of two Brownian paths starting from the centers of the instantons that approximate the initi al data.