BROWNIAN FLUCTUATIONS OF THE INTERFACE IN THE D=1 GINZBURG-LANDAU EQUATION WITH NOISE

We consider the Ginzburg-Landau equation in an interval of R, perturbe d by a white noise and with Neumann boundary conditions. The initial d atum is close to the stationary solution (that we call instanton) of t he equation without noise. We prove that, as the variance of the noise goes to zero and the length of the interval is proportional to the in verse of this variance, then, the solution approaches an instanton whi ch moves as a Brownian motion.