Interface Fluctuations and Couplings in the D=1 Ginzburg–Landau Equation with Noise

We consider a Ginzburg–Landau equation in the interval [−ε−κ, ε−κ], ε>0, κ≥1, with Neumann boundary conditions, perturbed by an additive white noise of strength √ε We prove that if the initial datum is close to an "instanton" then, in the limit ε→0+, the solution stays close to some instanton for times that may grow as fast as any inverse power of ε, as long as “the center of the instanton is far from the endpoints of the interval”. We prove that the center of the instanton, suitably normalized, converges to a Brownian motion. Moreover, given any two initial data, each one close to an instanton, we construct a coupling of the corresponding processes so that in the limit ε→0+ the time of success of the coupling (suitably normalized) converges in law to the first encounter of two Brownian paths starting from the centers of the instantons that approximate the initial data.