Stability of the instanton under small random perturbations

AbstractWe consider the process obtained as solution of the Allen-Cahn equation perturbed by white noise with initial condition an instanton. (That is, the stationary solution of the Allen-Cahn equation which interpolates between the two constant stationary solutions.) We show that, as the intensity of the noise goes to zero, the process remains close to the initial instanton for times that go to infinity conveniently. At the scale we consider, one sees an infinitesimal shift of the instanton, given by a Brownian motion