White noise perturbation of the viscous shock fronts of the burgers equation

We study the front dynamics of solutions of the initial value problem of the Burgers equation with initial data being the viscous shock front plus the white noise perturbation. In the sense of distribution, the solutions propagate with the same speed as the unperturbed front, however, the front location is random and satisfies a central limit theorem with the variance proportional to the time t, as t goes to infinity. With probability arbitrarily close to one, the front width is O(1) for large time.