The stochastic Allen-Cahn equation with Dobrushin boundary conditions
Seminar Room 1, Newton Institute
We consider the solution of the Allen-Cahn equation perturbed by a space-time white noise of intensity epsilon, imposing boundary conditions that fix the two different phases at the extremes of an interval that grows conveniently with epsilon. We study the dynamics of the interface and the behaviour of the invariant measure as epsilon goes to zero. We show that the motion of the latter is described by a one dimensional diffusion with a strong drift repelling from infinite, and the invariant measure, in the convenient scaling, converges to a non-trivial non translation invariant measure concentrated on an invariant set for the infinite volume equation. This is a joint work with L Bertini and P Butta.