Boundary effects on the interface dynamics for the stochastic Allen-Cahn equation

We consider a stochastic perturbation of the Allen-Cahn equation in a bounded interval [-a, b] with boundary conditions fixing the different phases at a and b. We investigate the asymptotic behavior of the front separating the two stable phases in the limit epsilon -> 0, when the intensity of the noise is root epsilon and a, b -> infinity with epsilon. In particular, we prove that it is possible to choose a = a(epsilon) such that in a suitable time scaling limit, the front evolves according to a one-dimensional diffusion process with a nonlinear drift accounting for a "soft" repulsion from a. We finally show that a "hard" repulsion can be obtained by an extra diffusive scaling.