Soft and Hard Wall in a Stochastic Reaction Diffusion Equation

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a “soft” repulsion from the boundary. We finally show how a “hard” repulsion can be obtained by an extra diffusive scaling.