Minimal cost for the forced displacement of an interface

We study the probability of the motion of an interface between two stable phases of a ferromagnetic system from an initial to a final position within fixed time. We work with a stochastic microscopic system of Ising spins with Kac interaction evolving in time according to the Glauber (non-conservative) dynamics. We derive the cost functional penalizing all possible transitions and we minimize it to find the most probable profile which corresponds to the motion of the interface in the macroscopic scale. This is joint work with N. Dirr and G. Manzi.