  We consider the stochastic evolution of a 1+1-dimensional interface (or
polymer) in presence of a substrate. This stochastic process is a dynamical
version of the homogeneous pinning model. We start from a configuration far
from equilibrium: a polymer with a non-trivial macroscopic height profile, and
look at the evolution of a space-time rescaled interface. In two cases, we
prove that this rescaled interface has a scaling limit on the diffusive scale
(space rescaled by $L$ in both dimensions and time rescaled by $L^2$ where $L$
denotes the length of the interface) which we describe: when the interaction
with the substrate is such that the system is unpinned at equilibrium, then the
scaling limit of the height profile is given by the solution of the heat
equation with Dirichlet boundary condition ; when the attraction to the
substrate is infinite, the scaling limit is given a free-boundary problem which
belongs to the class of Stefan problems with contracting boundary, also
referred to as Stefan freezing problems. In addition, we prove the existence
and regularity of the solution to this problem until a maximal time, where the
boundaries collide.
