  We consider a model for the propagation of a driven interface through a
random field of obstacles. The evolution equation, commonly referred to as the
Quenched Edwards-Wilkinson model, is a semilinear parabolic equation with a
constant driving term and random nonlinearity to model the influence of the
obstacle field. For the case of isolated obstacles centered on lattice points
and admitting a random strength with exponential tails, we show that the
interface propagates with a finite velocity for sufficiently large driving
force. The proof consists of a discretization of the evolution equation and a
supermartingale estimate akin to the study of branching random walks.
