  For a model for the propagation of a curvature sensitive interface in a time
independent random medium, as well as for a linearized version which is
commonly referred to as Quenched Edwards-Wilkinson equation, we prove existence
of a stationary positive supersolution at non-vanishing applied load. This
leads to the emergence of a hysteresis that does not vanish for slow loading,
even though the local evolution law is viscous (in particular, the velocity of
the interface in the model is linear in the driving force).
