  Useful theories for growth of surfaces under random deposition of material
have been developed by several authors. The simplest theory is that introduced
by Edwards and Wilkinson (EW), which is linear and soluble. Its non linear
generalization by Kardar, Parisi and Zhang (KPZ), resulted in many subsequent
studies. Yet both theories EW and KPZ contain an unphysical feature. When
deposition of material is stopped both theories predict that as time tends to
infinity, the surface becomes flat. In fact, of course, the final surface is
not flat, but simply has no gradients larger than the gradient related to the
angle of repose. We modify the EW and KPZ to accommodate this feature and study
the consequences for the simpler system which is a modification of the EW
equation. In spite of the fact that the equation describing the evolution of
the surface is not linear, we find that the steady state in the presence of
noise is not very different in the long wave length limit from that of the
linear EW. The situation is quite different from that of EW when deposition
stops. Initially there is still some rearrangement of the surface but that
stops as everywhere on the surface the gradient is less than that related to
the angle of repose. The most interesting feature observed after deposition
stops is the emergence of history-dependent steady state distributions.
