What is the connection between ballistic deposition and the Kardar-Parisi-Zhang equation?

Ballistic deposition (BD) is believed to belong to the Kardar-Parisi-Zhang (KPZ) universality class. In this paper we study the validity of this belief by rigorously deriving a continuum equation from the BD microscopic rules, which deviates from the KPZ equation. We show that in one dimension and in the presence of noise the deviation is not important. This is not the case in the absence of noise. In more than one dimension and in the presence of noise we obtain an equation that superficially seems to be a continuum equation but in which the symmetry under rotations around the growth direction is broken.