  A functional integral technique is used to study the ultraviolet or short
distance properties of the Kardar-Parisi-Zhang (KPZ) equation with white
Gaussian noise. We apply this technique to calculate the one-loop effective
potential for the KPZ equation. The effective potential is (at least) one-loop
ultraviolet renormalizable in 1, 2, and 3 space dimensions, but
non-renormalizable in 4 or higher space dimensions. This potential is
intimately related to the probability distribution function (PDF) for the
spacetime averaged field. For the restricted class of field configurations
considered here, the KPZ equation exhibits dynamical symmetry breaking (DSB)
via an analog of the Coleman-Weinberg mechanism in 1 and 2 space dimensions,
but not in 3 space dimensions.
