We investigate the Kardar--Parisi--Zhang (KPZ) equation in d spatial
dimensions with Gaussian spatially long--range correlated noise ---
characterized by its second moment R(xβxβ²)ββ£xβxβ²β£2Οβd --- by means of dynamic field theory and the
renormalization group. Using a stochastic Cole--Hopf transformation we derive
{\em exact} exponents and scaling functions for the roughening transition and
the smooth phase above the lower critical dimension dcβ=2(1+Ο). Below
the lower critical dimension, there is a line Οββ(d) marking the stability
boundary between the short-range and long-range noise fixed points. For Οβ₯Οββ(d), the general structure of the renormalization-group equations
fixes the values of the dynamic and roughness exponents exactly, whereas above
Οββ(d), one has to rely on some perturbational techniques. We discuss the
location of this stability boundary Οββ(d) in light of the exact results
derived in this paper, and from results known in the literature. In particular,
we conjecture that there might be two qualitatively different strong-coupling
phases above and below the lower critical dimension, respectively.Comment: 21 pages, 15 figure