  We solve the Fokker-Planck equation for Brownian motion in a logarithmic
potential. When the diffusion constant is below a critical value the solution
approaches a non-normalizable scaling state, reminiscent of an infinite
invariant density. With this non-normalizable density we obtain the phase
diagram of anomalous diffusion for this important process. We briefly discuss
the consequence for a range of physical systems including atoms in optical
lattices and charges in vicinity of long polyelectrolytes. Our work explains in
what sense the infinite invariant density and not Boltzmann's equilibrium
describes the long time limit of these systems.
