  We prove a Harnack inequality for positive solutions of a parabolic equation
with slow anisotropic spatial diffusion. After identifying its natural
scalings, we reduce the problem to a Fokker-Planck equation and construct a
self-similar Barenblatt solution. We exploit translation invariance to obtain
positivity near the origin via a self-iteration method and deduce a sharp
anisotropic expansion of positivity. This eventually yields a scale invariant
Harnack inequality in an anisotropic geometry dictated by the speed of the
diffusion coefficients. As a corollary, we infer H\"older continuity, an
elliptic Harnack inequality and a Liouville theorem.
