  In the classical theory of thin disc accretion discs, the constraints of mass
and angular momentum conservation lead to a diffusion-like equation for the
turbulent evolution of the surface density. Here, we revisit this problem,
extending the Newtonian analysis to the regime of Kerr geometry relevant to
black holes. A diffusion-like equation once again emerges, but now with a
singularity at the radius at which the effective angular momentum gradient
passes through zero. The equation may be analysed using a combination of WKB,
local techniques, and matched asymptotic expansions. It is shown that imposing
the boundary condition of a vanishing stress tensor (more precisely the
radial-azimuthal component thereof) allows smooth stable modes to exist
external to the angular momentum singularity, the innermost stable circular
orbit, while smoothly vanishing inside this location. The extension of the disc
diffusion equation to the domain of general relativity introduces a new tool
for numerical and phenomenolgical studies of accretion discs, and may prove to
be a useful technique for understanding black hole X-ray transients.
