Relativistic diffusion of massless particles

We obtain a limit when mass tends to zero of the relativistic diffusion of Schay and Dudley. The diffusion process has the log-normal distribution. We discuss Langevin stochastic differential equations leading to an equilibrium distribution. We show that for the Jüttner equilibrium distribution the relativistic diffusion is a linear approximation to the Kompaneetz equation describing a photon diffusion in an electron gas. The stochastic equation corresponding to the Jüttner distribution is explicitly soluble. We relate the relativistic diffusion to imaginary time quantum mechanics. Some astrophysical applications (including the Sunyaev-Zeldovich effect) are briefly discussed.