Relativistic diffusion with friction on a pseudoriemannian manifold

We study a relativistic diffusion equation on the Riemannian phase space defined by Franchi and Le Jan. We discuss stochastic Ito (Langevin) differential equations (defining the diffusion) as a perturbation by noise of the geodesic equation. We show that the expectation value of the angular momentum and the energy grow exponentially fast. We discuss drifts leading to an equilibrium. It is shown that the diffusion process with a drift corresponding to the Jüttner or quantum equilibrium distributions has a bounded expectation value of angular momentum and energy. The energy and the angular momentum tend exponentially fast to their equilibrium values. As examples we discuss a particle in a plane fronted gravitational wave and a particle in de Sitter universe. It is shown that the relativistic diffusion of momentum in de Sitter space is the same as the relativistic diffusion on the Minkowski mass-shell with the temperature proportional to the de Sitter radius.