Relativistic diffusion with friction on a pseudo-Riemannian 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 as a perturbation by noise of the geodesic equation. We show that the expectation values of the angular momentum and the energy growexponentially fast. We discuss drifts leading to an equilibrium. As an example we consider a particle in the de Sitter universe. It is shown that the relativistic diffusion of momentum in the de Sitter space is the same as the relativistic diffusion on the Minkowski mass-shell with the temperature proportional to the de Sitter radius. We study a diffusion process with a drift corresponding to the Juttner or quantum equilibrium distributions. We show that such a diffusion has bounded expectation values of angular momentum and energy. The energy and the angular momentum tend exponentially fast to their equilibrium values.