  A relativistic kinetic Fokker-Planck equation that has been recently proposed
in the physical literature is studied. It is shown that, in contrast to other
existing relativistic models, the one considered in this paper is invariant
under Lorentz transformations in the absence of friction. A similar property
(invariance by Galilean transformations in the absence of friction) is verified
in the non-relativistic case. In the first part of the paper some fundamental
mathematical properties of the relativistic Fokker-Planck equation are
established. In particular, it is proved that the model is compatible with the
finite propagation speed of particles in relativity. In the second part of the
paper, two non-linear relativistic mean-field models are introduced. One is
obtained by coupling the relativistic Fokker-Planck equation to the Maxwell
equations of electrodynamics, and is therefore of interest in plasma physics.
The other mean-field model couples the Fokker-Planck dynamics to a relativistic
scalar theory of gravity (the Nordstr\"om theory) and is therefore of interest
in gravitational physics. In both cases the existence of steady states for all
possible prescribed values of the mass is established. In the gravitational
case this result is better than for the corresponding non-relativistic model,
the Vlasov-Poisson-Fokker-Planck system, for which existence of steady states
is known only for small mass.
