Relativistic Poisson-Kac processes and equilibrium Jüttner distribution

It is shown that relativistic Ornstein-Uhlenbeck processes driven by stochastic perturbations possessing finite propagation velocity, and specifically by Poisson-Kac processes, are consistent with the known equilibrium velocity distribution of a relativistic gas (Jüttner distribution). Some observations on the boundedness of acceleration are addressed from elementary quantum principles (quantization of the action variable).