The spatially one-dimensional relativistic Ornstein-Uhlenbeck process in an arbitrary inertial frame

The spatially one-dimensional relativistic Ornstein-Uhlenbeck process is st udied in an arbitrary inertial reference frame. In particular, we derive di rectly from the stochastic equations of motion in an arbitrary inertial fra me the transport equation for the distribution function of the diffusing pa rticles in phase-space. We explain why this result is not trivial and has, at the very least, methodological interest. We also show that this result o ffers a conceptually new proof of the well-known fact that the relativistic one-particle distribution function in phase-space is a Lorentz scalar.