A diffusion equation from the relativistic Ornstein-Uhlenbeck process

We derive, in the "hydrodynamic" limit (large space and time scales), an evolution equation for the particle density in physical space from the (special) relativistic Omstein-Uhlenbeck process introduced by Debbasch, Mallick, and Rivet. This equation turns out to be identical with the classical diffusion equation, without any relativistic correction. We prove that, in the "hydrodynamic" limit, this result is indeed compatible with special relativity.