A Relativistic Stochastic Process

We give a review of the relativistic stochastic process denominated with the acronym ROUP, standing for relativistic Ornstein-Ohlenbeck process. This stochastic process was introduced in 1997 by Debbasch, Mallick and Rivet [J. Stat. Phys. 88:945 966 ] as a simpli ed model of irreversibility in a relativistic framework. This allows an investigation of the paradox arising when examining the large time and space regime of relativistic transport equations (like the relativistic Boltzmann equation) via the Chapman-Enskog approach, which inexplicably gives back non-relativistic parabolic equations. We then prove the markovian irreversible character of this process even in an arbitrary curved space-time, by introducing a conditional entropy current based on the manifestly covariant Fokker-Planck formulation of the general relativistic ROUP.