Theory of Relativistic Brownian Motion: The (1+3)-Dimensional Case
  A theory for (1+3)-dimensional relativistic Brownian motion under the influence of external force fields is put forward. Starting out from a set of relativistically covariant, but multiplicative Langevin equations we describe the relativistic stochastic dynamics of a forced Brownian particle. The corresponding Fokker-Planck equations are studied in the laboratory frame coordinates. In particular, the stochastic integration prescription, i.e. the discretization rule dilemma, is elucidated (pre-point discretization rule {\it vs.} mid-point discretization rule {\it vs.} post-point discretization rule). Remarkably, within our relativistic scheme we find that the post-point rule (or the transport form) yields the only Fokker-Planck dynamics from which the relativistic Maxwell-Boltzmann statistics is recovered as the stationary solution. The relativistic velocity effects become distinctly more pronounced by going from one to three spatial dimensions. Moreover, we present numerical results for the asymptotic mean square displacement of a free relativistic Brownian particle moving in (1+3) dimensions. 
J\"orn Dunkel and Peter H\"anggi