Superluminal problem in diffusion of relativistic particles and its   phenomenological solution
  We discuss the superluminal problem in the diffusion of ultra high energy protons with energy losses taken into account. The phenomenological solution of this problem is found with help of the generalized J\"uttner propagator, originally proposed for relativization of the Maxwellian gas distribution. It is demonstrated that the generalized J\"uttner propagator gives the correct expressions in the limits of diffusive and rectilinear propagation of the charged particles in the magnetic fields, together with the intermediate regime, in all cases without superluminal velocities. This solution, very general for the diffusion, is considered for two particular cases: diffusion inside the stationary objects, like e.g. galaxies, clusters of galaxies etc, and for expanding universe. The comparison with the previously obtained solutions for propagation of UHE protons in magnetic fields is performed. 
R. Aloisio, V. Berezinsky, A. Gazizov