Particles with constant speed and random velocity in 3+1 space-time: separation of the space variables.

Abstract

We consider a particle in 3+1 space-time dimensions which constantly moves at speed of light c, randomly changing its velocity which can be represented by a point on the surface of a sphere of radius c. The velocity performs an isotropic Wiener process on this surface so that the velocity trajectories are almost everywhere continuous although not differentiable, while the position trajectories are continuous and differentiable. Together with the process that describes the particle in the ‘rest frame’, where the diffusion of velocity on the surface of the sphere is isotropic, the entire family of anisotropic processes which result from Lorentz boosts is also described. The focus of this article is on stochastic evolution in space. We identify a reduced set of variables whose stochastic evolution is autonomous from the remaining variables, but, nevertheless, carry all the relevant information concerning the spatial properties of the process. The associated stochastic equations as well the Forward Kolmogorov Equation (FKE) are considerably simplified compared to those of the complete set of position and velocity variables.