Random Motion with Uniformly Distributed Directions and Random Velocity

Abstract

In this paper we deal with uniformly distributed direction of motion or isotropic motion at random speed or velocity where the direction alternations occur according to the renewal epochs of a general distribution. We derive the renewal equation for the characteristic function of the transition density of the multidimensional motion. Then, by using the renewal equation, we study the behavior of the transition density near the sphere of its singularity for one-, two-, three-, and four-dimensional cases. To illustrate our solution methodology we present detailed calculations of many solvable examples.