Cyclic random motions in -space with n directions

Institut National des Sciences Appliquées de Lyon, Bâtiment Léonard de Vinci, 20 avenue Albert Einstein, 69621 Villeurbanne Cedex, France; aime.lachal@insa-lyon.fr; http://maths.univ-lyon1.Fr/~lachal

Received: 30 September 2005
Revised: 2 February 2006

Abstract

We study the probability distribution of the location of a particle performing a cyclic random motion in . The particle can take n possible directions with different velocities and the changes of direction occur at random times. The speed-vectors as well as the support of the distribution form a polyhedron (the first one having constant sides and the other expanding with time t). The distribution of the location of the particle is made up of two components: a singular component (corresponding to the beginning of the travel of the particle) and an absolutely continuous component.
We completely describe the singular component and exhibit an integral representation for the absolutely continuous one. The distribution is obtained by using a suitable expression of the location of the particle as well as some probability calculus together with some linear algebra. The particular case of the minimal cyclic motion (n=d+1) with Erlangian switching times is also investigated and the related distribution can be expressed in terms of hyper-Bessel functions with several arguments.