A cyclic random motion in R 3 with four directions and finite velocity

Abstract

In this paper, a cyclic random motion with four directions forming a regular tetrahedron in the Euclidean space R 3 is considered. Changes of direction are governed by a homogenous Poisson process. We obtain the explicit form of the absolutely continuous part of the distribution of the moving particle (X(t), Y(t), Z(t)), in terms of Bessel functions of fourth-order. We prove that the singular part of the distribution is uniformly distributed on the vertices (when no event occurs), on the edges (when one change of direction takes place) and on the faces when two events are recorded. We are also able to obtain (by solving suitable boundary-value problems for the governing equation of the probability generating function) the distribution of (X(t), Y(t), Z(t)) under the condition that the number of Poisson events is fixed. The results of this paper are compared with random motions in the plane and on the line for which the explicit distribution is known.