Random motions in R3 with orthogonal directions

This paper is devoted to the detailed analysis of three-dimensional motions in R3 with orthogonal directions switching at Poisson times and moving with constant speed c > 0. The study of the random position at an arbitrary time t > 0 on the surface of the support, forming an octahedron Sct, is completely carried out on the edges and faces (Fct). In particular, the motion on the faces Fct is analyzed by means of a transformation which reduces it to a three-directions planar random motion. This permits us to obtain an integral representation on Fct in terms of integral of products of first order Bessel functions. The investigation of the distribution of the position p = p(t, x, y, z) inside Sct implied the derivation of a sixth-order partial differential equation governing p (expressed in terms of the products of three D'Alembert operators). A number of results, also in explicit form, concern the time spent on each direction and the position reached by each coordinate as the motion develops. The analysis is carried out when the incoming direction is orthogonal to the ongoing one and also when all directions can be uniformly chosen at each Poisson event. If the switches are governed by a homogeneous Poisson process many explicit results are obtained.(c) 2023 Elsevier B.V. All rights reserved.