Random motions in $\mathbb{R}^3$ with orthogonal directions

TL;DR

This paper analyzes three-dimensional orthogonal random motions with Poisson-driven direction switches, deriving distributional properties, integral representations, and a sixth-order PDE for the position within an octahedral support.

Contribution

It provides a comprehensive analysis of 3D orthogonal motions with Poisson switching, including explicit distributional results and a new PDE characterization.

Findings

Distribution of position on the octahedral support surface

Integral representation involving Bessel functions

Sixth-order PDE governing the position distribution

Abstract

This paper is devoted to the detailed analysis of three-dimensional motions in $\mathbb{R}^3$ 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 $S_{ct}$, is completely carried out on the edges $E_{ct}$ and faces $F_{ct}$. In particular, the motion on the faces $F_{ct}$ is analysed by means of a transformation which reduces it to a three-directions planar random motion. This permits us to obtain an integral representation on $F_{ct}$ 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 $S_{ct}$ 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 coordinates as the motion devolpes. The analysis is carried out when the incoming direction is orthogonal to the ongoing one and also when all directions can be uniformely choosen at each Poisson event. If the switches are governed by homogeneus Poisson process many explicit results are obtained.