Reflecting random flights

TL;DR

This paper analyzes reflecting random flights in Euclidean space, deriving explicit probability distributions for particle positions with reflections on spheres and hyperplanes, involving solutions to differential equations and fractional Poisson processes.

Contribution

It provides explicit distributions for reflecting random flights with spherical and hyperplane reflections, incorporating fractional Poisson processes and solutions to Euler-Darboux-Poisson equations.

Findings

Explicit distributions for fixed number of direction changes.

Distributions involve solutions to Euler-Darboux-Poisson equations.

Unconditional distributions obtained via fractional Poisson process.

Abstract

We consider random flights in $\mathbb{R}^d$ reflecting on the surface of a sphere $\mathbb{S}^{d-1}_R,$ with center at the origin and with radius $R,$ where reflection is performed by means of circular inversion. Random flights studied in this paper are motions where the orientation of the deviations are uniformly distributed on the unit-radius sphere $\mathbb{S}^{d-1}_1$. We obtain the explicit probability distributions of the position of the moving particle when the number of changes of direction is fixed and equal to $n\geq 1$. We show that these distributions involve functions which are solutions of the Euler-Darboux-Poisson equation. The unconditional probability distributions of the reflecting random flights are obtained by suitably randomizing $n$ by means of a fractional-type Poisson process. Random flights reflecting on hyperplanes according to the optical reflection form are considered and the related distributional properties derived.