Finite velocity planar random motions driven by inhomogeneous fractional Poisson distributions

TL;DR

This paper investigates finite velocity planar random motions with an infinite number of directions, where the change of directions follows an inhomogeneous fractional Poisson process, leading to explicit probability laws involving Mittag-Leffler functions.

Contribution

It introduces a novel fractional Poisson distribution model for random motions and derives explicit probability laws using Mittag-Leffler functions.

Findings

Explicit probability laws for fractional Poisson-driven motions

Connection between fractional Poisson processes and random flights

Extension to motions with Dirichlet-distributed velocities

Abstract

In this paper we study finite velocity planar random motions with an infinite number of possible directions, where the number of changes of direction is randomized by means of an inhomogeneous fractional Poisson distribution. We first discuss the properties of the distributions of the generalized fractional inhomogeneous Poisson process. Then we show that the explicit probability law of the planar random motions where the number of changes of direction is governed by this fractional distribution can be obtained in terms of Mittag-Leffler functions. We also consider planar random motions with random velocities obtained from the projection of random flights with Dirichlet displacements onto the plane, randomizing the number of changes of direction with a suitable adaptation of the fractional Poisson distribution.