Stochastic velocity motions and processes with random time

TL;DR

This paper analyzes a class of stochastic particle motions with random velocities influenced by Poisson events and friction, deriving explicit distributions and exploring effects of random time changes like Bessel and Gamma times.

Contribution

It introduces a detailed analysis of random motions with velocity changes driven by Poisson processes, including explicit distributions and the impact of random time changes such as Bessel and Gamma times.

Findings

Explicit probability distributions for particle positions in specific cases.

Connections established between random motions and random flights.

Analysis of moments and probabilistic interpretations with random time changes.

Abstract

The aim of this paper is to analyze a class of random motions which models the motion of a particle on the real line with random velocity and subject to the action of the friction. The speed randomly changes when a Poissonian event occurs. We study the characteristic and the moment generating function of the position reached by the particle at time $t>0$. We are able to derive the explicit probability distributions in few cases for which discuss the connections with the random flights. The moments are also widely analyzed. For the random motions having an explicit density law, further interesting probabilistic interpretations emerge if we deal with them varying up a random time. Essentially, we consider two different type of random times, namely Bessel and Gamma times, which contain, as particular cases, some important probability distributions (e.g. Gaussian, Exponential). In particular, for the random processes built by means of these compositions, we derive the probability distributions fixed the number of Poisson events. Some remarks on the possible extensions to the random motions in higher spaces are proposed. We focus our attention on the persistent planar random motion.