Random flights connecting Porous Medium and Euler-Poisson-Darboux equations

TL;DR

This paper explores the connection between porous medium equations and Euler-Poisson-Darboux equations through random flights, extending to fractional and higher-order versions, and analyzing their probabilistic solutions.

Contribution

It establishes a novel relationship between porous medium solutions and random flights via Euler-Poisson-Darboux equations, including fractional and higher-order cases.

Findings

Solution of space-fractional Euler-Poisson-Darboux equation presented

Probabilistic interpretation of fractional porous medium solutions

Extension to higher-order equations with pseudoprocesses

Abstract

In this paper we consider the Porous Medium Equation and establish a relationship between its Kompanets-Zel'dovich-Barenblatt solution $u(\xd,t), \xd\in \mathbb R^d,t>0$ and random flights. The time-rescaled version of $u(\xd,t)$ is the fundamental solution of the Euler-Poisson-Darboux equation which governs the distribution of random flights performed by a particle whose displacements have a Dirichlet probability distribution and choosing directions uniformly on a $d$-dimensional sphere (see, e.g., \cite{dgo}). We consider the space-fractional version of the Euler-Poisson-Darboux equation and present the solution of the related Cauchy problem in terms of the probability distributions of random flights governed by the classical Euler-Poisson-Darboux equation. Furthermore, this research is also aimed at studying the relationship between the solutions of a fractional Porous Medium Equation and the fractional Euler-Poisson-Darboux equation. A considerable part of the paper is devoted to the analysis of the probabilistic tools of the solutions of the fractional equations. Also the extension to higher-order Euler-Poisson-Darboux equation is considered and the solutions interpreted as compositions of laws of pseudoprocesses.