Random flights connecting porous medium and Euler-Poisson-Darboux equations

In this paper, we consider the porous medium equation and establish a relationship between its Kompanets-Zel'dovich-Barenblatt solution u(xd,t),xd∈Rd,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. 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 this paper is devoted to the analysis of the probabilistic tools of the solutions of the fractional equations. The extension to the higher-order Euler-Poisson-Darboux equation is considered, and the solutions are interpreted as compositions of laws of pseudoprocesses.