  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.
