Ricerc@Sapienza

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.

The space-fractional and the time-fractional Poisson processes are two well-known models of fractional evolution. They can be constructed as standard Poisson processes with the time variable replaced by a stable subordinator and its inverse, respectively. The aim of this paper is to study nonhomogeneous versions of such models, which can be defined by means of the so-called multistable subordinator (a jump process with nonstationary increments), denoted by H:=H(t), t≥0.

It is well-known that compositions of Markov processes with inverse subordinators are governed by integro-differential equations of generalized fractional type. This kind of processes are of wideinterest in statistical physics as they are connected to anomalous diffusions. In this paper we consider a generalization; more precisely we mean componentwise compositions of R^d -valued Markov processes