Continuous time random walk and diffusion with generalized fractional Poisson process

Abstract

A non-Markovian counting process, the `generalized fractional Poisson process' (GFPP) introduced by Cahoy and Polito in 2013 is analyzed. The GFPP contains two index parameters $0<\beta\leq 1$, $\alpha >0$ and a time scale parameter. Generalizations to Laskin's fractional Poisson distribution and to the fractional Kolmogorov-Feller equation are derived. We develop a continuous time random walk subordinated to a GFPP in the infinite integer lattice $\mathbb{Z}^d$. For this stochastic motion, we deduce a `generalized fractional diffusion equation'. In a well-scaled diffusion limit this motion is governed by the same type of fractional diffusion equation as with the fractional Poisson process exhibiting subdiffusive $t^{\beta}$-power law for the mean-square displacement. In the special cases $\alpha=1$ with $0<\beta<1$ the equations of the Laskin fractional Poisson process and for $\alpha=1$ with $\beta=1$ the classical equations of the standard Poisson process are recovered. The remarkably rich dynamics introduced by the GFPP opens a wide field of applications in anomalous transport and in the dynamics of complex systems.