Continuous time random walk and diffusion with generalized fractional Poisson process

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 $0<β\leq 1$, $α>0$ 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 $t^β$-power law for the mean-square displacement. In the special cases $α=1$ with $0<β<1$ the equations of the Laskin fractional Poisson process and for $α=1$ with $β=1$ 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.

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