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<β≤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 Zd. For this stochastic motion, we deduce a ‘generalized fractional diffusion equation’. For long observations, the generalized fractional diffusion exhibits the same power laws as fractional diffusion with fat-tailed waiting time densities and subdiffusive tβ-power law for the expected number of arrivals. However, in short observation times, the GFPP exhibits distinct power-law patterns, namely tαβ−1-scaling of the waiting time density and a tαβ-pattern for the expected number of arrivals. The latter exhibits for αβ>1 superdiffusive behavior when the observation time is short. 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.