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 Z d . 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 t β -power law for the expected number of arrivals. However, in short observation times, the GFPP exhibits distinct power-law patterns, namely t αβ−1 -scaling of the waiting time density and a t αβ -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.

Generalized Space–Time Fractional Dynamics in Networks and Lattices

“…In a Poisson renewal process the expected number of arrivals increases linearly in time. The exponential decay in the distributions related to the Poisson process make this process memoryless with the Markovian property [17,20].…”

“…In anomalous diffusion one has for the average number of arrivals instead of the linear behavior (22) a power law ∼ t β with 0 < β < 1 [5,17,18], among others. To describe such anomalous power-law behavior a 'fractional generalization' of the classical Poisson renewal process was introduced and analyzed by Laskin [14,15] and others [13,20,25].…”

“…This process which was to our knowledge first introduced by Repin and Saichev [13], was developed and analyzed by Laskin who called this process the 'fractional Poisson process' [14,15]. The Laskin's fractional Poisson process was further generalized in order to obtain greater flexibility to adopt real-world situations [16,17,18]. We refer this renewal process to as 'generalized fractional Poisson process' (GFPP).…”

“…We refer this renewal process to as 'generalized fractional Poisson process' (GFPP). Recently we developed a CTRW model of a normal random walk subordinated to a GFPP [17,18].The purpose of the present paper is to explore space fractional random walks that are subordinated to a GFPP. We analyze such motions in undirected networks and as a special application in the multidimensional infinite integer lattice Z d .…”

“…We refer this renewal process to as 'generalized fractional Poisson process' (GFPP). Recently we developed a CTRW model of a normal random walk subordinated to a GFPP [17,18].…”