Continuous time random walk and diffusion with generalized fractional Poisson process

Thomas Michelitsch (1) , Alejandro Perez Riascos (2)

1 DALEMBERT - Institut Jean Le Rond d'Alembert
2 UNAM - Universidad Nacional Autónoma de México = National Autonomous University of Mexico

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'. 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.

Mots clés: fractional Kolmogorov-Feller equation; Renewal process; generalized fractional diffusion; fractional Poisson process and distribution; continuous time random walk

Domaines: Matière Condensée [cond-mat]

Thomas Michelitsch, Alejandro Perez Riascos. Continuous time random walk and diffusion with generalized fractional Poisson process. Physica A: Statistical Mechanics and its Applications, 2020, ⟨10.1016/j.physa.2019.123294⟩. ⟨hal-02336450⟩