GENERALIZED FRACTIONAL POISSON PROCESS AND RELATED STOCHASTIC DYNAMICS

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

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

We survey the 'generalized fractional Poisson process' (GFPP). The GFPP is a renewal process generalizing Laskin's fractional Poisson counting process and was first introduced by Cahoy and Polito. The GFPP contains two index parameters with admissible ranges 0 < β ≤ 1, α > 0 and a parameter characterizing the time scale. The GFPP involves Prab-hakar generalized Mittag-Leffler functions and contains for special choices of the parameters the Laskin fractional Poisson process, the Erlang process and the standard Poisson process. We demonstrate this by means of explicit formulas. We develop the Montroll-Weiss continuous-time random walk (CTRW) for the GFPP on undirected networks which has Prabhakar distributed waiting times between the jumps of the walker. For this walk, we derive a generalized fractional Kolmogorov-Feller equation which involves Prabhakar generalized fractional operators governing the stochastic motions on the network. We analyze in d dimensions the 'well-scaled' diffusion limit and obtain a fractional diffusion equation which is of the same type as for a walk with Mittag-Leffler distributed waiting times. The GFPP has the potential to capture various aspects in the dynamics of certain complex systems. MSC 2010 : Primary 60K05, 33E12, 26A33; Secondary 60J60, 65R10, 60K40

Mots clés: and Phrases: Prabhakar fractional calculus; renewal process; Mittag-Leffler type functions; generalized fractional Poisson process; fractional ordinary and partial differential equations; generalized fractional Kolmogorov-Feller equation; continuous-time random walk; Non-Markovian random walks

Domaines: Mécanique statistique [cond-mat.stat-mech]

Thomas Michelitsch, Alejandro Pérez Riascos. GENERALIZED FRACTIONAL POISSON PROCESS AND RELATED STOCHASTIC DYNAMICS. Fractional Calculus and Applied Analysis, 2020, 23, ⟨10.1515/fca-2020-0034⟩. ⟨hal-02888237⟩