# Generalized fractional Poisson process and related stochastic dynamics

## TL;DR

This paper surveys the generalized fractional Poisson process (GFPP), a flexible renewal process involving Prabhakar functions, and develops related stochastic models including CTRWs and fractional diffusion equations, extending classical processes.

## Contribution

It introduces the GFPP with two parameters, derives explicit formulas, and develops new stochastic dynamics models including fractional Kolmogorov equations and diffusion limits.

## Key findings

- GFPP generalizes classical Poisson and Erlang processes.
- Derived explicit formulas for GFPP and related stochastic models.
- Established fractional diffusion equations from GFPP-based random walks.

## Abstract

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 $0<\beta\leq 1$, $\alpha >0$ and a parameter characterizing the time scale. The GFPP involves Prabhakar 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.