On the fractional Poisson process and the discretized stable subordinator

TL;DR

This paper analyzes the fractional Poisson and Wright processes, their inverse processes, and their diffusion limits, providing insights into fractional diffusion modeling and simulation.

Contribution

It offers a detailed analysis of the fractional Poisson and Wright processes, including their inverse processes and diffusion limits, advancing understanding of fractional diffusion processes.

Findings

Derived diffusion limits for the processes

Analyzed the counting and Erlang processes

Provided insights into fractional diffusion simulation

Abstract

The fractional Poisson process and the Wright process (as discretization of the stable subordinator) along with their diffusion limits play eminent roles in theory and simulation of fractional diffusion processes. Here we have analyzed these two processes, concretely the corresponding counting number and Erlang processes, the latter being the processes inverse to the former. Furthermore we have obtained the diffusion limits of all these processes by well-scaled refinement of waiting times and jumps