The fractional Poisson process and the inverse stable subordinator

TL;DR

This paper demonstrates that a Poisson process with a random time change by an inverse stable subordinator is equivalent to the fractional Poisson process, unifying key approaches in fractional diffusion modeling.

Contribution

It establishes the equivalence between the fractional Poisson process and a time-changed Poisson process using inverse stable subordinators, linking two main stochastic approaches.

Findings

Equivalence between fractional Poisson process and time-changed Poisson process.

Extension of the equivalence to renewal processes for fractional diffusion.

Connection between fractional Poisson process and Brownian time.

Abstract

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also establishes an interesting connection between the fractional Poisson process and Brownian time.