The Fractional Poisson Process and the Inverse Stable Subordinator

Mark Meerschaert,P Vellaisamy,Erkan Nane

doi:10.1214/ejp.v16-920

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 {discusses the relation between} the fractional Poisson process and Brownian time.

Highlights

The fractional Poisson process (FPP) was introduced and studied by Repin and Saichev [42], Jumarie [25], Laskin [30], Mainardi et al [32, 33], Uchaikin et al [46] and Beghin and Orsingher [6, 7]
The FPP is a natural generalization of the usual Poisson process, with an interesting connection to fractional calculus
This section develops some interesting connections between the fractional Poisson process and fractional calculus

Summary

Introduction

The fractional Poisson process (FPP) was introduced and studied by Repin and Saichev [42], Jumarie [25], Laskin [30], Mainardi et al [32, 33], Uchaikin et al [46] and Beghin and Orsingher [6, 7]. The FPP is a natural generalization of the usual Poisson process, with an interesting connection to fractional calculus. This renewal process has IID waiting times Jn that satisfy. We will prove that the FPP and the FTPP are the same process, by showing that the waiting times between jumps in the FTPP are IID Mittag-Leffler. This strong connection between the FPP and the FTPP unifies the two main approaches in the stochastic theory of fractional diffusion. The FPP approach was used recently in the work of Behgin and Orsingher [6], while the inverse stable subordinator is a key ingredient in [39]

Two equivalent formulations

Fractional calculus

Renewal processes and inverse subordinators

CTRW scaling limits and governing equations