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2011 The Fractional Poisson Process and the Inverse Stable Subordinator
Mark Meerschaert, Erkan Nane, P. Vellaisamy
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Electron. J. Probab. 16: 1600-1620 (2011). DOI: 10.1214/EJP.v16-920


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.


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Mark Meerschaert. Erkan Nane. P. Vellaisamy. "The Fractional Poisson Process and the Inverse Stable Subordinator." Electron. J. Probab. 16 1600 - 1620, 2011.


Accepted: 28 August 2011; Published: 2011
First available in Project Euclid: 1 June 2016

zbMATH: 1245.60084
MathSciNet: MR2835248
Digital Object Identifier: 10.1214/EJP.v16-920

Primary: 60K05
Secondary: 26A33 , 33E12

Keywords: Caputo fractional derivative , Continuous time random walk limit , Di , Fractional difference-differential equations , Fractional Poisson process , generalized Mittag-Leffler function , Inverse stable subordinator , Mittag-Leffler waiting time , Renewal process

Vol.16 • 2011
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