The fractional Poisson process and the inverse stable subordinator

Details The fractional Poisson process and the inverse stable subordinator The fractional Poisson process and the inverse stable subordinator

2010-07-28

arxiv.org/abs/1007.5051v2

Electron. J. Probab. Vol. 16 (2011), no. 59, 1600-1620

Mathematics

Probability

22 pages, version submitted on December 2 2010

Scientific paper

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.

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