The Fractional Poisson Process and the Inverse Stable Subordinator

Mark M Meerschaert (Michigan State University)
Erkan Nane (Auburn University)
P. Vellaisamy (Indian Institute of Technology Bombay)

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.

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Pages: 1600-1620

Publication Date: August 28, 2011

DOI: 10.1214/EJP.v16-920

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