The fractional Poisson process and the inverse stable subordinator. (English) Zbl 1245.60084

Meerschaert, Mark M.; Nane, Erkan; Vellaisamy, P.

Electron. J. Probab. 16, Paper No. 59, 1600-1620 (2011).

Summary: 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.

MSC:

60K05	Renewal theory
33E12	Mittag-Leffler functions and generalizations
26A33	Fractional derivatives and integrals

Keywords:

fractional Poisson process; inverse stable subordinator; renewal process; Mittag- Leffler waiting time; fractional difference-differential equations; Caputo fractional derivative; generalized Mittag-Leffler function; continuous time random walk limit; distributed order derivative; tempered fractional derivative