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.
Citation
Mark Meerschaert. Erkan Nane. P. Vellaisamy. "The Fractional Poisson Process and the Inverse Stable Subordinator." Electron. J. Probab. 16 1600 - 1620, 2011. https://doi.org/10.1214/EJP.v16-920
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