@article{EJP920,
author = {Mark Meerschaert and Erkan Nane and P. Vellaisamy},
title = {The Fractional Poisson Process and the Inverse Stable Subordinator},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {16},
year = {2011},
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; Di},
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.},
pages = {no. 59, 1600-1620},
issn = {1083-6489},
doi = {10.1214/EJP.v16-920},
url = {http://ejp.ejpecp.org/article/view/920}}