This book provides a systematic, comprehensive, and self-contained treatment of the fractional Poisson process and the fractional Poisson distribution, covering both their theory and wide-ranging applications. It explores four main approaches to studying the fractional Poisson process: the fractional generalization of the Kolmogorov-Feller equation; renewal theory, replacing the exponential waiting time distribution of a Poisson process with one modeled by the Mittag-Leffler function; the inverse stable subordinator method, which constructs a fractional Poisson process from a known Poisson process by subordinating the time variable to an independent inverse stable subordinator; and the direct design of the probability mass function using the complete monotonicity of the Mittag-Leffler function and its generalizations, without relying on fractional differential or integral operators.
The book demonstrates the use of various Mittag-Leffler type functions, Fox-Wright and Fox's H-functions, the Lévy α-stable distribution, and techniques such as the Laplace and Mellin transforms and the Mellin-Barnes representation. These tools are shown to be especially effective for analyzing fractional stochastic processes and their probability mass and density functions.
This is the first book devoted to the fractional Poisson process, written by its inventor, who introduced the concept in 2003.