A fractional generalization of the Poisson processes

By Francesco Mainardi, Rudolf Gorenflo, Enrico Scalas. 2007

226 citations recorded by the source corpus.

It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the basic renewal theory including its fundamental concepts like waiting time between events, the survival probability, the counting function. If the waiting time is exponentially distributed we have a Poisson process, which is Markovian. However, other waiting time distributions are also relevant in applications, in particular such ones with a fat tail caused by a power law decay of its density. In this context we analyze a non-Markovian renewal process with a waiting time distribution described by the MittagLeffler function. This distribution, containing the exponential as particular case, is shown to play a fundamental role in the infinite thinning procedure of a generic renewal process governed by a power-asymptotic waiting time. We then consider the renewal theory with reward that implies a random walk subordinated to a renewal process.

Subfields: Mathematical Physics, Modeling and Simulation, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/e8f4e1cae4dcd6cbcf4834826b103f92a97a99a4

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Citations in the corpus

The space-fractional Poisson process (2011, methodology): “…studied so far have been obtained either by considering renewal processes with intertimes between events represented by Mittag–Leffler distributions [Mainardi et al., 2004, Beghin and Orsingher, 2009] or by replacing the time derivative in the equations governing the state probabilities with the…”
Poisson-type processes governed by fractional and higher-order recursive differential equations (2009, background): “[17]-[18]-[19], where a deep analysis of the related process is performed: it turns out to be a true renewal process, loosing however the Markovian property.”
Caputo-Type Modification of the Erdélyi-Kober Fractional Derivative (2007, background): “[18], [19], Matignon and Montseny [20], Metzler et al.”
The Fractional Poisson Process and the Inverse Stable Subordinator (2010, unknown): “[32, 33], Uchaikin et al.”
Parameter estimation for fractional Poisson processes (2010, methodology): “More recently, Mainardi et al. (2004, 2005) provided an approach to fPp based on analysis of the survival probability function Θ(t) = P (T > t).”
The fractional non-homogeneous Poisson process (2016, methodology): “The renewal approach fully characterizes a speciﬁc process and was used in Mainardi et al. (2004).”