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Prabhakar discrete-time generalization of the time-fractional Poisson process and related random walks

Abstract : In recent years a huge interdisciplinary field has emerged which is devoted to the 'complex dynamics' of anomalous transport with long-time memory and non-markovian features. It was found that the framework of fractional calculus and its generalizations are able to capture these phenomena. Many of the classical models are based on continuous-time renewal processes and use the Montroll-Weiss continuoustime random walk (CTRW) approach. On the other hand their discretetime counterparts are rarely considered in the literature despite their importance in various applications. The goal of the present paper is to give a brief sketch of our recently introduced discrete-time Prabhakar generalization of the fractional Poisson process and the related discrete-time random walk (DTRW) model. We show that this counting process is connected with the continuous-time Prabhakar renewal process by a ('wellscaled') continuous-time limit. We deduce the state probabilities and discrete-time generalized fractional Kolmogorov-Feller equations governing the Prabhakar DTRW and discuss effects such as long-time memory (non-markovianity) as a hallmark of the 'complexity' of the process.
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Contributor : Thomas Michelitsch Connect in order to contact the contributor
Submitted on : Thursday, May 27, 2021 - 10:24:37 AM
Last modification on : Wednesday, April 27, 2022 - 3:08:57 AM
Long-term archiving on: : Saturday, August 28, 2021 - 6:25:13 PM


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Thomas M Michelitsch, Federico Polito, Alejandro P Riascos. Prabhakar discrete-time generalization of the time-fractional Poisson process and related random walks. Proceedings of the International Conference on Fractional Differentiation and its Applications (ICFDA’21). ICFDA 2021. Lecture Notes in Networks and Systems, Sep 2021, Warsaw, Poland. ⟨10.1007/978-3-031-04383-3_14⟩. ⟨hal-03238477⟩



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