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 continuous time random walk (CTRW) approach. On the other hand their discrete time 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 (well scaled) 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 (nonmarkovianity) as a hallmark of the complexity of the process.