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

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 (non-markovianity) as a hallmark of the `complexity' of the process.