On discrete time Prabhakar-generalized fractional Poisson processes and related stochastic dynamics

Recently the so-called Prabhakar generalization of the fractional Poisson counting process attracted much interest for his flexibility to adapt to real world situations. In this renewal process the waiting times between events are IID continuous random variables. In the present paper we analyze discrete-time counterparts: Renewal processes with integer IID interarrival times which converge in well-scaled continuous-time limits to the Prabhakar-generalized fractional Poisson process. These processes exhibit non-Markovian features and long-time memory effects. We recover for special choices of parameters the discrete-time versions of classical cases, such as the fractional Bernoulli process and the standard Bernoulli process as discrete-time approximations of the fractional Poisson and the standard Poisson process, respectively. We derive difference equations of generalized fractional type that govern these discrete time-processes where in well-scaled continuous-time limits known evolution equations of generalized fractional Prabhakar type are recovered. We also develop in Montroll–Weiss fashion the “Prabhakar Discrete-time random walk (DTRW)” as a random walk on a graph time-changed with a discrete-time version of Prabhakar renewal process. We derive the generalized fractional discrete-time Kolmogorov–Feller difference equations governing the resulting stochastic motion. Prabhakar-discrete-time processes open a promising field capturing several aspects in the dynamics of complex systems. •Discrete-time renewal processes are analyzed and general well-scaled continuous-time limit procedures are developed.•A discrete-time generalization of fractional Poisson processes, the Prabhakar discrete-time counting process (PDTP) is introduced.•The PDTP state probabilities and their continuous-time limits are derived. Classical cases such as fractional Bernoulli and standard Bernoulli are shown to be contained by means of explicit formulas.•Generalized Kolmogorov–Feller equations are derived governing the state probabilities and related stochastic dynamics.•The resulting stochastic motion on undirected graphs is analyzed: The “Prabhakar discrete-time random walk” (Prabhakar DTRW) is developed.