Semi-Markovian discrete-time telegraph process with generalized Sibuya waiting times

TL;DR

This paper introduces a generalized Sibuya distribution-based semi-Markovian discrete-time telegraph process, analyzing its anomalous diffusion behaviors and providing new mathematical representations for its generating functions.

Contribution

It extends the discrete-time telegraph process by incorporating a generalized Sibuya renewal process, revealing diverse anomalous diffusion regimes and deriving new generating function representations.

Findings

Exhibits multiple regimes of anomalous diffusion depending on the GSD moments.

Contains the standard Sibuya distribution as a special case.

Provides new mathematical representations for generating functions related to GSD.

Abstract

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred this random walk to as squirrel random walk (SRW). The SRW is a discrete-time random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the `generalized Sibuya distribution' (GSD) is such that the moments are finite up to a certain order $r\leq m-1$ ($m \geq 1$) and diverging for orders $r \geq m$ capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case ($m=1$). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order $m$ of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics.