Semi-Markovian Discrete-Time Telegraph Process with Generalized Sibuya Waiting Times

• Published in: Mathematics (Basel) Vol. 11; no. 2; p. 471
• Main Authors: Michelitsch, Thomas M., Polito, Federico, Riascos, Alejandro P.
• Format: Journal Article
• Language: English
• Published: Basel MDPI AG 01-01-2023; MDPI
• Subjects: Condensed Matter; discrete-time renewal process; generalized Sibuya distribution; Markov analysis; Mathematics; non-markovian random walk; Physics; Probability; Random variables; Random walk; Statistical Mechanics; Switches; telegraph (Cattaneo) process; Velocity; discrete-time renewal process; generalized Sibuya distribution; Non-Markovian random walk telegraph (Cattaneo) process generalized Sibuya distribution discrete-time renewal process; telegraph (Cattaneo) process; Non-Markovian random walk
• ISSN: 2227-7390; 2227-7390
• DOI: 10.3390/math11020471

In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘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≤m−1 (m≥1) and diverging for orders r≥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.