Squirrels can remember little: A random walk with jump reversals induced by a discrete-time renewal process

TL;DR

This paper introduces a discrete-time random walk model with reversals governed by a renewal process, analyzing its asymptotic behavior, convergence to the telegraph process, and various diffusion regimes including anomalous superdiffusion.

Contribution

It provides exact formulas for the propagator, explores the effects of different waiting time distributions, and introduces time-changed versions leading to new classes of anomalous diffusion models.

Findings

Walk converges to the telegraph process with geometric waiting times.

Finite mean waiting times lead to localization; fat-tailed distributions cause superdiffusion.

Explicit variance formulas for Sibuya-distributed waiting times.

Abstract

We consider a class of discrete-time random walks with directed unit steps on the integer line. The direction of the steps is reversed at the time instants of events in a discrete-time renewal process and is maintained at uneventful time instants. This model represents a discrete-time semi-Markovian generalization of the telegraph process. We derive exact formulae for the propagator using generating functions. We prove that for geometrically distributed waiting times in the diffusive limit, this walk converges to the classical telegraph process. We consider the large-time asymptotics of the expected position: For waiting time densities with finite mean the walker remains in the average localized close to the departure site whereas escapes for fat-tailed waiting-time densities (i.e. densities with infinite mean) by a sublinear power-law. We explore anomalous diffusion features by accounting for the `aging effect' as a hallmark of non-Markovianity where the discrete-time version of the `aging renewal process' comes into play. By deriving pertinent distributions of this process we obtain explicit formulae for the variance when the waiting-times are Sibuya-distributed. In this case and generally for fat-tailed waiting time PDFs a $t^2$-ballistic superdiffusive scaling emerges in the large time limit. In contrast if the waiting time PDF between the step reversals is light-tailed (`narrow' with finite mean and variance) the walk exhibits normal diffusion and for `broad' waiting time PDFs (with finite mean and infinite variance) superdiffusive large time scaling. We also consider time-changed versions where the walk is subordinated to a continuous-time point process such as the time-fractional Poisson process. This defines a new class of biased continuous-time random walks exhibiting several regimes of anomalous diffusion.