A continuous-time random walk extension of the Gillis model

TL;DR

This paper extends the Gillis random walk model to continuous time with heavy-tailed waiting times, revealing subdiffusive behavior, ergodicity breaking, and providing exact analytical results supported by simulations.

Contribution

It introduces a continuous-time version of the Gillis model with heavy-tailed waiting times, offering new analytical insights into its diffusive and ergodic properties.

Findings

Heavy-tailed waiting times induce subdiffusion.

The model exhibits ergodicity breaking.

Analytical results are validated by numerical simulations.

Abstract

We consider a continuous-time random walk which is the generalization, by means of the introduction of waiting periods on sites, of the one-dimensional nonhomogeneous random walk with a position-dependent drift known in the mathematical literature as Gillis random walk. This modified stochastic process allows to significantly change local, non-local and transport properties in the presence of heavy-tailed waiting-time distributions lacking the first moment: we provide here exact results concerning hitting times, first-time events, survival probabilities, occupation times, the moments spectrum and the statistics of records. Specifically, normal diffusion gives way to subdiffusion and we are witnessing the breaking of ergodicity. Furthermore we also test our theoretical predictions with numerical simulations.