First-return time in fractional kinetics

Dahlenburg, Marcus; Pagnini, Gianni

doi:10.15559/25-VMSTA288

Modern Stochastics: Theory and Applications

First-return time in fractional kinetics

Marcus Dahlenburg

Gianni Pagnini

https://doi.org/10.15559/25-VMSTA288

Pub. online: 13 November 2025 Type: Research Article

Open Access

Received
2 July 2025

Revised
14 October 2025

Accepted
31 October 2025

Published
13 November 2025

Abstract

The first-return time is the time that it takes a random walker to go back to the initial position for the first time. In this paper, the first-return time is studied when random walkers perform fractional kinetics, specifically fractional diffusion, that is modelled within the framework of the continuous-time random walk on homogeneous space in the uncoupled formulation with Mittag-Leffler distributed waiting-times. Both the Markovian and non-Markovian settings are considered, as well as any kind of symmetric jump-size distributions, namely with finite or infinite variance. It is shown that the first-return time density is indeed independent of the jump-size distribution when it is symmetric, and therefore it is affected only by the waiting-time distribution that embodies the memory of the process. The analysis is performed in two cases: first jump then wait and first wait then jump, and several exact results are provided, including the relation between results in the Markovian and non-Markovian settings and the difference between the two cases.

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Keywords

First-return time fractional kinetics fractional diffusion continuous-time random walk first-passage time Sparre Andersen theorem

MSC2020

82C05 60K50 60J76

Funding

This research is supported by the Basque Government through the BERC 2022–2025 program and by the Ministry of Science and Innovation: BCAM Severo Ochoa accreditation CEX2021-001142-S / MICIN / AEI / 10.13039/501100011033.

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