First-return time in fractional kinetics

TL;DR

This paper investigates the first-return time in fractional diffusion processes modeled by continuous-time random walks with Mittag-Leffler waiting times, revealing its independence from jump-size distribution and dependence on waiting-time memory effects.

Contribution

It provides exact analytical results for first-return time distributions in fractional kinetics, covering both Markovian and non-Markovian cases with symmetric jump distributions.

Findings

First-return time density is independent of jump-size distribution for symmetric jumps.

Memory effects in waiting times significantly influence the first-return time.

Derived relations between Markovian and non-Markovian first-return time results.

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. We study the first-return time 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. We consider both Markovian and non-Markovian settings, as well as any kind of symmetric jump-size distributions, namely with finite or infinite variance. We show 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. We perform our analysis in two cases: first jump then wait and first wait then jump, and we provide several exact results, including the relation between results in the Markovian and non-Markovian settings and the difference between the two cases.