[Paper Review] First-return time in fractional kinetics

M. Dahlenburg G. Pagnini|arXiv (Cornell University)|Mar 15, 2026

Diffusion and Search Dynamics0 citations

TL;DR

The paper analyzes the first-return time (FRT) for symmetric continuous-time random walks with fractional kinetics, showing universality w.r.t. jump-size distributions under symmetric jumps, and contrasts Markovian and non-Markovian (Mittag–Leffler waiting times) cases for two formulations: first jump then wait and first wait then jump.

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.

Motivation & Objective

• Motivate the study of first-return time (FRT) for fractional kinetics within CTRWs.
• Show that FRT statistics are universal w.r.t. symmetric jump-size distributions.
• Differentiate and relate Markovian and non-Markovian waiting-time structures.
• Provide exact results for two measurement formulations: first jump then wait (jw) and first wait then jump (wj).
• Offer analytic connections between Markovian and non-Markovian cases and between jw and wj formulations.

Proposed method

• Frame the problem via the Sparre Andersen theorem in the CTRW setting.
• Use Wiener–Hopf/Pollaczek–Spitzer formalism to relate survival and waiting-time distributions.
• Employ Laplace transforms to derive integral equations for survival probability and FRT densities.
• Specialize to Mittag–Leffler waiting times to model non-Markovian memory.
• Derive exact FRT densities for jw and wj cases and establish universality w.r.t. jump-size distribution.
• Utilize Efros/Wright and Mainardi functions to express inverse transforms and asymptotics.

Experimental results

Research questions

• RQ1 Does the first-return time density depend on the tail of the jump-size distribution when the distribution is symmetric?
• RQ2 How do Markovian and non-Markovian waiting-time statistics (exponential vs Mittag–Leffler) affect the FRT in jw and wj formulations?
• RQ3 What are the exact relationships between the FRT densities in jw and wj, and between jw/wj and their Markovian counterparts?
• RQ4 What are the asymptotic behaviors (short- and long-time) of the FRT density under Mittag–Leffler waiting times?

Key findings

• The first-return time density is independent of the jump-size distribution when the jump distribution is symmetric (universality).
• In the jw case with symmetric CTRW, the FRT density is independent of the jump-size distribution and can be expressed in closed form for non-Markovian waiting times.
• In the non-Markovian setting with Mittag–Leffler waiting times, the FRT density has heavy tails with f(t) ~ 2 t^{-β/2 -1} for large t and infinite mean FRT, while the short-time behavior scales as t^{β-1}.
• In the Markovian limit (β=1), explicit FRT densities are obtained (e.g., f_M with known closed form), and the asymptotics recover 1/t behavior consistent with diffusion-like processes.
• The jw and wj formulations yield related but distinct FRT densities; via Efros’ formula, jw and wj results are connected through stable-law convolutions, scaling with β in a consistent way.
• Theorem 1 establishes unconditional survival independence from the jump-size distribution when the initial position matches the jump distribution; Theorem 2 and Theorem 4 provide the jw and wj relations between Mittag–Leffler and exponential waiting times; Corollary 1 and Theorem 3 give cumulative and distributional links via Mainardi/Wright functions.