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First-return time in fractional kinetics

M. Dahlenburg G. Pagnini

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.

First-return time in fractional kinetics

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.
Paper Structure (9 sections, 9 theorems, 70 equations)

This paper contains 9 sections, 9 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Sparre Andersen theorem
  4. From the discrete- to the continuous-time setting
  5. The Sparre Andersen theorem in the framework of the CTRW
  6. First-return time for CTRW
  7. Problem formulation and definitions
  8. The (jw)-case
  9. The (wj)-case

Key Result

Theorem 1

The unconditional survival probability of a symmetric CTRW is independent of the jump-size distribution if this is equal to the distribution of the initial position.

Theorems & Definitions (13)

  • Theorem 1
  • Remark 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Remark 2
  • Remark 3
  • Theorem 4
  • Proposition 1
  • Proposition 2
  • ...and 3 more