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Sub-diffusive behavior of a recurrent Axis-Driven Random Walk

Pierre Andreoletti, Pierre Debs

TL;DR

The paper analyzes a recurrent axis-driven random walk in the plane, revealing subdiffusive behavior in the regions between axes. By developing a second-order renewal framework for axis excursions and studying the second-order fluctuations of axis-entry times, it derives a log-Laplace characterization of the limiting distribution and establishes a joint limit for the axis-entrance process. The main result describes the asymptotics of the number of axis excursions and the residual time, with a $1$-stable tail and a $3$-D Bessel process interpretation, while the overall position exhibits a mixed diffusion-subdiffusion limit governed by Bessel processes and a $1$-stable mixing law. These findings provide a precise, explicit description of the long-time, subdiffusive behavior and connect to classical random walks in cones, enriching the understanding of constrained diffusion under axis-directed forces.

Abstract

We study the second order of the number of excursions of a simple random walk with a bias that drives a return toward the origin along the axes introduced by P. Andreoletti and P. Debs \cite{AndDeb3}. This is a crucial step toward deriving the asymptotic behavior of these walks, whose limit is explicit and reveals various characteristics of the process: the invariant probability measure of the extracted coordinates away from the axes, the 1-stable distribution arising from the tail distribution of entry times on the axes, and finally, the presence of a Bessel process of dimension 3, which implies that the trajectory can be interpreted as a random path conditioned to stay within a single quadrant.

Sub-diffusive behavior of a recurrent Axis-Driven Random Walk

TL;DR

The paper analyzes a recurrent axis-driven random walk in the plane, revealing subdiffusive behavior in the regions between axes. By developing a second-order renewal framework for axis excursions and studying the second-order fluctuations of axis-entry times, it derives a log-Laplace characterization of the limiting distribution and establishes a joint limit for the axis-entrance process. The main result describes the asymptotics of the number of axis excursions and the residual time, with a -stable tail and a -D Bessel process interpretation, while the overall position exhibits a mixed diffusion-subdiffusion limit governed by Bessel processes and a -stable mixing law. These findings provide a precise, explicit description of the long-time, subdiffusive behavior and connect to classical random walks in cones, enriching the understanding of constrained diffusion under axis-directed forces.

Abstract

We study the second order of the number of excursions of a simple random walk with a bias that drives a return toward the origin along the axes introduced by P. Andreoletti and P. Debs \cite{AndDeb3}. This is a crucial step toward deriving the asymptotic behavior of these walks, whose limit is explicit and reveals various characteristics of the process: the invariant probability measure of the extracted coordinates away from the axes, the 1-stable distribution arising from the tail distribution of entry times on the axes, and finally, the presence of a Bessel process of dimension 3, which implies that the trajectory can be interpreted as a random path conditioned to stay within a single quadrant.

Paper Structure

This paper contains 10 sections, 9 theorems, 108 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['SONn']}
  3. A second order for $(\rho_m)_m$
  4. Proof of Theorem \ref{['SONn']}
  5. Convergence of the main process
  6. Preliminaries
  7. The limit of $\mathbb{P}(|X_{n}|\geq a b_n)$, with $a=(a_1,a_2)\in (\mathbb{R}_+^*)^2$
  8. Appendix
  9. Proof of Lemma \ref{['leLem']}
  10. Facts about the random walk in the cone and on the axis

Key Result

Theorem 1.1

Assume $\alpha>3$, for any $a=(a_1,a_2) \in (\mathbb{R}_+^*)^2$ with $\mathfrak{R}_{s}=(\mathfrak{R}_{s}^1,\mathfrak{R}_s^2)$, $\mathfrak{R}^1$ and $\mathfrak{R}^2$ are two independent Bessel processes of dimension 3 and $\mu_{\mathcal{S}_1^0}$ is a 1-stable distribution which is described in the second theorem below.

Figures (1)

  • Figure 1: probabilities of transition of $\mathbf{X}$

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 1.2
  • Theorem 1.3
  • Corollary 2.1
  • Proposition 2.2
  • Lemma 2.4
  • Lemma 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • ...and 4 more