Random walks in a one-dimensional L\'evy random environment

TL;DR

This paper studies a one-dimensional stochastic process modeling particle motion in a random environment with long-tailed spacing, proving central limit theorems for both the discrete and continuous-time versions.

Contribution

It extends the analysis of the Le9vy-Lorentz gas by establishing CLTs and moment convergence for the process in a long-tailed random environment.

Findings

Proved quenched CLT for the random walk on the point process.

Established annealed CLT for the continuous-time process.

Demonstrated convergence of rescaled moments.

Abstract

We consider a generalization of a one-dimensional stochastic process known in the physical literature as L\'evy-Lorentz gas. The process describes the motion of a particle on the real line in the presence of a random array of marked points, whose nearest-neighbor distances are i.i.d. and long-tailed (with finite mean but possibly infinite variance). The motion is a continuous-time, constant-speed interpolation of a symmetric random walk on the marked points. We first study the quenched random walk on the point process, proving the CLT and the convergence of all the accordingly rescaled moments. Then we derive the quenched and annealed CLTs for the continuous-time process.