Anomalous recurrence properties of many-dimensional zero-drift random walks

TL;DR

This paper investigates how non-homogeneous, elliptic random walks in multiple dimensions can exhibit either recurrence or transience, challenging classical results for homogeneous walks by analyzing their asymptotic covariance structures.

Contribution

It introduces a class of elliptic random walks with adjustable recurrence properties, extending classical recurrence results to non-homogeneous, anisotropic settings.

Findings

Elliptic random walks can be tuned to be recurrent or transient in any dimension.

Recurrence classification is achieved using Lamperti's criteria.

The work generalizes classical results for homogeneous walks to non-homogeneous, anisotropic cases.

Abstract

Famously, a $d$-dimensional, spatially homogeneous random walk whose increments are non-degenerate, have finite second moments, and have zero mean is recurrent if $d \in \{1,2\}$ but transient if $d \geq 3$. Once spatial homogeneity is relaxed, this is no longer true. We study a family of zero-drift spatially non-homogeneous random walks (Markov processes) whose increment covariance matrix is asymptotically constant along rays from the origin, and which, in any ambient dimension $d \geq 2$, can be adjusted so that the walk is either transient or recurrent. Natural examples are provided by random walks whose increments are supported on ellipsoids that are symmetric about the ray from the origin through the walk's current position; these \emph{elliptic random walks} generalize the classical homogeneous Pearson--Rayleigh walk (the spherical case). Our proof of the recurrence classification is based on fundamental work of Lamperti.