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

Nicholas Georgiou, Mikhail V. Menshikov, Aleksandar Mijatović and Andrew R. Wade

Advances in Applied Probability, 48, issue A, July 2016, 99–118.

Supported by EPSRC award Non-homogeneous random walks (EP/J021784/1).

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 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.

Further remarks

This paper is part of a broader project on non-homogeneous random walks. Here are two pictures (by Nic Georgiou) from the paper, showing paths of two zero-drift random walks in the plane whose increments have an elliptical covariance structure (see the paper for details). The first is recurrent and the second is transient.