Conservative and semiconservative random walks: recurrence and transience

TL;DR

This paper introduces conservative and semiconservative random walks in integer lattices, classifies them, and provides new methods to analyze their recurrence or transience, including examples in various dimensions.

Contribution

It defines new classes of random walks, offers a classification framework, and constructs explicit recurrent and transient examples across different dimensions.

Findings

Symmetric random walks are conservative and include simple random walks.

New classification reduces analysis to birth-and-death processes.

Constructed recurrent walks in all dimensions ≥3 and transient walks in 2D.

Abstract

In the present paper we define conservative and semiconservative random walks in $\mathbb{Z}^d$ and study different families of random walks. The family of symmetric random walks is one of the families of conservative random walks, and simple (P\'olya) random walks are their representatives. The classification of random walks given in the present paper enables us to provide a new approach to random walks in $\mathbb{Z}^d$ by reduction to birth-and-death processes. We construct nontrivial examples of recurrent random walks in $\mathbb{Z}^d$ for any $d\geq3$ and transient random walks in $\mathbb{Z}^2$.