Some results and problems for anisotropic random walks on the plane

TL;DR

This paper surveys asymptotic behaviors of anisotropic random walks on the 2D lattice, discussing recent findings on strong approximations, local times, and range, and highlighting open problems in the field.

Contribution

It compiles recent research results on anisotropic random walks and presents new open problems for further investigation.

Findings

Analysis of strong approximation properties

Results on local times of the walk

Insights into the range of the walk

Abstract

This is an expository paper on the asymptotic results concerning path behaviour of the anisotropic random walk on the two-dimensional square lattice Z^2. In recent years Mikl\'os and the authors of the present paper investigated the properties of this random walk concerning strong approximations, local times and range. We give a survey of these results together with some further problems.