Harnack Inequalities of Hitting Distributions of Projections of Planar Symmetric Random Walks on the Lattice Torus

TL;DR

This paper establishes Harnack inequalities for hitting distributions of symmetric random walks on the 2D lattice and their projections onto the torus, extending previous frameworks and results to a broader setting.

Contribution

It generalizes existing Harnack inequality results for simple random walks to a wider class of symmetric walks on the lattice and their toral projections.

Findings

Harnack inequalities hold for a large family of symmetric random walks

Results extend previous work from simple random walks to more general walks

Framework applies to projections on the lattice torus

Abstract

We give Harnack inequalities for the hitting distributions of a large family of symmetric random walks on $\Z^2$, and their projections onto the lattice torus $\Z^2_K$. This extends a framework for the simple random walk in Dembo, et al. (2006), and generalizes the results in Bass, Rosen (2007) to the toral projection.