Elliptic Harnack Inequality for ${\mathbb{Z}}^d$

TL;DR

This paper provides a simplified, accessible probabilistic proof of the scale-invariant Elliptic Harnack Inequality for non-negative harmonic functions on integer lattices, using classical probabilistic tools.

Contribution

It offers a self-contained, undergraduate-level proof of EHI on ${bZ}^d$ utilizing the Local Central Limit Theorem and classical potential theory tools.

Findings

Establishes Gaussian bounds for random walk probabilities

Proves scale-invariant EHI for harmonic functions on ${bZ}^d$

Provides a simplified proof accessible to undergraduates

Abstract

We prove the scale invariant Elliptic Harnack Inequality (EHI) for non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of EHI in ${\mathbb{Z}}^d$ that is accessible at the undergraduate level. We use the Local Central Limit Theorem for simple symmetric random walks on ${\mathbb{Z}}^d$ to establish Gaussian bounds for the $n$-step probability function. The uniform Green inequality and the classical Balayage formula then imply the EHI.