Elliptic Harnack inequality for Zᵈ

Abstract

We prove the scale-invariant elliptic Harnack inequality (EHI) for nonnegative harmonic functions on Zᵈ. The purpose of this note is to provide a simplified self-contained probabilistic proof of the EHI in Zᵈ that is accessible at the undergraduate level. We use the local central limit theorem for simple symmetric random walks on Zᵈ to establish Gaussian bounds for the n-step probability function. The uniform Green inequality and the classical balayage formula then imply the EHI.