Elliptic Harnack inequality for ℤd

Abstract

We prove the scale-invariant elliptic Harnack inequality (EHI) for nonnegative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of the 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.