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

Vol. 15 (2022), No. 4, 687–708
##### Abstract

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

##### Keywords
random walk, harmonic function, Harnack inequality, Gaussian bounds, balayage
##### Mathematical Subject Classification
Primary: 05C81
Secondary: 31C05, 31C20