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

Siva Athreya, Nitya Gadhiwala and Ritvik R. Radhakrishnan

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

We prove the scale-invariant elliptic Harnack inequality (EHI) for nonnegative harmonic functions on d . The purpose of this note is to provide a simplified self-contained probabilistic proof of the EHI in d that is accessible at the undergraduate level. We use the local central limit theorem for simple symmetric random walks on 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
Milestones
Received: 21 September 2021
Revised: 22 November 2021
Accepted: 26 January 2022
Published: 7 January 2023

Communicated by Amarjit Singh Budhiraja
Authors
Siva Athreya
International Centre for Theoretical Sciences of the Tata Institute of Fundamental Research
Bengaluru
India
Indian Statistical Institute, Bangalore Centre
Bengaluru
India
Nitya Gadhiwala
Department of Mathematics
University of British Columbia
Vancouver, BC
Canada
Ritvik R. Radhakrishnan
Department of Mathematics
ETH Zurich
Switzerland