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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
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Abstract |
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We prove the scale-invariant elliptic Harnack inequality (EHI) for nonnegative harmonic functions on . The purpose of this note is to provide a simplified self-contained probabilistic proof of the EHI in that is accessible at the undergraduate level. We use the local central limit theorem for simple symmetric random walks on to establish Gaussian bounds for the -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
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Mathematical Subject Classification
Primary: 05C81
Secondary: 31C05, 31C20
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Milestones
Received: 21 September 2021
Revised: 22 November 2021
Accepted: 26 January 2022
Published: 7 January 2023
Communicated by Amarjit Singh Budhiraja
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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 |
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| Ritvik R. Radhakrishnan | |
| Department of Mathematics ETH Zurich Switzerland |
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