Elliptic Harnack inequality for Z^d

Authors

Siva Athreya, Indian Statistical Institute Bangalore
Nitya Gadhiwala, The University of British Columbia
Ritvik R. Radhakrishnan, ETH Zürich

Article Type

Research Article

Publication Title

Involve

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 un-dergraduate 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.

First Page

687

Last Page

708

DOI

10.2140/involve.2022.15.687

Publication Date

1-1-2022

Comments

Open Access, Green

Recommended Citation

Athreya, Siva; Gadhiwala, Nitya; and Radhakrishnan, Ritvik R., "Elliptic Harnack inequality for Z^d" (2022). Journal Articles. 3302.
https://digitalcommons.isical.ac.in/journal-articles/3302