Elliptic Harnack Inequality for ${\mathbb{Z}}^d$
| Autor: | Athreya, Siva, Gadhiwala, Nitya, Radhakrishnan, Ritvik R. |
|---|---|
| Rok vydání: | 2022 |
| Předmět: | |
| Zdroj: | Involve 15 (2022) 687-708 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.2140/involve.2022.15.687 |
| Popis: | We prove the scale invariant Elliptic Harnack Inequality (EHI) for non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of 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. Comment: To appear in Involve-a journal of mathematics |
| Databáze: | arXiv |
| Externí odkaz: |
|