Harnack inequality for $p$-harmonic functions: improved dimension dependence via tug of war
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866909029191647232 |
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| author | Peres, Yuval Wang, Han |
| author_facet | Peres, Yuval Wang, Han |
| contents | Let $p>1$. The Harnack inequality and Hölder continuity for $p$-harmonic functions in bounded domains in $\mathbb{R}^d$ are usually proved via Moser iteration. In 2013 Luiro, Parviainen and Saksman showed that tug-of-war games can also be used to derive these inequalities. We refine their analysis and obtain improved dependence on $p$ and the dimension $d$ by probabilistic methods. In particular, we show that for all $p>1$, the constant in Harnack's inequality is $O(\exp(C_p d\log d))$ as $d\rightarrow\infty$, which improves the constant derived from Moser iteration. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10142 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Harnack inequality for $p$-harmonic functions: improved dimension dependence via tug of war Peres, Yuval Wang, Han Probability Analysis of PDEs 91A15, 35J92, 35B65 Let $p>1$. The Harnack inequality and Hölder continuity for $p$-harmonic functions in bounded domains in $\mathbb{R}^d$ are usually proved via Moser iteration. In 2013 Luiro, Parviainen and Saksman showed that tug-of-war games can also be used to derive these inequalities. We refine their analysis and obtain improved dependence on $p$ and the dimension $d$ by probabilistic methods. In particular, we show that for all $p>1$, the constant in Harnack's inequality is $O(\exp(C_p d\log d))$ as $d\rightarrow\infty$, which improves the constant derived from Moser iteration. |
| title | Harnack inequality for $p$-harmonic functions: improved dimension dependence via tug of war |
| topic | Probability Analysis of PDEs 91A15, 35J92, 35B65 |
| url | https://arxiv.org/abs/2604.10142 |