Radial isoperimetry and absence of harmonic functions with $\ell^p$-gradient

Antoine Gournay

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arXiv · 2509.22731

Radial isoperimetry and absence of harmonic functions with $\ell^p$-gradient

Antoine Gournay

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Abstract

In this paper we show that groups for which the probability of return of a random walk is bounded below by $K_1 exp(-K_2n^c)$ have no non-constant harmonic functions with gradient in $\ell^p$. The proof relies on results from $\ell^p$-cohomology, a form of radial isoperimetry, transport patterns and revisiting some results of F{\o}lner.

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Antoine Gournay. 2025-09-25. Radial isoperimetry and absence of harmonic functions with $\ell^p$-gradient. https://arxiv.org/abs/2509.22731

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