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arXiv · 2507.13696

Characterizations of $p$-Parabolicity on Graphs

Andrea Adriani·Florian Fischer·Alberto G. Setti
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Abstract

We study $p$-energy functionals on infinite locally summable graphs for $p\in (1,\infty)$ and show that many well-known characterizations for a parabolic space are also true in this discrete, non-local and non-linear setting. Among the characterizations are an Ahlfors-type, a Kelvin-Nevanlinna-Royden-type, a Khas'minski\u{\i}-type and a Poincar\'{e}-type characterization. We also illustrate some applications and describe examples of graphs which are locally summable but not locally finite. Finally, we study the obstacle problem for the $p$-Laplacian using an approximation procedure by finite graphs in the summable, not necessarily locally finite, case. This is then utilized to give an alternative proof of the Khas'minski\u{\i}-type characterization.

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Andrea Adriani, Florian Fischer, Alberto G. Setti. 2025-07-18. Characterizations of $p$-Parabolicity on Graphs. https://arxiv.org/abs/2507.13696

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