Characterizations of $p$-Parabolicity on Graphs

TL;DR

This paper extends classical characterizations of $p$-parabolicity to infinite, locally summable graphs, exploring their properties, applications, and the obstacle problem for the $p$-Laplacian.

Contribution

It generalizes well-known parabolicity characterizations to non-local, non-linear graph settings and introduces new methods for the obstacle problem.

Findings

Many classical characterizations hold in the graph setting.

Examples of graphs that are locally summable but not locally finite.

An alternative proof of the Khas'minskii-type characterization.

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.