Characterizations of $p$-Parabolicity on Graphs

We study $p$-energy functionals on infinite locally summable graphs for $p\in (1,\mathrm{\infty })$ 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’minskiĭ-type and a Poincaré-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’minskiĭ-type characterization.