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Characterizations of $p$-Parabolicity on Graphs

Andrea Adriani, Florian Fischer, Alberto G. Setti

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'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.

Characterizations of $p$-Parabolicity on Graphs

Abstract

We study -energy functionals on infinite locally summable graphs for 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 -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.

Paper Structure

This paper contains 22 sections, 36 theorems, 171 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Setting the Scene
  3. Graphs and Laplace Operators
  4. Energy Functionals Associated with Graphs
  5. Characterizations of $p$-Parabolicity
  6. Green's Functions
  7. The Functional-Theoretic Perspective
  8. The Superharmonic Functions of Finite Energy Perspective
  9. Kelvin-Nevanlinna-Royden-type characterization
  10. Poincaré-type characterization
  11. Ahlfors-type characterization
  12. A Lenz-Puchert-Schmidt-type characterization
  13. Boundaries, limits, and capacities
  14. Small perturbations
  15. Intrinsic and path pseudometrics
  16. ...and 7 more sections

Key Result

Lemma 2.1

Let $p\in [1,\infty)$, then $\mathcal{E}_p$ satisfies the second Beurling-Deny criterion, that is, for all $f,g \in D^p$ and all normal contractions $C\colon \mathbb{R} \to \mathbb{R}$, we have In particular, $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Example of $k_-(x)$, $k_+(x)$, $\partial B(1)$ w.r.t. $x_0$
  • Figure 2: Graphs of Examples \ref{['ex:star']} - \ref{['ex:starline']}
  • Figure 3: Graphs of Examples \ref{['ex:T']} - \ref{['ex:AT']}

Theorems & Definitions (85)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Remark 2.3
  • Example 2.4
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • proof
  • ...and 75 more