A Compactness Theorem for functions on Poisson point clouds

TL;DR

This paper establishes a compactness theorem for functions on Poisson point clouds with bounded non-local p-Dirichlet energy, focusing on an intermediate-interaction regime, advancing understanding of discrete function behavior in stochastic geometric settings.

Contribution

It introduces a novel compactness theorem for discrete functions on Poisson point clouds in an intermediate-interaction regime, expanding the theoretical framework for non-local energies.

Findings

Proves compactness for sequences with bounded non-local p-Dirichlet energy.

Identifies the intermediate-interaction regime as critical for analysis.

Provides new tools for studying functions on random geometric structures.

Abstract

In this work we show a compactness Theorem for discrete functions on Poisson point clouds. We consider sequences with equibounded non-local $p$-Dirichlet energy: the novelty consists in the intermediate-interaction regime at which the non-local energy is computed.