On generalized divergence and Laplace operators as a matter of division of distributions

TL;DR

This paper introduces a generalized framework for divergence and Laplace operators based on coupling measures and graph structures, unifying classical and novel instances within a broad mathematical setting.

Contribution

It develops a unified approach to divergence and Laplace operators using coupling measures, extending classical concepts to new generalized forms.

Findings

Provides a new perspective on Laplace operators via coupling measures.

Introduces novel instances of divergence and Laplace operators.

Unifies classical and new approaches within a common framework.

Abstract

Starting from the approach to the Laplacian with respect to coupling measures and undirected weighted graphs, we provide a setting for a general point of view for a Kirchhoff type divergence and a Laplace operators built on the trivial gradient of order zero $f(y)-f(x)$. We consider some particular classical and new instances of this approach.