Graph Laplace and Markov operators on a measure space

TL;DR

This paper develops a measurable analogue of weighted network theory for infinite measure spaces, introducing Laplace operators and semigroups with applications across various mathematical fields.

Contribution

It extends the theory of weighted networks to measure spaces, defining new Hilbert spaces and spectral theorems for Laplace operators in this setting.

Findings

Explicit spectral and potential theoretic theorems

Two realizations of Laplace operators and semigroups

Energy space is crucial for studying transient Markov processes

Abstract

The main goal of this paper is to build a measurable analogue to the theory of weighted networks on infinite graphs. Our basic setting is an infinite $\sigma$-finite measure space $(V, \mathcal B, \mu)$ and a symmetric measure $\rho$ on $(V\times V, \mathcal B\times \mathcal B)$ supported by a measurable symmetric subset $E\subset V\times V$. This applies to such diverse areas as optimization, graphons (limits of finite graphs), symbolic dynamics, measurable equivalence relations, to determinantal processes, to jump-processes; and it extends earlier studies of infinite graphs $G = (V, E)$ which are endowed with a symmetric weight function $c_{xy}$ defined on the set of edges $E$. As in the theory of weighted networks, we consider the Hilbert spaces $L^2(\mu), L^2(c\mu)$ and define two other Hilbert spaces, the dissipation space $Diss$ and finite energy space $\mathcal H_E$. Our main results include a number of explicit spectral theoretic and potential theoretic theorems that apply to two realizations of Laplace operators, and the associated jump-diffusion semigroups, one in $L^2(\mu)$, and, the second, its counterpart in $\mathcal H_E$. We show in particular that it is the second setting (the energy-Hilbert space and the dissipation Hilbert space) which is needed in a detailed study of transient Markov processes.