Analysis of unbounded operators and random motion

TL;DR

This paper investigates the structure of infinite weighted graphs and reproducing kernel Hilbert spaces, focusing on limits at infinity, associated unbounded operators, and their applications to network analysis and statistical mechanics.

Contribution

It introduces a novel framework for analyzing limits at infinity in infinite graphs using reproducing kernel Hilbert spaces and associated unbounded operators, with new subspace characterizations.

Findings

Identifies two closed subspaces measuring limits at infinity in RKHS.

Generalizes finite-energy harmonic functions in the context of infinite graphs.

Establishes a connection between unbounded operators and boundary concepts at infinity.

Abstract

We study infinite weighted graphs with view to \textquotedblleft limits at infinity,\textquotedblright or boundaries at infinity. Examples of such weighted graphs arise in infinite (in practice, that means \textquotedblleft very\textquotedblright large) networks of resistors, or in statistical mechanics models for classical or quantum systems. But more generally our analysis includes reproducing kernel Hilbert spaces and associated operators on them. If $X$ is some infinite set of vertices or nodes, in applications the essential ingredient going into the definition is a reproducing kernel Hilbert space; it measures the differences of functions on $X$ evaluated on pairs of points in $X$. And the Hilbert norm-squared in $\mathcal{H}(X)$ will represent a suitable measure of energy. Associated unbounded operators will define a notion or dissipation, it can be a graph Laplacian, or a more abstract unbounded Hermitian operator defined from the reproducing kernel Hilbert space under study. We prove that there are two closed subspaces in reproducing kernel Hilbert space $\mathcal{H}(X)$ which measure quantitative notions of limits at infinity in $X$, one generalizes finite-energy harmonic functions in $\mathcal{H}(X)$, and the other a deficiency index of a natural operator in $\mathcal{H}(X)$ associated directly with the diffusion. We establish these results in the abstract, and we offer examples and applications. Our results are related to, but different from, potential theoretic notions of \textquotedblleft boundaries\textquotedblright in more standard random walk models. Comparisons are made.