The main objective of this paper is to develop the concepts and ideas used in the theory of weighted networks on infinite graphs. Our basic ingredients are a standard Borel space (V,B)\documentclass[12pt]{minimal}
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\begin{document}$$(V, {{\mathcal {B}}})$$\end{document} and symmetric σ\documentclass[12pt]{minimal}
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\begin{document}$$\sigma $$\end{document}-finite measure ρ\documentclass[12pt]{minimal}
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\begin{document}$$\rho $$\end{document} on (V×V,B×B)\documentclass[12pt]{minimal}
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\begin{document}$$(V\times V, {{\mathcal {B}}}\times {{\mathcal {B}}})$$\end{document} supported by a symmetric Borel subset E⊂V×V\documentclass[12pt]{minimal}
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\begin{document}$$E \subset V\times V$$\end{document}. This setting is analogous to that of weighted networks and are used in various areas of mathematics such as Dirichlet forms, graphons, Borel equivalence relations, Guassian processes, determinantal point processes, joinings, etc. We define and study the properties of a measurable analogue of the graph Laplacian, finite energy Hilbert space, dissipation Hilbert space, and reversible Markov processes associated directly with the measurable framework. In the settings of networks on infinite graphs and standard Borel space, our present focus will be on the following two dichotomies: (i) Discrete vs continuous (or rather the measurable category); and (ii) deterministic vs stochastic. Both (i) and (ii) will be made precise inside our paper; in fact, this part constitutes a separate issue of independent interest. A main question we shall address in connection with both (i) and (ii) is that of limits. For example, what are the useful notions, and results, which yield measurable limits; more precisely, Borel structures arising as “limits” of systems of graph networks? We shall study it both in the abstract, and via concrete examples, especially models built on Bratteli diagrams. The question of limits is of basic interest; both as part of the study of dynamical systems on path- space; and also with regards to numerical considerations; and design of simulations. The particular contexts for the study of this question entails a careful design of appropriate Hilbert spaces, as well as operators acting between them. Our main results include: an explicit spectral theory for measure theoretic graph-Laplace operators; new properties of Green’s functions and corresponding reproducing kernel Hilbert spaces; structure theorems of the set of harmonic functions in L2\documentclass[12pt]{minimal}
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\begin{document}$$L^2$$\end{document}-spaces and finite energy Hilbert spaces; the theory of reproducing kernel Hilbert spaces related to Laplace operators; a rigorous analysis of the Laplacian on Borel equivalence relations; a new decomposition theory; irreducibility criteria; dynamical systems governed by endomorphisms and measurable fields; and path-space measures and induced dissipation Hilbert spaces. We discuss also several applications of our results to other fields such as machine learning problems, ergodic theory, and reproducing kernel Hilbert spaces.
