Graph Laplace and Markov operators on a measure space

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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 σ-finite measure space (V, B, μ) and a symmetric measure ρ on (V× V, B× B) supported by a measurable symmetric subset E⊂ V× 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 cxy defined on the set of edges E. As in the theory of weighted networks, we consider the Hilbert spaces L²(μ), L²(cμ) and define two other Hilbert spaces, the dissipation space Diss and finite energy space Hₑ. 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 L²(μ), and, the second, its counterpart in Hₑ. 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.