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.Comment: 72 page