2018/01/13 by Bezuglyi, Sergey, Jorgensen, Palle E. T.
#Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.1801.04459
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, \mathcal B, μ) and a symmetric measure ρ on (V× V, \mathcal B× \mathcal 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 \mathcal 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 \mathcal 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.