Operator Theory Seminar
The setting is analogous to that of weighted networks and are used in various areas such as Dirichlet forms, graphons, Borel equivalence relations, Gaussian processes, determinantal point processes, joinings, etc. We 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 spaces, 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. A main question is connection for 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 discuss concrete examples, especially models built on Bratteli diagrams. This is joint work with Sergii Bezuglyi.