Laplace operators in finite energy and dissipation spaces

Recent applications of large network models to machine learning, and to neural network suggest a need for a systematic study of the general correspondence, (i) discrete vs (ii) continuous. Even if the starting point is (i), limit considerations lead to (ii), or, more precisely, to a measure theoretic framework which we make precise. Our motivation derives from graph analysis, e.g., studies of (infinite) electrical networks of resistors, but our focus will be (ii), i.e., the measure theoretic setting. In electrical networks of resistors, one considers pairs (of typically countably infinite), sets $V$ (vertices), $E$ (edges) a suitable subset of $V \times V$, and prescribed positive symmetric functions $c$ on $E$ . A conductance function $c$ is defined on $E$ (edges), or on $V \times V$, but with $E$ as its support. From an initial triple $(V, E, c)$ , one gets graph-Laplacians, generalized Dirichlet spaces (also called energy Hilbert spaces), dipoles, relative reproducing kernel-theory, dissipation spaces, reversible Markov chains, and more. Our main results include: spectral theory and Green's functions for measure theoretic graph-Laplace operators; 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; orbit equivalence criteria; and path-space measures and induced dissipation Hilbert spaces. We consider several applications of our results to other fields such as machine learning problems, reproducing kernel Hilbert spaces, Gaussian and determinantal processes, and joinings.