After the Explosion: Dirichlet Forms and Boundary Problems for Infinite Graphs

TL;DR

This paper studies Laplace operators on resistance networks with vertex weights, addressing boundary problems and constructing self-adjoint operators via Dirichlet forms, with applications to infinite graphs.

Contribution

It introduces a new connectivity hypothesis for completed graphs and solves the Dirichlet problem for resistance networks with boundary conditions.

Findings

Constructed self-adjoint Laplace operators for infinite graphs.

Solved the Dirichlet problem in the context of resistance networks.

Established conditions for the existence of probability semigroups.

Abstract

Formal Laplace operators are analyzed for a large class of resistance networks with vertex weights. The graphs are completed with respect to the minimal resistance path metric. Compactness and a novel connectivity hypothesis for the completed graphs play an essential role. A version of the Dirichlet problem is solved. Self adjoint Laplace operators and the probability semigroups they generate are constructed using reflecting and absorbing conditions on subsets of the graph boundary.