Laplacians on infinite graphs: Dirichlet and Neumann boundary conditions

TL;DR

This paper investigates Laplacians on graphs, characterizing Dirichlet and Neumann types, their properties, differences, and conditions for uniqueness, self-adjointness, and stochastic completeness, with a focus on both finite and infinite graphs.

Contribution

It provides a comprehensive characterization of Dirichlet and Neumann Laplacians on graphs, including their differences, conditions for equality, and properties related to self-adjointness and stochastic completeness.

Findings

Characterization of Dirichlet and Neumann Laplacians as Markovian restrictions.

Conditions under which Dirichlet and Neumann Laplacians agree.

Basic estimates on the long-term behavior of the associated semigroup.

Abstract

We study Laplacians associated to a graph and single out a class of such operators with special regularity properties. In the case of locally finite graphs, this class consists of all selfadjoint, non-negative restrictions of the standard formal Laplacian and we can characterize the Dirichlet and Neumann Laplacians as the largest and smallest Markovian restrictions of the standard formal Laplacian. In the case of general graphs, this class contains the Dirichlet and Neumann Laplacians and we describe how these may differ from each other, characterize when they agree, and study connections to essential selfadjointness and stochastic completeness. Finally, we study basic common features of all Laplacians associated to a graph. In particular, we characterize when the associated semigroup is positivity improving and present some basic estimates on its long term behavior. We also discuss some situations in which the Laplacian associated to a graph is unique and, in this context, characterize its boundedness.