New York Journal of Mathematics
Volume 17 (2011) 745-781


Palle E. T. Jorgensen and Erin P. J. Pearse

Gel'fand triples and boundaries of infinite networks

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Published: November 8, 2011
Keywords: Dirichlet form, graph energy, discrete potential theory, graph Laplacian, weighted graph, trees, spectral graph theory, electrical resistance network, effective resistance, resistance forms, Markov process, random walk, transience, Martin boundary, boundary theory, boundary representation, harmonic analysis, Hilbert space, orthogonality, unbounded linear operators, reproducing kernels.
Subject: Primary: 05C50, 05C75, 31C20, 46E22, 47B25, 47B32, 60J10; secondary: 31C35, 47B39, 82C41

We study the boundary theory of a connected weighted graph G from the viewpoint of stochastic integration. For the Hilbert space HE of Dirichlet-finite functions on G, we construct a Gel'fand triple S ⊆ HE ⊆ S'. This yields a probability measure P on S' and an isometric embedding of HE into L2(S',P), and hence gives a concrete representation of the boundary as a certain class of "distributions'' in S'. In a previous paper, we proved a discrete Gauss-Green identity for infinite networks which produces a boundary representation for harmonic functions of finite energy, given as a certain limit. In this paper, we use techniques from stochastic integration to make the boundary bd G precise as a measure space, and obtain a boundary integral representation as an integral over S'.


The work of PETJ was partially supported by NSF grant DMS-0457581. The work of EPJP was partially supported by the University of Iowa Department of Mathematics NSF VIGRE grant DMS-0602242.

Author information

Palle E. T. Jorgensen:
University of Iowa, Iowa City, IA 52246-1419 USA

Erin P. J. Pearse:
University of Oklahoma, Norman OK 73019-0315 USA