 

Palle E. T. Jorgensen and Erin P. J. Pearse
Gel'fand triples and boundaries of infinite networks view print


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 


Abstract
We study the boundary theory of a connected weighted graph G from the viewpoint of stochastic integration. For the Hilbert space
H_{E} of Dirichletfinite functions on G, we construct a Gel'fand triple
S ⊆ H_{E} ⊆ S'. This yields a probability measure P on S' and an isometric embedding of H_{E} into L^{2}(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 GaussGreen 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'.


Acknowledgements
The work of PETJ was partially supported by NSF grant DMS0457581. The work of EPJP was partially supported by the University of Iowa Department of Mathematics NSF VIGRE grant DMS0602242.


Author information
Palle E. T. Jorgensen:
University of Iowa, Iowa City, IA 522461419 USA
pallejorgensen@uiowa.edu
Erin P. J. Pearse:
University of Oklahoma, Norman OK 730190315 USA
ep@ou.edu

