Journal article
Gel'fand triples and boundaries of infinite networks
New York journal of mathematics, Vol.17, pp.745-781
01/01/2011
Abstract
We study the boundary theory of a connected weighted graph G from the viewpoint of stochastic integration. For the Hilbert space H-epsilon of Dirichlet-finite functions on G, we construct a Gel'fand triple S subset of H-epsilon subset of S'. This yields a probability measure P on S' and an isometric embedding of H-epsilon 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 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'.
Details
- Title: Subtitle
- Gel'fand triples and boundaries of infinite networks
- Creators
- Palle E. T Jorgensen - Univ Iowa, Iowa City, IA 52246 USAErin P. J Pearse - Univ Oklahoma, Norman, OK 73019 USA
- Resource Type
- Journal article
- Publication Details
- New York journal of mathematics, Vol.17, pp.745-781
- Publisher
- ELECTRONIC JOURNALS PROJECT
- ISSN
- 1076-9803
- eISSN
- 1076-9803
- Number of pages
- 37
- Grant note
- DMS-0602242 / University of Iowa Department of Mathematics NSF VIGRE grant DMS-0457581 / NSF; National Science Foundation (NSF)
- Language
- English
- Date published
- 01/01/2011
- Academic Unit
- Mathematics
- Record Identifier
- 9984240765902771
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