Resistance Boundaries of Infinite Networks [chapter]

Palle E. T. Jorgensen, Erin P. J. Pearse
2011 Random Walks, Boundaries and Spectra  
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 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
more » ... te 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 .
doi:10.1007/978-3-0346-0244-0_7 fatcat:fmv7g6nh3vebdghgctgeswip6a