Resistance Boundaries of Infinite Networks

Abstract

A resistance network is a connected graph (G, c). The conductance function \(c_{xy}\) weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form ε produces a Hilbert space structure h ε on the space of functions of finite energy.The relationship between the natural Dirichlet form \( \rm{\varepsilon}\)and the discrete Laplace operator \( \rm{\Delta}\) on a finite network is given by \( {{\varepsilon(u,\,v)}}\, = \, {\langle{u},\,\Delta {v}\rangle}2, \) where the latter is the usual l 2 inner product. We describe a reproducing kernel v x for ε which allows one to extend the discrete Gauss-Green identity to infinite networks: \( {{\varepsilon(u,\,v)}}\, = \, {\sum}_{G}\, {u\Delta v}+{\sum}_{bd\,\,G} \,\,{u}\,\frac{\partial {v}} {\partial {n}},\,\, \) where the latter sum is understood in a limiting sense, analogous to a Riemann sum. This formula yields a boundary sum representation for the harmonic functions of finite energy.Techniques from stochastic integration allow one to make the boundary bdG precise as a measure space, and give a boundary integral representation (in a sense analogous to that of Poisson or Martin boundary theory). This is done in terms of a Gel’fand triple \( {S}\,\, \subseteq \, \, {H_\varepsilon}\,\, \subseteq\,\,{{S}^\prime} {\rm{and}\, {gives}\, {a}\, {probability}\, {measure}\,\mathbb{P}}\, {\rm{and}\, {an}\, {isometric}\, {embedding}\,{of}\,{H_\varepsilon}\,\,{into}}\,\,{{S}^\prime},\,\mathbb{P},\) and yields a concrete representation of the boundary as a set of linear functionals on S.KeywordsDirichlet formgraph energydiscrete potential theorygraph Laplacianweighted graphtreeelectrical resistance networkeffective resistanceresistance formMarkov processrandom walktransienceMartin boundaryboundary theoryboundary representationharmonic analysisHilbert spaceorthogonalityunbounded linear operatorsreproducing kernel