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.

Mathematics Subject Classification (2000). Primary: 05C50, 05C75, 31C20, 46E22, 47B25, 47B32, 60J10, Secondary: 31C35, 47B39, 82C41.