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Resistance Boundaries of Infinite Networks
Book chapter

Resistance Boundaries of Infinite Networks

Palle Jorgensen and Erin Pearse
Random Walks, Boundaries and Spectra, pp.111-142
Progress in Probability, Springer Basel
2011
DOI: 10.1007/978-3-0346-0244-0_7

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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.
 Probability Theory and Stochastic Processes Mathematics

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