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19 April 2024
 
  » arxiv » 0906.2745

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Resistance boundaries of infinite networks
Palle E. T. Jorgensen ; Erin P. J. Pearse ;
Date 15 Jun 2009
AbstractA resistance network is a connected graph $(G,c)$ with edges (and edge weights) determined by the conductance function $c_{xy}$. The Dirichlet energy form $mathcal E$ produces a Hilbert space structure (which we call the energy space ${mathcal H}_{mathcal E}$) on the space of functions of finite energy. In a previous paper, we constructed a reproducing kernel ${v_x}$ for this Hilbert space and used it to prove a discrete Gauss-Green identity [{mathcal E}(u,v) = sum_{G} u Delta v + sum_{operatorname{bd}G} u frac{partial}{partial mathbf{n}} v,] where the latter sum is understood in a limiting sense. Applying this formula to a harmonic function $u in {mathcal H}_{mathcal E}$ gives a boundary representation [u(x) = sum_{operatorname{bd}G} u frac{partial}{partial mathbf{n}} v + u(o),] where $o$ is a fixed reference vertex.
In this paper, we use techniques from stochastic integration to make the boundary $operatorname{bd}G$ precise as a measure space, and replace the latter formula with a boundary integral representation (in a sense analogous to that of Poisson or Martin boundary theory). We construct a Gel’fand triple $S ci {mathcal H}_{mathcal E} ci S’$ and obtain a probability measure $mathbb{P}$ and an isometric embedding of ${mathcal H}_{mathcal E}$ into $L^2(S’,mathbb{P})$. This gives a concrete representation of the boundary as a certain subset of $S’$.
Source arXiv, 0906.2745
Other source [GID 1251224] 0909.1518
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