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29 March 2024
 
  » arxiv » 0909.1518

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Resistance boundaries of infinite networks
Palle E. T. Jorgensen ; Erin P. J. Pearse ;
Date 8 Sep 2009
AbstractA 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 $mathcal E$ produces a Hilbert space structure ${mathcal H}_{mathcal E}$ on the space of functions of finite energy.
The relationship between the natural Dirichlet form $mathcal E$ and the discrete Laplace operator $Delta$ on a finite network is given by $mathcal E(u,v) = la u, Lap v a_2$, where the latter is the usual $ell^2$ inner product. We describe a reproducing kernel ${v_x}$ for $mathcal E$ and used it to extends the discrete Gauss-Green identity to infinite networks: [{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, analogous to a Riemann sum. This formula immediately yields a boundary sum representation for the harmonic functions of finite energy.
Techniques from stochastic integration allow one to make the boundary $operatorname{bd}G$ 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 $S ci {mathcal H}_{mathcal E} ci S’$ and gives a probability measure $mathbb{P}$ and an isometric embedding of ${mathcal H}_{mathcal E}$ into $L^2(S’,mathbb{P})$, and yields a concrete representation of the boundary as a set of linear functionals on $S$.
Source arXiv, 0909.1518
Other source [GID 1251224] 0906.2745
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