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Resistance boundaries of infinite networks | Palle E. T. Jorgensen
; Erin P. J. Pearse
; | Date: |
15 Jun 2009 | Abstract: | A 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 | Services: | Forum | Review | PDF | Favorites |
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