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Resistance boundaries of infinite networks | Palle E. T. Jorgensen
; Erin P. J. Pearse
; | Date: |
8 Sep 2009 | 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 $mathcal E$ 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 | Services: | Forum | Review | PDF | Favorites |
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