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Resistance boundaries of infinite networks. (English) Zbl 1223.05175
Lenz, Daniel (ed.) et al., Random walks, boundaries and spectra. Proceedings of the workshop on boundaries, Graz, Austria, June 29–July 3, 2009 and the Alp-workshop, Sankt Kathrein, Austria, July 4–5, 2009. Basel: Birkhäuser (ISBN 978-3-0346-0243-3/hbk; 978-3-0346-0244-0/ebook). Progress in Probability 64, 111-142 (2011).
Summary: 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 e and the discrete Laplace operator \(\Delta\) on a finite network is given by \(\mathcal{E}(u,v) = \langle u,\Delta v\rangle_2\), where the latter is the usual \(\ell^{2}\) inner product. We describe a reproducing kernel \(\{v_{x}\}\) for \(\mathcal{E}\) which allows one to extend the discrete Gauss-Green identity to infinite networks: \[ \mathcal{E}(u,v) = \sum_G u\Delta v + \sum_{\text{bd}G} u\frac{\partial v}{\partial n}, \] 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 \(\text{bd}G\) 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 \(S \subseteq {\mathcal{H}}_{\mathcal{E}} \subseteq S'\) 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 \(S\).
For the entire collection see [Zbl 1214.05001].

MSC:
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C75 Structural characterization of families of graphs
31C20 Discrete potential theory
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
47B25 Linear symmetric and selfadjoint operators (unbounded)
47B32 Linear operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces)
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
31C35 Martin boundary theory
47B39 Linear difference operators
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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