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].