Resistance boundaries of infinite networks

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 $\mathcal E(u,v) = \la u, \Lap v\ra_2$, 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: \[{\mathcal E}(u,v) = \sum_{G} u \Delta v + \sum_{\operatorname{bd}G} u \tfrac{\partial}{\partial \mathbf{n}} v,\] 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 $S$.