Gel'fand triples and boundaries of infinite networks

We study the boundary theory of a connected weighted graph $G$ from the viewpoint of stochastic integration. For the Hilbert space \HE of Dirichlet-finite functions on $G$, we construct a Gel'fand triple $S \ci {\mathcal H}_{\mathcal E} \ci S'$. This yields a probability measure $\mathbb{P}$ on $S'$ and an isometric embedding of ${\mathcal H}_{\mathcal E}$ into $L^2(S',\mathbb{P})$, and hence gives a concrete representation of the boundary as a certain class of "distributions" in $S'$. In a previous paper, we proved a discrete Gauss-Green identity for infinite networks which produces a boundary representation for harmonic functions of finite energy, given as a certain limit. In this paper, we use techniques from stochastic integration to make the boundary $\operatorname{bd}G$ precise as a measure space, and obtain a boundary integral representation as an integral over $S'$.