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 ⊆ HE ⊆ S′. This yields a probability measure P on S′ and an isometric embedding of HE into L2(S′,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 bdG precise as a measure space, and obtain a boundary integral representation as an integral over S′.