Gel'fand triples and boundaries of infinite networks

Abstract

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 $S^{\prime }$ and an isometric embedding of ${\mathcal{H}}_{\mathcal{E}}$ into $L^2(S^{\prime },\mathbb{P})$, and hence gives a concrete representation of the boundary as a certain class of "distributions" in $S^{\prime }$. 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 $\mathrm{bd}G$ precise as a measure space, and obtain a boundary integral representation as an integral over $S^{\prime }$.

arXiv comment35 pages, 1 figure. arXiv admin note: text overlap with arXiv:0806.3881, arXiv:0909.1518

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