Stochastic integration and boundaries of infinite networks

Details Stochastic integration and boundaries of infinite networks Stochastic integration and boundaries of infinite networks

2009-06-15

arxiv.org/abs/0906.2745v3

Mathematics

Functional Analysis

29 pages, no figures

Scientific paper

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 Gel'fand triples $S \ci {\mathcal H}_{\mathcal E} \ci S'$. This yields a probability measure $\mathbb{P}$ 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 a harmonic function $u$: $u(x) = \sum_{\operatorname{bd}G} u \frac{\partial v_x}{\partial \mathbf{n}}$, where the sum is understood in a limiting sense. In this paper, we use techniques from stochastic integration to make the boundary $\operatorname{bd}G$ precise as a measure space, and replace the sum with an integral over $S'$, thus obtaining a boundary integral representation for the harmonic function $u$.

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