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arXiv · 0906.1586

A discrete Gauss-Green identity for unbounded Laplace operators and transience of random walks

Palle E. T. Jorgensen·Erin P. J. Pearse
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Abstract

A resistance network is a connected graph $(G,c)$. The conductance function $c_{xy}$ weights the edges, which are then interpreted as resistors of possibly varying strengths. The relationship between the natural Dirichlet form $\mathcal E$ and the discrete Laplace operator $Δ$ 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 extend this formula to infinite networks, where a new (boundary) term appears. The Laplace operator is typically unbounded in this context; we construct a reproducing kernel for the space of functions of finite energy which allows us to specify a dense domain for $Δ$ and give several criteria for the transience of the random walk on the network. The extended Gauss-Green identity and the reproducing kernel are the foundation for a boundary integral representation for harmonic functions of finite energy, akin to that of Martin boundary theory.

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Palle E. T. Jorgensen, Erin P. J. Pearse. 2010-02-18. A discrete Gauss-Green identity for unbounded Laplace operators and transience of random walks. https://arxiv.org/abs/0906.1586

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