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- W2951409664 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 $Delta$ on a finite network is given by $mathcal E(u,v) = la u, Lap vra_2$, 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 $Delta$ 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." @default.
- W2951409664 created "2019-06-27" @default.
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- W2951409664 date "2009-06-08" @default.
- W2951409664 modified "2023-09-27" @default.
- W2951409664 title "A discrete Gauss-Green identity for unbounded Laplace operators and transience of random walks" @default.
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