Mathematics – Functional Analysis
Scientific paper
2009-06-08
Mathematics
Functional Analysis
31 pages
Scientific paper
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 $\mathcal E(u,v) = \la u, \Lap v\ra_2$, 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 $\Delta$ 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.
Jorgensen Palle E. T.
Pearse Erin P. J.
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