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

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 $\mathcal E(u,v) = \la u, \Lap v\ra_2$, 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.

31 pages

References in corpus (6)

• Stochastic Completeness of Graphs
• Laplacians on the basilica Julia set
• Operator theory of electrical resistance networks
• Electric currents in infinite networks
• A Hilbert space approach to effective resistance metric
• Heat Kernel and Essential Spectrum of Infinite Graphs

Cited by in corpus (4)

• Symmetric pairs of unbounded operators in Hilbert space, and their applications in mathematical physics
• A Hilbert space approach to effective resistance metric
• Self-adjoint extensions of network Laplacians and applications to resistance metrics
• Unbounded containment in the energy space of a network and the Krein extension of the energy Laplacian