A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks
By Palle E. T. Jorgensen, Erin P. J. Pearse. 2009
48 citations recorded by the source corpus.
On a finite network (connected weighted undirected graph), the relationship between the natural Dirichlet form E and the discrete Laplace operator Δ is given by $$\varepsilon (u,v) = {\langle u,\Delta v\rangle _{{\ell ^2}}}$$, where the latter is the usual ℓ2 inner product. This formula is not generally true for infinite networks; earlier authors have given various conditions under which this formula remains valid. Instead, we extend this formula to arbitrary infinite networks (including the case when Δ is unbounded) by including a new (boundary) term, in parallel with the classical Gauss-Green identity. This tool allows for detailed study of the boundary of the network. 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 also yield a boundary integral representation for harmonic functions of finite energy. The boundary representation is developed further in [24].
Subfields: Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/785ca613d2be9622fa11c03e7d927243b04f0d09
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Citations in the corpus
- A Hilbert Space Approach to Effective Resistance Metric (2009, methodology): “resistance metrics globally, i.e., directly in terms of Hilbert space norms and independent of any limits taken with respect to finite sub networks. §1 contains a brief encapsulation of the results of [JP09b] which will be necessary for the current study. In particular, the orthogonal decomposition o”