Journal article
A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks
Israel Journal of Mathematics, Vol.196(1), pp.113-160
08/2013
DOI: 10.1007/s11856-012-0165-2
Abstract
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 ℓ 2 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].
Details
- Title: Subtitle
- A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks
- Creators
- Palle Jorgensen - Department of Mathematics University of Iowa Iowa City IA 52246-1419 USAErin Pearse - Department of Mathematics University of Iowa Iowa City IA 52246-1419 USA
- Resource Type
- Journal article
- Publication Details
- Israel Journal of Mathematics, Vol.196(1), pp.113-160
- DOI
- 10.1007/s11856-012-0165-2
- ISSN
- 0021-2172
- eISSN
- 1565-8511
- Publisher
- Springer US; Boston
- Language
- English
- Date published
- 08/2013
- Academic Unit
- Mathematics
- Record Identifier
- 9983985987802771
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