naa a22 4500 510783139 CHVBK 20180411083254.0 cr unu---uuuuu 180411e20130801xx s 000 0 eng 10.1007/s11856-012-0165-2 doi (NATIONALLICENCE)springer-10.1007/s11856-012-0165-2 A discrete Gauss-Green identity for unbounded Laplace operators, and the transience of random walks [Elektronische Daten] [Palle Jorgensen, Erin Pearse] 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]. Hebrew University Magnes Press, 2013 Jorgensen Palle Department of Mathematics, University of Iowa, 52246-1419, Iowa City, IA, USA aut Pearse Erin Department of Mathematics, University of Iowa, 52246-1419, Iowa City, IA, USA aut Israel Journal of Mathematics Springer US; http://www.springer-ny.com 196/1(2013-08-01), 113-160 0021-2172 196:1<113 2013 196 11856 https://doi.org/10.1007/s11856-012-0165-2 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s11856-012-0165-2 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Jorgensen Palle Department of Mathematics, University of Iowa, 52246-1419, Iowa City, IA, USA aut NATIONALLICENCE 700 1- Pearse Erin Department of Mathematics, University of Iowa, 52246-1419, Iowa City, IA, USA aut NATIONALLICENCE 773 0- Israel Journal of Mathematics Springer US; http://www.springer-ny.com 196/1(2013-08-01), 113-160 0021-2172 196:1<113 2013 196 11856 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer