Graphs of finite measure
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Publication:2345413
DOI10.1016/J.MATPUR.2014.10.006zbMATH Open1314.31017arXiv1309.3501OpenAlexW2963567655MaRDI QIDQ2345413
Author name not available (Why is that?)
Publication date: 22 May 2015
Published in: (Search for Journal in Brave)
Abstract: We consider weighted graphs with an infinite set of vertices. We show that boundedness of all functions of finite energy can be seen as a notion of `relative compactness' for such graphs and study sufficient and necessary conditions for this property in terms of various metrics. We then equip graphs satisfying this property with a finite measure and investigate the associated Laplacian and its semigroup. In this context, our results include the trace class property for the semigroup, uniqueness and existence of solutions to the Dirichlet problem with boundary arising from the natural compactification, an explicit description of the domain of the Dirichlet Laplacian, convergence of the heat semigroup for large times as well as stochastic incompleteness and transience of the corresponding random walk in continuous time.
Full work available at URL: https://arxiv.org/abs/1309.3501
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