Graphs of finite measure
By Agelos Georgakopoulos, Sebastian Haeseler, Matthias Keller, Daniel Lenz, Radoslaw K. Wojciechowski. 2013
47 citations recorded by the source corpus.
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.
Subfields: Computational Theory and Mathematics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/6b63c5c29cc66fa200e0aaf1b36616ebed3e2d7d
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In Computational Theory and Mathematics
In Mathematical Physics
In Statistics and Probability
Citations in the corpus
- The Kazdan–Warner equation on canonically compactifiable graphs (2017, methodology): “These graphs, which can be seen as discrete analogues to open pre-compact manifolds, were studied systematically in [14], see also [24].”