Graphs of finite measure
Graphs of finite measure
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DOI：
10.1016/j.matpur.2014.10.006
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发表时间：
2013-09
期刊：
影响因子：
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通讯作者：
Agelos Georgakopoulos;Sebastian Haeseler;M. Keller;D. Lenz;Radoslaw K. Wojciechowski
中科院分区：
文献类型：
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作者：
Agelos Georgakopoulos;Sebastian Haeseler;M. Keller;D. Lenz;Radoslaw K. Wojciechowski
我们考虑具有无限顶点集的赋权图。我们证明了有限能量函数的有界性可以看作是这类图的“相对紧性”的概念，并用各种度量研究了这一性质的充要条件。然后，我们给满足这一性质的图配备了一个有限测度，并研究了相关的拉普拉斯算子及其半群。在此背景下，我们的结果包括半群的迹类性质，自然紧化产生的带边界的Dirichlet问题解的唯一性和存在性，Dirichlet拉普拉斯空间的显式刻画，热半群的大时间收敛，以及相应的随机游动在连续时间的随机不完备性和瞬时性。
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.