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• W2963567655 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. On considère des graphes avec poids ayant une infinité de sommets. On montre que la majoration des fonctions d'énergie finie peut être vue comme une « compacité relative » pour de tels graphes et on étudie des conditions nécessaires et suffisantes pour que cette propriété ait lieu et ceci pour différrentes métriques. On munit ces graphes satisfaisant cette propriété d'une mesure finie, on étudie le laplacien associé et son semi-groupe. Dans ce contexte, les résultats obtenus incluent la propriété de la trace, l'unicité et l'existence de solutions du problème de Dirichlet avec bord issu de la compactification naturelle, une description explicite du domaine du laplacien avec condition de Dirichlet, la convergence du semi-groupe de la chaleur pour des temps assez grands ainsi que l'incomplétude stochastique et le caractère transitoire de la marche aléatoire correspondante en temps continu." @default.
• W2963567655 created "2019-07-30" @default.
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• W2963567655 date "2015-05-01" @default.
• W2963567655 modified "2023-10-12" @default.
• W2963567655 title "Graphs of finite measure" @default.
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• W2963567655 doi "https://doi.org/10.1016/j.matpur.2014.10.006" @default.
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