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Stochastically Incomplete Manifolds and Graphs

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Book cover Random Walks, Boundaries and Spectra

Part of the book series: Progress in Probability ((PRPR,volume 64))

Abstract

We survey geometric properties which imply the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to the case of Riemannian manifolds, there exist examples of stochastically incomplete graphs of polynomial volume growth.

Mathematics Subject Classification (2000). Primary 39A12; Secondary 58J65.

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Authors and Affiliations

  1. Grupo de Física-Matemática, Complexo Interdisciplinar da Universidade de Lisboa, Av. Prof. Gama Pinto, 2, PT–1649–003, Lisboa, Portugal

    Radosław Krzysztof Wojciechowski

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  1. Radosław Krzysztof Wojciechowski
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Correspondence to Radosław Krzysztof Wojciechowski .

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Editors and Affiliations

  1. Mathematisches Inst., Universität Jena, Ernst-Abbe-Platz 2, Jena, 07737, Germany

    Daniel Lenz

  2. Inst. Mathematik C, TU Graz, Steyrergasse 30, Graz, 8010, Austria

    Florian Sobieczky

  3. Inst. Mathematik C, TU Graz, Steyrergasse 30, Graz, 8010, Austria

    Wolfgang Woess

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Wojciechowski, R.K. (2011). Stochastically Incomplete Manifolds and Graphs. In: Lenz, D., Sobieczky, F., Woess, W. (eds) Random Walks, Boundaries and Spectra. Progress in Probability, vol 64. Springer, Basel. https://doi.org/10.1007/978-3-0346-0244-0_9

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