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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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On the uniqueness class, stochastic completeness and volume growth for graphs
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by Xueping Huang, Matthias Keller and Marcel Schmidt PDF
Trans. Amer. Math. Soc. 373 (2020), 8861-8884 Request permission

Abstract:

In this note we prove an optimal volume growth condition for stochastic completeness of graphs under very mild assumptions. This is realized by proving a uniqueness class criterion for the heat equation which is an analogue to a corresponding result of Grigor’yan on manifolds. This uniqueness class criterion is shown to hold for graphs that we call globally local, i.e., graphs where we control the jump size far outside. The transfer from general graphs to globally local graphs is then carried out via so-called refinements.
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Additional Information
  • Xueping Huang
  • Affiliation: School of Mathematics and Statistics, Nanjing University of Information Science and Technology, No. 219, Ninglu Road, Nanjing, Jiangsu, 210044, People’s Republic of China; and School of Mathematics and Statistics, Jiangsu Normal University, 221116, Xuzhou, People’s Republic of China
  • Email: hxp@nuist.edu.cn
  • Matthias Keller
  • Affiliation: Universität Potsdam, Institut für Mathematik, 14476 Potsdam, Germany
  • MR Author ID: 886028
  • Email: matthias.keller@uni-potsdam.de
  • Marcel Schmidt
  • Affiliation: Mathematisches Institut, Friedrich-Schiller-Universität Jena, 07743 Jena, Germany
  • MR Author ID: 1091878
  • ORCID: 0000-0002-7918-0715
  • Email: schmidt.marcel@uni-jena.de
  • Received by editor(s): May 16, 2019
  • Received by editor(s) in revised form: May 5, 2020
  • Published electronically: October 5, 2020
  • Additional Notes: The first author was supported by the National Natural Science Foundation of China (Grant No. 11601238) and The Startup Foundation for Introducing Talent of Nanjing University of Information Science and Technology (Grant No. 2015r053).
    The second and third authors acknowledge the financial support of the DFG
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 8861-8884
  • MSC (2010): Primary 47D06, 60J27; Secondary 58J65
  • DOI: https://doi.org/10.1090/tran/8211
  • MathSciNet review: 4177278