On the uniqueness class, stochastic completeness and volume growth for graphs
HTML articles powered by AMS MathViewer
- 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.References
- Robert Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193–240. MR 356254
- E. B. Davies, Heat kernel bounds, conservation of probability and the Feller property, J. Anal. Math. 58 (1992), 99–119. Festschrift on the occasion of the 70th birthday of Shmuel Agmon. MR 1226938, DOI 10.1007/BF02790359
- Matthew Folz, Volume growth and stochastic completeness of graphs, Trans. Amer. Math. Soc. 366 (2014), no. 4, 2089–2119. MR 3152724, DOI 10.1090/S0002-9947-2013-05930-2
- Matthew P. Gaffney, The conservation property of the heat equation on Riemannian manifolds, Comm. Pure Appl. Math. 12 (1959), 1–11. MR 102097, DOI 10.1002/cpa.3160120102
- A. A. Grigor′yan, Stochastically complete manifolds, Dokl. Akad. Nauk SSSR 290 (1986), no. 3, 534–537 (Russian). MR 860324
- A. A. Grigor′yan, Stochastically complete manifolds and summable harmonic functions, Izv. Akad. Nauk SSSR Ser. Mat. 52 (1988), no. 5, 1102–1108, 1120 (Russian); English transl., Math. USSR-Izv. 33 (1989), no. 2, 425–432. MR 972099
- Alexander Grigor′yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135–249. MR 1659871, DOI 10.1090/S0273-0979-99-00776-4
- Alexander Grigor’yan, Xueping Huang, and Jun Masamune, On stochastic completeness of jump processes, Math. Z. 271 (2012), no. 3-4, 1211–1239. MR 2945605, DOI 10.1007/s00209-011-0911-x
- Xueping Huang, On stochastic completeness of graphs, Ph.D. Thesis, Bielefeld. 2011.
- Xueping Huang, Stochastic incompleteness for graphs and weak Omori-Yau maximum principle, J. Math. Anal. Appl. 379 (2011), no. 2, 764–782. MR 2784357, DOI 10.1016/j.jmaa.2011.02.009
- Xueping Huang, On uniqueness class for a heat equation on graphs, J. Math. Anal. Appl. 393 (2012), no. 2, 377–388. MR 2921681, DOI 10.1016/j.jmaa.2012.04.026
- Xueping Huang, A note on the volume growth criterion for stochastic completeness of weighted graphs, Potential Anal. 40 (2014), no. 2, 117–142. MR 3152158, DOI 10.1007/s11118-013-9342-0
- Xueping Huang, Matthias Keller, Jun Masamune, and Radosław K. Wojciechowski, A note on self-adjoint extensions of the Laplacian on weighted graphs, J. Funct. Anal. 265 (2013), no. 8, 1556–1578. MR 3079229, DOI 10.1016/j.jfa.2013.06.004
- Xueping Huang and Yuichi Shiozawa, Upper escape rate of Markov chains on weighted graphs, Stochastic Process. Appl. 124 (2014), no. 1, 317–347. MR 3131296, DOI 10.1016/j.spa.2013.08.004
- Leon Karp and Peter Li, The heat equation on complete Riemannian manifolds, unpublished.
- Matthias Keller, Intrinsic metrics on graphs: a survey, Mathematical technology of networks, Springer Proc. Math. Stat., vol. 128, Springer, Cham, 2015, pp. 81–119. MR 3375157, DOI 10.1007/978-3-319-16619-3_{7}
- M. Keller and D. Lenz, Unbounded Laplacians on graphs: basic spectral properties and the heat equation, Math. Model. Nat. Phenom. 5 (2010), no. 4, 198–224. MR 2662456, DOI 10.1051/mmnp/20105409
- Matthias Keller and Daniel Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, J. Reine Angew. Math. 666 (2012), 189–223. MR 2920886, DOI 10.1515/CRELLE.2011.122
- Karl-Theodor Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and $L^p$-Liouville properties, J. Reine Angew. Math. 456 (1994), 173–196. MR 1301456, DOI 10.1515/crll.1994.456.173
- Masayoshi Takeda, On a martingale method for symmetric diffusion processes and its applications, Osaka J. Math. 26 (1989), no. 3, 605–623. MR 1021434
- Radoslaw Krzysztof Wojciechowski, Stochastic completeness of graphs, ProQuest LLC, Ann Arbor, MI, 2008. Thesis (Ph.D.)–City University of New York. MR 2711706
- Radoslaw Krzysztof Wojciechowski, Stochastically incomplete manifolds and graphs, Random walks, boundaries and spectra, Progr. Probab., vol. 64, Birkhäuser/Springer Basel AG, Basel, 2011, pp. 163–179. MR 3051698, DOI 10.1007/978-3-0346-0244-0_{9}
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