Electronic Journal of Probability

Anomalous threshold behavior of long range random walks

Mathav Murugan and Laurent Saloff-Coste

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Abstract

We consider weighted graphs satisfying sub-Gaussian estimate for the natural random walk.  On such graphs, we study symmetric Markov chains with heavy tailed jumps. We establish a threshold behavior of such Markov chains when the index governing the tail heaviness (or jump index) equals the escape time exponent (or walk dimension) of the sub-Gaussian estimate. In a certain sense, this generalizes the classical threshold corresponding to the second moment condition.

Article information

Source
Electron. J. Probab., Volume 20 (2015), paper no. 74, 21 pp.

Dates
Accepted: 28 June 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067180

Digital Object Identifier
doi:10.1214/EJP.v20-3989

Mathematical Reviews number (MathSciNet)
MR3371433

Zentralblatt MATH identifier
1321.60090

Subjects
Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 60J75: Jump processes 60J15

Keywords
sub-Gaussian estimate heavy-tailed random walk

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Murugan, Mathav; Saloff-Coste, Laurent. Anomalous threshold behavior of long range random walks. Electron. J. Probab. 20 (2015), paper no. 74, 21 pp. doi:10.1214/EJP.v20-3989. https://projecteuclid.org/euclid.ejp/1465067180


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References

  • Barlow, Martin T. Which values of the volume growth and escape time exponent are possible for a graph? Rev. Mat. Iberoamericana 20 (2004), no. 1, 1–31.
  • Barlow, Martin T.; Bass, Richard F.; Kumagai, Takashi. Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps. Math. Z. 261 (2009), no. 2, 297–320.
  • Barlow, Martin T.; Coulhon, Thierry; Kumagai, Takashi. Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs. Comm. Pure Appl. Math. 58 (2005), no. 12, 1642–1677.
  • Barlow, Martin T.; Grigor'yan, Alexander; Kumagai, Takashi. Heat kernel upper bounds for jump processes and the first exit time. J. Reine Angew. Math. 626 (2009), 135–157.
  • Bass, Richard F.; Levin, David A. Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc. 354 (2002), no. 7, 2933–2953 (electronic).
  • Bendikov, A.; Saloff-Coste, L. Random walks on groups and discrete subordination. Math. Nachr. 285 (2012), no. 5-6, 580–605.
  • Bendikov, Alexander; Saloff-Coste, Laurent. Random walks driven by low moment measures. Ann. Probab. 40 (2012), no. 6, 2539–2588.
  • Bingham, N. H.; Goldie, C. M.; Teugels, J. L. Regular variation. Encyclopedia of Mathematics and its Applications, 27. Cambridge University Press, Cambridge, 1987. xx+491 pp. ISBN: 0-521-30787-2
  • Coulhon, Thierry. Ultracontractivity and Nash type inequalities. J. Funct. Anal. 141 (1996), no. 2, 510–539.
  • Chen, Zhen-Qing; Kumagai, Takashi. Heat kernel estimates for stable-like processes on $d$-sets. Stochastic Process. Appl. 108 (2003), no. 1, 27–62.
  • Chen, Zhen-Qing; Kumagai, Takashi. Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Related Fields 140 (2008), no. 1-2, 277–317.
  • Carlen, E. A.; Kusuoka, S.; Stroock, D. W. Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 245–287.
  • Coulhon, Thierry; Grigor'yan, Alexander. On-diagonal lower bounds for heat kernels and Markov chains. Duke Math. J. 89 (1997), no. 1, 133–199.
  • Delmotte, Thierry. Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoamericana 15 (1999), no. 1, 181–232.
  • M. Murugan, L. Saloff-Coste, Transition probability estimates for long range random walks, preprint arXiv:1411.2706
  • Pittet, Ch.; Saloff-Coste, L. On the stability of the behavior of random walks on groups. J. Geom. Anal. 10 (2000), no. 4, 713–737.
  • Ch. Pittet, L. Saloff-Coste, A survey on the relationships between volume growth, isoperimetry and the behavior of simple random walk on Cayley graphs, with examples (in preparation)
  • Graczyk, Piotr; Stós, Andrzej. Transition density estimates for stable processes on symmetric spaces. Pacific J. Math. 217 (2004), no. 1, 87–100.
  • Grigor'yan, Alexander; Hu, Jiaxin; Lau, Ka-Sing. Heat kernels on metric spaces with doubling measure. Fractal geometry and stochastics IV, 3–44, Progr. Probab., 61, Birkhäuser Verlag, Basel, 2009.
  • Grigor'yan, Alexander; Telcs, András. Sub-Gaussian estimates of heat kernels on infinite graphs. Duke Math. J. 109 (2001), no. 3, 451–510.
  • Grigor'yan, Alexander; Telcs, András. Harnack inequalities and sub-Gaussian estimates for random walks. Math. Ann. 324 (2002), no. 3, 521–556.
  • T. Kumagai, Anomalous random walks and diffusions: From fractals to random media, Proceedings of 2014 ICM, Seoul.
  • Saloff-Coste, L. A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices 1992, no. 2, 27–38.
  • L. Saloff-Coste, T. Zheng, On some random walks driven by spread-out measures, preprint arXiv:1309.6296.