Journal of the Mathematical Society of Japan

Harnack inequalities and local central limit theorem for the polynomial lower tail random conductance model

Omar BOUKHADRA, Takashi KUMAGAI, and Pierre MATHIEU

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Abstract

We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near 0. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local central limit theorems, parabolic Harnack inequalities and Gaussian bounds for the heat kernel. Some of the arguments are robust and applicable for random walks on general graphs. Such results are stated under a general setting.

Article information

Source
J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1413-1448.

Dates
First available in Project Euclid: 27 October 2015

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1445951155

Digital Object Identifier
doi:10.2969/jmsj/06741413

Mathematical Reviews number (MathSciNet)
MR3417502

Zentralblatt MATH identifier
1332.60065

Subjects
Primary: 60G50: Sums of independent random variables; random walks 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60K37: Processes in random environments

Keywords
Markov chains random walk random environments random conductances percolation

Citation

BOUKHADRA, Omar; KUMAGAI, Takashi; MATHIEU, Pierre. Harnack inequalities and local central limit theorem for the polynomial lower tail random conductance model. J. Math. Soc. Japan 67 (2015), no. 4, 1413--1448. doi:10.2969/jmsj/06741413. https://projecteuclid.org/euclid.jmsj/1445951155


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References

  • S. Andres, M. T. Barlow, J.-D. Deuschel and B. M. Hambly, Invariance principle for the random conductance model, Probab. Theory Related Fields, 156 (2013), 535–580.
  • S. Andres, J.-D. Deuschel and M. Slowik, Harnack inequalities on weighted graphs and some applications for the random conductance model, to appear Probab. Theory Related Fields.
  • M. T. Barlow, R. F. Bass and T. Kumagai, Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps, Math. Z., 261 (2009), 297–320.
  • M. T. Barlow, Random walks on supercritical percolation clusters, Ann. Probab., 32 (2004), 3024–3084.
  • M. T. Barlow, Diffusions on fractals, Lecture Notes in Math., 1690, Ecole d'été de probabilités de Saint-Flour XXV–1995, Springer, New York, 1998.
  • M. T. Barlow and R. F. Bass, Construction of Brownian motion on the Sierpinski carpet, Ann. Inst. H. Poincaré, 25 (1989), 225–257.
  • M. T. Barlow and J.-D. Deuschel, Invariance principle for the random conductance model with unbounded conductances, Ann. Probab., 38 (2010), 234–276.
  • M. T. Barlow, A. Grigor'yan and T. Kumagai, On the equivalence of parabolic Harnack inequalities and heat kernel estimates, J. Math. Soc. Japan, 64 (2012), 1091–1146.
  • M. T. Barlow and B. M. Hambly, Parabolic Harnack inequality and local limit theorem for percolation clusters, Electron. J. Probab., 14 (2009), 1–27.
  • N. Berger and M. Biskup, Quenched invariance principle for simple random walk on percolation clusters, Probab. Theory Related Fields, 137 (2007), 83–120.
  • N. Berger, M. Biskup, C. E. Hoffman and G. Kozma, Anomalous heat-kernel decay for random walk among bounded random conductances, Ann. Inst. Henri Poincaré Probab. Stat., 44 (2008), 374–392.
  • M. Biskup, Recent progress on the Random Conductance Model, Prob. Surveys, 8 (2011), 294–373.
  • M. Biskup and O. Boukhadra, Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models, J. London Math. Soc., 86 (2012), 455–481.
  • M. Biskup and T. M. Prescott, Functional CLT for random walk among bounded random conductances, Electron. J. Probab., 12 (2007), 1323–1348.
  • M. Biskup, O. Louidor, A. Rozinov and A. Vandenberg-Rodes, Trapping in the random conductance model, J. Statist. Phys., 150 (2011), 66–87.
  • O. Boukhadra, Heat-kernel estimates for random walk among random conductances with heavy tail, Stochastic Process. Appl., 120 (2010), 182–194.
  • O. Boukhadra, Standard spectral dimension for the polynomial lower tail random conductances model, Electron. J. Probab., 15 (2010), 2069–2086.
  • X. Chen, Pointwise upper estimates for transition probability of continuous time random walks on graphs, arXiv:1310.2680.
  • T. Coulhon, A. Grigor'yan and F. Zucca, The discrete integral maximum principle and its applications, Tohoku Math. J., 57 (2005), 559–587.
  • E. B. Davies, Large deviations for heat kernels on graphs, J. London Math. Soc. (2), 47 (1993), 65–72.
  • T. Delmotte, Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana, 15 (1999), 181–232.
  • R. Durrett and R. H. Schonmann, Large deviations for the contact process and two-dimensional percolation, Probab. Theory Related Fields, 77 (1988), 583–603.
  • M. Folz, Gaussian upper bounds for heat kernels of continuous time simple random walks, Elect. J. Probab., 16 (2011), 1693–1722.
  • L. R. G. Fontes and P. Mathieu, On symmetric random walks with random conductances on ${\Bbb Z}^d$, Probab. Theory Related Fields, 134 (2006), 565–602.
  • A. Grigor'yan, Gaussian upper bounds for heat kernel on arbitrary manifolds, J. Differ. Geom., 45 (1997), 33–52.
  • A. Grigor'yan and A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Annalen, 324 (2002), 521–556.
  • A. Grigor'yan and A. Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, Duke Math. J., 109 (2001), 451–510.
  • G. Grimmett, Percolation, second edition, Grundlehren der Mathematischen Wissenschaften, 321, Springer, Berlin, 1999.
  • T. Kumagai, Random Walks on Disordered Media and their Scaling Limits, Lect. Notes in Math., 2101, École d'Été de Probabilités de Saint-Flour XL–2010. Springer, New York, 2014.
  • P. Mathieu, Quenched invariance principles for random walks with random conductances, J. Stat. Phys., 130 (2008), 1025–1046.
  • P. Mathieu and A. Piatnitski, Quenched invariance principles for random walks on percolation clusters, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2287–2307.
  • P. Mathieu and E. Remy, Isoperimetry and heat kernel decay on percolation clusters, Ann. Probab., 32 (2004), 100–128.
  • V. Sidoravicius and A.-S. Sznitman, Quenched invariance principles for walks on clusters of percolation or among random conductances, Probab. Theory Related Fields, 129 (2004), 219–244.