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Harnack inequalities and local central limit theorem for the polynomial lower tail random conductance model. (English) Zbl 1332.60065

Summary: 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.

MSC:

60G50 Sums of independent random variables; random walks
60F05 Central limit and other weak theorems
60K37 Processes in random environments
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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References:

[1] 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. · Zbl 1356.60174
[2] 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. · Zbl 1336.31021
[3] 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. · Zbl 1159.60021
[4] M. T. Barlow, Random walks on supercritical percolation clusters, Ann. Probab., 32 (2004), 3024-3084. · Zbl 1067.60101
[5] 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. · Zbl 0916.60069
[6] M. T. Barlow and R. F. Bass, Construction of Brownian motion on the Sierpinski carpet, Ann. Inst. H. Poincaré, 25 (1989), 225-257. · Zbl 0691.60070
[7] M. T. Barlow and J.-D. Deuschel, Invariance principle for the random conductance model with unbounded conductances, Ann. Probab., 38 (2010), 234-276. · Zbl 1189.60187
[8] 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. · Zbl 1281.58016
[9] M. T. Barlow and B. M. Hambly, Parabolic Harnack inequality and local limit theorem for percolation clusters, Electron. J. Probab., 14 (2009), 1-27. · Zbl 1192.60107
[10] N. Berger and M. Biskup, Quenched invariance principle for simple random walk on percolation clusters, Probab. Theory Related Fields, 137 (2007), 83-120. · Zbl 1107.60066
[11] 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. · Zbl 1187.60034
[12] M. Biskup, Recent progress on the Random Conductance Model, Prob. Surveys, 8 (2011), 294-373. · Zbl 1245.60098
[13] 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. · Zbl 1260.60186
[14] M. Biskup and T. M. Prescott, Functional CLT for random walk among bounded random conductances, Electron. J. Probab., 12 (2007), 1323-1348. · Zbl 1127.60093
[15] M. Biskup, O. Louidor, A. Rozinov and A. Vandenberg-Rodes, Trapping in the random conductance model, J. Statist. Phys., 150 (2011), 66-87. · Zbl 1259.82042
[16] O. Boukhadra, Heat-kernel estimates for random walk among random conductances with heavy tail, Stochastic Process. Appl., 120 (2010), 182-194. · Zbl 1185.60116
[17] O. Boukhadra, Standard spectral dimension for the polynomial lower tail random conductances model, Electron. J. Probab., 15 (2010), 2069-2086. · Zbl 1231.60037
[18] X. Chen, Pointwise upper estimates for transition probability of continuous time random walks on graphs, arXiv: arXiv: arXiv:1310.2680 · Zbl 1361.60030
[19] T. Coulhon, A. Grigor’yan and F. Zucca, The discrete integral maximum principle and its applications, Tohoku Math. J., 57 (2005), 559-587. · Zbl 1096.60023
[20] E. B. Davies, Large deviations for heat kernels on graphs, J. London Math. Soc. (2), 47 (1993), 65-72. · Zbl 0799.58086
[21] T. Delmotte, Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana, 15 (1999), 181-232. · Zbl 0922.60060
[22] R. Durrett and R. H. Schonmann, Large deviations for the contact process and two-dimensional percolation, Probab. Theory Related Fields, 77 (1988), 583-603. · Zbl 0621.60108
[23] M. Folz, Gaussian upper bounds for heat kernels of continuous time simple random walks, Elect. J. Probab., 16 (2011), 1693-1722. · Zbl 1244.60099
[24] L. R. G. Fontes and P. Mathieu, On symmetric random walks with random conductances on \({\mathbb Z}^d\), Probab. Theory Related Fields, 134 (2006), 565-602. · Zbl 1086.60066
[25] A. Grigor’yan, Gaussian upper bounds for heat kernel on arbitrary manifolds, J. Differ. Geom., 45 (1997), 33-52. · Zbl 0865.58042
[26] A. Grigor’yan and A. Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Annalen, 324 (2002), 521-556. · Zbl 1011.60021
[27] A. Grigor’yan and A. Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, Duke Math. J., 109 (2001), 451-510. · Zbl 1010.35016
[28] G. Grimmett, Percolation, second edition, Grundlehren der Mathematischen Wissenschaften, 321, Springer, Berlin, 1999. · Zbl 0926.60004
[29] 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. · Zbl 1360.60003
[30] P. Mathieu, Quenched invariance principles for random walks with random conductances, J. Stat. Phys., 130 (2008), 1025-1046. · Zbl 1214.82044
[31] 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. · Zbl 1131.82012
[32] P. Mathieu and E. Remy, Isoperimetry and heat kernel decay on percolation clusters, Ann. Probab., 32 (2004), 100-128. · Zbl 1078.60085
[33] 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. · Zbl 1070.60090
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