Abstract
We study a continuous time random walk X in an environment of i.i.d. random conductances \({\mu_{e} \in [0,\infty)}\) in \({\mathbb{Z}^d}\) . We assume that \({\mathbb{P}(\mu_{e} > 0) > p_c}\) , so that the bonds with strictly positive conductances percolate, but make no other assumptions on the law of the μ e . We prove a quenched invariance principle for X, and obtain Green’s functions bounds and an elliptic Harnack inequality.
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Antal P., Pisztora A.: On the chemical distance for supercritical Bernoulli percolation. Ann. Probab. 24(2), 1036–1048 (1996)
Barlow M.T.: Random walks on supercritical percolation clusters. Ann. Probab. 32, 3024–3084 (2004)
Barlow M.T., Černý J.: Convergence to fractional kinetics for random walks associated with unbounded conductances. Probab. Theory Relat. Fields 149(3–4), 639673 (2011)
Barlow M.T., Deuschel J.-D.: Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38, 234–276 (2010)
Barlow, M.T., Hambly, B.M.: Parabolic Harnack inequality and local limit theorem for percolation clusters. Electron. J. Probab. 14, paper(1) 1–26 (2009)
Barlow M.T., Zheng X.: The random conductance model with Cauchy tails. Ann. Appl. Probab. 20(3), 869–889 (2010)
Bass R.F.: On Aronsen’s upper bounds for heat kernels. Bull. Lond. Math. Soc. 34, 415–419 (2002)
Ben Arous G., Černý J., Mountford T.: Aging in two-dimensional Bouchaud’s model. Probab. Theory Relat. Fields 134(1), 1–43 (2006)
Ben Arous G., Černý J.: Scaling limit for trap models on \({\mathbb{Z}^d}\) . Ann. Probab. 35(6), 2356–2384 (2007)
Ben Arous G., Černý J.: The arcsine law as a universal aging scheme for trap models. Commun. Pure Appl. Math. 61(3), 289–329 (2008)
Berger N., Biskup M.: Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Relat. Fields 137(1-2), 83–120 (2007)
Berger N., Biskup M., Hoffman C.E., Kozma G.: Anomalous heat-kernel decay for random walk among bounded random conductances. Ann. IHP Probab. Stat. 44(2), 374–392 (2008)
Biskup, M., Boukhadra, O.: Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models. J. Lond. Math. Soc. Preprint: arXiv:1010.5542v1 (2010)
Biskup, M., Prescott, T.M.: Functional CLT for random walk among bounded random conductances. Elec. J. Prob. 12, paper 49, 1323–1348 (2007)
Carne T.K.: A transmutation formula for Markov chains. Bull. Sci. Math. 109, 399–405 (1985)
Černý J.: On two-dimensional random walk among heavy-tailed conductances. Electron. J. Probab. 16(10), 293–313 (2011)
Coulhon T., Grigoryan A., Zucca F.: The discrete integral maximum principle and its applications. Tohoku Math. J. (2) 4(57), 559–587 (2005)
Croydon D.A., Hambly B.M.: Local limit theorems for sequences of simple random walks on graphs. Potential Anal. 29(4), 351–389 (2008)
Davies E.B.: Large deviations for heat kernels on graphs. J. Lond. Math. Soc. 47(2), 65–72 (1993)
Delmotte T.: Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Math. Iberoamericana 15, 181–232 (1999)
De Masi A., Ferrari P.A., Goldstein S., Wick W.D.: An invariance principle for reversible Markov processes.Applications to random motions in random environments. J. Stat. Phys. 55, 787–855 (1989)
Durrett R.: Probability: Theory and Examples (4th Edition). Cambridge University Press, Cambridge (2010)
Fabes E.B., Stroock D.W.: A new proof of Moser’s parabolic Harnack inequality via the old ideas of Nash. Arch. Mech. Ration. Anal. 96, 327–338 (1986)
Fannjiang A., Komorowski T.: An invariance principle for diffusion in turbulence. Ann. Probab. 27, 751–781 (1999)
Folz M.: Gaussian upper bounds for heat kernels of continuous time simple random walks. Electron. J. Probab. 16(62), 1693–1722 (2011)
Grigor’yan A.: Gaussian heat kernel bounds on arbitrary manifolds. J. Differ. Geom. 45, 33–52 (1997)
Helland, I.: Central limit theorems for martingales with discrete or continuous time. Scand. J. Stat. 9, 79–94
Keynes H.B., Markley N.G., Sears M.: Ergodic averages and integrals of cocycles. Acta Math. Univ. Comemanae LXIV, 123–139 (1995)
Kozlov S.: The method of averaging and walks in inhomogeneous environments. Russ. Math. Surv. 40(2), 73–145 (1985)
Liggett T.M., Schonmann R.H., Stacey A.M.: Domination by product measures. Ann. Probab. 25, 71–95 (1997)
Mathieu P., Remy E.: Isoperimetry and heat kernel decay on percolation clusters. Ann. Probab. 32(1A), 100–128 (2004)
Mathieu P., Piatnitski A.: Quenched invariance principles for random walks on percolation clusters. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 463(2085), 2287–2307 (2007)
Mathieu P.: Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130(5), 1025–1046 (2008)
Nash J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Osada, H.: Homogenization of diffusion processes with random stationary coefficients. In: Probability Theory and Mathematical Statistics, Tbilissi, 1982. Lecture Notes in Mathematics, vol. 1021, pp. 507–517. Springer, Berlin (1983)
Pete G.: A note on percolation on \({\mathbb{Z}^d}\) : isoperimetric profile via exponential cluster repulsion. Electron. Commun. Probab. 13, 377–392 (2008)
Penrose M.D., Pisztora A.: Large deviations for discrete and continuous percolation. Adv. Appl. Probab. 28, 29–52 (1996)
Pisztora A.: Surface order large deviations for Ising, Potts and percolation models. Probab. Theory Relat. Fields 104, 427–466 (1996)
Sidoravicius V., Sznitman A.-S.: Quenched invariance principles for walks on clusters of percolation or among random conductances. Probab. Theory Relat. Fields 129(2), 219–244 (2004)
Varopoulos N.Th.: Long range estimates for Markov chains. Bull. Sci. Math. 2e Ser. 109, 225–252 (1985)
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S. Andres and M. T. Barlow were partially supported by NSERC (Canada), J.-D. Deuschel was partially supported by DFG (Germany).
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Andres, S., Barlow, M.T., Deuschel, JD. et al. Invariance principle for the random conductance model. Probab. Theory Relat. Fields 156, 535–580 (2013). https://doi.org/10.1007/s00440-012-0435-2
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DOI: https://doi.org/10.1007/s00440-012-0435-2