Invariance principle for the random conductance model with dynamic bounded conductances
Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 352-374.

Nous étudions une chaîne de Markov en temps continu X dans un environnement dynamique de conductances aléatoires dans d . Nous supposons que les conductances sont stationnaires ergodiques, uniformément positives et polynomialement mélangeantes en espace et en temps. Nous montrons un principe d’invariance << quenched >> pour X, et nous obtenons des bornes sur les fonctions de Green et un théorème limite local. Nous discutons aussi les liens avec les modèles d’interfaces aléatoires.

We study a continuous time random walk X in an environment of dynamic random conductances in d . We assume that the conductances are stationary ergodic, uniformly bounded and bounded away from zero and polynomially mixing in space and time. We prove a quenched invariance principle for X, and obtain Green’s functions bounds and a local limit theorem. We also discuss a connection to stochastic interface models.

DOI : 10.1214/12-AIHP527
Classification : 60K37, 60F17, 82C41
Mots clés : random conductance model, dynamic environment, invariance principle, ergodic, corrector, point of view of the particle, stochastic interface model
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Andres, Sebastian. Invariance principle for the random conductance model with dynamic bounded conductances. Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 2, pp. 352-374. doi : 10.1214/12-AIHP527. http://www.numdam.org/articles/10.1214/12-AIHP527/

[1] S. Andres, M. T. Barlow, J.-D. Deuschel and B. Hambly. Invariance principle for the random conductance model. Preprint. Probab. Theory Related Fields. To appear. Available at DOI:10.1007/s00440-012-0435-2. | MR

[2] A. Bandyopadhyay and O. Zeitouni. Random walk in dynamic Markovian random environment. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006) 205-224. | MR | Zbl

[3] M. T. Barlow and J.-D. Deuschel. Invariance principle for the random conductance model with unbounded conductances. Ann. Probab. 38 (2010) 234-276. | MR | Zbl

[4] M. T. Barlow and B. M. Hambly. Parabolic Harnack inequality and local limit theorem for percolation clusters. Electron. J. Probab. 14 (2009) 1-16. | MR | Zbl

[5] N. Berger and M. Biskup. Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 (2007) 83-120. | MR | Zbl

[6] M. Biskup. Recent progress on the random concuctance model. Probab. Surv. 8 (2011) 294-373. | MR | Zbl

[7] M. Biskup and T. M. Prescott. Functional CLT for random walk among bounded random conductances. Electron J. Probab. 12 (2007) 1323-1348. | MR | Zbl

[8] T. Bodineau and B. Graham. Helffer-Sjöstrand representation for conservative dynamics. Markov Process. Related Fields 18 (2012) 71-88. | MR | Zbl

[9] C. Boldrighini, R. A. Minlos and A. Pellegrinotti. Random walks in quenched i.i.d. space-time random environment are always a.s. diffusive. Probab. Theory Related Fields 129 (2004) 133-156. | MR | Zbl

[10] C. Boldrighini, R. A. Minlos and A. Pellegrinotti. Discrete-time random motion in a continuous random medium. Stochastic Process. Appl. 119 (2009) 3285-3299. | MR | Zbl

[11] T. Delmotte and J.-D. Deuschel. On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to ϕ interface model. Probab. Theory Related Fields 133 (2005) 358-390. | MR | Zbl

[12] Y. Derriennic and M. Lin. Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123 (2001) 93-130. | MR | Zbl

[13] D. Dolgopyat and C. Liverani. Non-perturbative approach to random walk in Markovian environment. Electron. Commun. Probab. 14 (2009) 245-251. | MR | Zbl

[14] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Univ. Press, Cambridge, 2010. | MR | Zbl

[15] S. Ethier and T. Kurtz. Markov Processes. Wiley Series in Probability and Mathematical Statistics. Wiley, New York, 1986. | MR | Zbl

[16] T. Funaki. Stochastic Interface Models. In Ecole d'été de probabilités de Saint Flour 2003 103-274. Lecture Notes in Mathematics 1869. Springer, Berlin, 2005. | MR | Zbl

[17] T. Funaki and H. Spohn. Motion by mean curvature from the Ginzburg-Landau ϕ interface models. Commun. Math. Phys. 185 (1997) 1-36. | MR | Zbl

[18] G. Giacomin, S. Olla and H. Spohn. Equilibrium fluctuations for ϕ interface model. Ann. Probab. 29 (2001) 1138-1172. | MR | Zbl

[19] B. Helffer and J. Sjöstrand. On the correlation for Kac-like models in the convex case. J. Stat. Phys. 74 (1994) 349-409. | MR | Zbl

[20] M. Joseph and F. Rassoul-Agha. Almost sure invariance principle for continuous-space random walk in dynamic random environment. ALEA Lat. Am. J. Probab. Math. Stat. 8 (2011) 43-57. | MR | Zbl

[21] T. Komorowski, C. Landim and S. Olla. Fluctuations in Markov processes. Time Symmetry and Martingale Approximation. Grundlehren der Mathematischen Wissenschaften 345. Springer, Heidelberg, 2012. | MR

[22] P. Mathieu. Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130 (2008) 1025-1046. | MR | Zbl

[23] M. Maxwell and M. Woodroofe. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000) 713-724. | MR | Zbl

[24] J.-C. Mourrat. Variance decay for functionals of the environment viewed by the particle. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 294-327. | Numdam | MR | Zbl

[25] F. Rassoul-Agha and T. Seppäläinen. An almost sure invariance principle for random walks in a space-time random environment. Probab. Theory Related Fields 133 (2005) 299-314. | MR | Zbl

[26] F. Rassoul-Agha and T. Seppäläinen. Almost sure functional central limit theorem for ballistic random walk in random environment. Ann. Inst. Henri Poincaré Probab. Stat. 45 (2009) 373-420. | Numdam | MR | Zbl

[27] F. Redig and F. Völlering, Limit theorems for random walks in dynamic random environment. Preprint. Available at arXiv:1106.4181v2. | Zbl

[28] W. Rudin. Functional Analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York, 1973. | MR | Zbl

[29] 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. | MR | Zbl

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