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

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.

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.

DOI : https://doi.org/10.1214/12-AIHP527
Classification:  60K37,  60F17,  82C41
Keywords: random conductance model, dynamic environment, invariance principle, ergodic, corrector, point of view of the particle, stochastic interface model
@article{AIHPB_2014__50_2_352_0,
     author = {Andres, Sebastian},
     title = {Invariance principle for the random conductance model with dynamic bounded conductances},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {50},
     number = {2},
     year = {2014},
     pages = {352-374},
     doi = {10.1214/12-AIHP527},
     zbl = {1290.60109},
     mrnumber = {3189075},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2014__50_2_352_0}
}
Andres, Sebastian. Invariance principle for the random conductance model with dynamic bounded conductances. Annales de l'I.H.P. Probabilités et statistiques, Volume 50 (2014) no. 2, pp. 352-374. doi : 10.1214/12-AIHP527. http://www.numdam.org/item/AIHPB_2014__50_2_352_0/

[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 3078279 | Zbl pre06207007

[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 2249655 | Zbl 1115.60104

[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 2599199 | Zbl 1189.60187

[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 2471657 | Zbl 1192.60107

[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 2278453 | Zbl 1107.60066

[6] M. Biskup. Recent progress on the random concuctance model. Probab. Surv. 8 (2011) 294-373. | MR 2861133 | Zbl 1245.60098

[7] M. Biskup and T. M. Prescott. Functional CLT for random walk among bounded random conductances. Electron J. Probab. 12 (2007) 1323-1348. | MR 2354160 | Zbl 1127.60093

[8] T. Bodineau and B. Graham. Helffer-Sjöstrand representation for conservative dynamics. Markov Process. Related Fields 18 (2012) 71-88. | MR 2952020 | Zbl 1273.60113

[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 2052866 | Zbl 1062.60044

[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 2568274 | Zbl 1175.60086

[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 2198017 | Zbl 1083.60082

[12] Y. Derriennic and M. Lin. Fractional Poisson equations and ergodic theorems for fractional coboundaries. Israel J. Math. 123 (2001) 93-130. | MR 1835290 | Zbl 0988.47009

[13] D. Dolgopyat and C. Liverani. Non-perturbative approach to random walk in Markovian environment. Electron. Commun. Probab. 14 (2009) 245-251. | MR 2507753 | Zbl 1189.60188

[14] R. Durrett. Probability: Theory and Examples, 4th edition. Cambridge Univ. Press, Cambridge, 2010. | MR 2722836 | Zbl 1202.60001

[15] S. Ethier and T. Kurtz. Markov Processes. Wiley Series in Probability and Mathematical Statistics. Wiley, New York, 1986. | MR 838085 | Zbl 0592.60049

[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 2228384 | Zbl 1119.60081

[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 1463032 | Zbl 0884.58098

[18] G. Giacomin, S. Olla and H. Spohn. Equilibrium fluctuations for ϕ interface model. Ann. Probab. 29 (2001) 1138-1172. | MR 1872740 | Zbl 1017.60100

[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 1257821 | Zbl 0946.35508

[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 2748407 | Zbl 1276.60125

[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 2952852 | Zbl pre06028501

[22] P. Mathieu. Quenched invariance principles for random walks with random conductances. J. Stat. Phys. 130 (2008) 1025-1046. | MR 2384074 | Zbl 1214.82044

[23] M. Maxwell and M. Woodroofe. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28 (2000) 713-724. | MR 1782272 | Zbl 1044.60014

[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 2779406 | Zbl 1213.60163

[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 2198014 | Zbl 1088.60094

[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 2521407 | Zbl 1176.60087

[27] F. Redig and F. Völlering, Limit theorems for random walks in dynamic random environment. Preprint. Available at arXiv:1106.4181v2. | Zbl 1277.82051

[28] W. Rudin. Functional Analysis. McGraw-Hill Series in Higher Mathematics. McGraw-Hill, New York, 1973. | MR 365062 | Zbl 0867.46001

[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 2063376 | Zbl 1070.60090