Variance decay for functionals of the environment viewed by the particle
Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, p. 294-327

For the random walk among random conductances, we prove that the environment viewed by the particle converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that the conductances are bounded away from zero. The basis of our method is the establishment of a Nash inequality, followed either by a comparison with the simple random walk or by a more direct analysis based on a martingale decomposition. As an example of application, we show that under certain conditions, our results imply an estimate of the speed of convergence of the mean square displacement of the walk towards its limit.

Pour la marche aléatoire en conductances aléatoires, nous montrons que l'environnement vu par la particule converge vers l'équilibre à une vitesse polynomiale au sens de la variance, notre hypothèse principale étant que les conductances sont uniformément minorées. Notre méthode se base sur l'établissement d'une inégalité de Nash, suivie soit d'une comparaison avec la marche aléatoire simple, soit d'une analyse plus directe fondée sur une méthode de martingale. Comme exemple d'application, nous montrons que sous certaines conditions, ces résultats permettent d'estimer la vitesse de convergence vers sa limite du déplacement quadratique moyen de la marche.

DOI : https://doi.org/10.1214/10-AIHP375
Classification:  60K37,  82C41,  35B27
Keywords: algebraic convergence to equilibrium, random walk in random environment, environment viewed by the particle, homogenization
@article{AIHPB_2011__47_1_294_0,
     author = {Mourrat, Jean-Christophe},
     title = {Variance decay for functionals of the environment viewed by the particle},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {47},
     number = {1},
     year = {2011},
     pages = {294-327},
     doi = {10.1214/10-AIHP375},
     zbl = {1213.60163},
     mrnumber = {2779406},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2011__47_1_294_0}
}
Mourrat, Jean-Christophe. Variance decay for functionals of the environment viewed by the particle. Annales de l'I.H.P. Probabilités et statistiques, Volume 47 (2011) no. 1, pp. 294-327. doi : 10.1214/10-AIHP375. http://www.numdam.org/item/AIHPB_2011__47_1_294_0/

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