Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Variance decay for functionals of the environment viewed by the particle

Jean-Christophe Mourrat

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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.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 1 (2011), 294-327.

First available in Project Euclid: 4 January 2011

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Primary: 60K37: Processes in random environments 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50]

Algebraic convergence to equilibrium Random walk in random environment Environment viewed by the particle Homogenization


Mourrat, Jean-Christophe. Variance decay for functionals of the environment viewed by the particle. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 1, 294--327. doi:10.1214/10-AIHP375.

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