Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Quantitative homogenization of degenerate random environments

Arianna Giunti and Jean-Christophe Mourrat

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We study discrete linear divergence-form operators with random coefficients, also known as random conductance models. We assume that the conductances are bounded, independent and stationary; the law of a conductance may depend on the orientation of the associated edge. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive, in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector.


Nous étudions des opérateurs linéaires discrets sous forme divergence à coefficients aléatoires, aussi appelés modèles de conductances aléatoires. Nous supposons que les conductances sont bornées, indépendantes et stationnaires ; la loi d’une conductance peut dépendre de l’orientation de l’arète associée. Nous donnons une condition nécessaire et suffisante simple pour que la relaxation de l’environnement vu par la particule soit diffusive, au sens de tous les moments polynomiaux. Comme conséquence, nous estimons les moments polynomiaux du correcteur.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 54, Number 1 (2018), 22-50.

Received: 5 March 2016
Revised: 7 September 2016
Accepted: 12 September 2016
First available in Project Euclid: 19 February 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35K65: Degenerate parabolic equations 60K37: Processes in random environments

Quantitative homogenization Environment viewed by the particle Mixing of Markov chains Corrector estimate


Giunti, Arianna; Mourrat, Jean-Christophe. Quantitative homogenization of degenerate random environments. Ann. Inst. H. Poincaré Probab. Statist. 54 (2018), no. 1, 22--50. doi:10.1214/16-AIHP793.

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