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

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.

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.

DOI : 10.1214/10-AIHP375
Classification : 60K37, 82C41, 35B27
Mots-clés : algebraic convergence to equilibrium, random walk in random environment, environment viewed by the particle, homogenization 
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Mourrat, Jean-Christophe. Variance decay for functionals of the environment viewed by the particle. Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 1, pp. 294-327. doi : 10.1214/10-AIHP375. http://www.numdam.org/articles/10.1214/10-AIHP375/

[1] P. Baldi, N. Lohoué and J. Peyrière. Sur la classification des groupes récurrents. C. R. Acad. Sci. Paris Sér. A 285 (1977) 1103-1104. | MR | Zbl

[2] L. Bertini and B. Zegarlinski. Coercive inequalities for Kawasaki dynamics: The product case. Markov Process. Related Fields 5 (1999) 125-162. | MR | Zbl

[3] D. Boivin. Tail estimates for homogenization theorems in random media. ESAIM Probab. Stat. 13 (2009) 51-69. | Numdam | MR | Zbl

[4] A. Bourgeat and A. Piatnitski. Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 153-165. | Numdam | MR | Zbl

[5] P. Caputo and D. Ioffe. Finite volume approximation of the effective diffusion matrix: The case of independent bond disorder. Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 505-525. | Numdam | MR | Zbl

[6] M. Cudna and T. Komorowski. A finite dimensional approximation of the effective diffusivity for a symmetric random walk in a random environment. J. Comput. Appl. Math. 213 (2008) 186-204. | MR | Zbl

[7] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick. An invariance principle for reversible Markov processes. Applications to random motions in random environments. J. Statist. Phys. 55 (1989) 787-855. | MR | Zbl

[8] J.-P. Deuschel. Algebraic L2 decay of attractive critical processes on the lattice. Ann. Probab. 22 (1994) 264-283. | MR | Zbl

[9] A. Gloria and F. Otto. An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Preprint, 2009. Available at hal.archives-ouvertes.fr/hal-00383953_v2. | MR | Zbl

[10] A. Gloria and F. Otto. An optimal error estimate in stochastic homogenization of discrete elliptic equations. Preprint, 2010. Available at hal.archives-ouvertes.fr/inria-00457020_v1.

[11] V. V. Jikov, S. M. Kozlov and O. A. Oleinik. Homogenization of Differential Operators and Integral Functionals. Springer, Berlin, 1994. | MR | Zbl

[12] C. Kipnis and S. R. S. Varadhan. Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Comm. Math. Phys. 104 (1986) 1-19. | MR | Zbl

[13] S. M. Kozlov. Averaging of random structures. Dokl. Akad. Nauk SSSR 241 (1978) 1016-1019. English transl.: Soviet Math. Dokl. 19 (1978) 950-954. | MR | Zbl

[14] S. M. Kozlov. The averaging method and walks in inhomogeneous environments. Uspekhi Mat. Nauk 40 (1985) 61-120. English transl.: Russian Math. Surveys 40 (1985) 73-145. | MR | Zbl

[15] R. Künnemann. The diffusion limit for reversible jump processes on ℤd with ergodic random bond conductivities. Comm. Math. Phys. 90 (1983) 27-68. | MR | Zbl

[16] T. M. Liggett. L2 rates of convergence for attractive reversible nearest particle systems: The critical case. Ann. Probab. 19 (1991) 935-959. | MR | Zbl

[17] J. C. Maxwell. Medium in which small spheres are uniformly disseminated. In A Treatise on Electricity and Magnetism, 3rd edition, Part II, Chapter IX, Article 314. Clarendon Press, Oxford, 1891.

[18] S. A. Molchanov. Ideas in the theory of random media. Acta Appl. Math. 22 (1991) 139-282. | MR | Zbl

[19] J.-C. Mourrat. Scaling limit of the random walk among random traps on ℤd. Preprint, 2010. Available at arXiv:1001.2459.

[20] H. Osada. Homogenization of diffusion processes with random stationary coefficients. Probability Theory and Mathematical Statistics (Tbilisi, 1982) 507-517. Lecture Notes in Math. 1021. Springer, Berlin, 1983. | MR | Zbl

[21] G. C. Papanicolaou and S. R. S. Varadhan. Boundary value problems with rapidly oscillating random coefficients. Random Fields (Esztergom, 1979) 835-873. Colloq. Math. Soc. János Bolyai 27. North-Holland, Amsterdam, 1981. | MR | Zbl

[22] G. C. Papanicolaou and S. R. S. Varadhan. Diffusions with random coefficients. In Statistics and Probability: Essays in Honor of C. R. Rao 547-552. North-Holland, 1982. | MR | Zbl

[23] J. W. Strutt (3d Baron Rayleigh). On the influence of obstacles arranged in rectangular order upon the properties of a medium. Philos. Mag. 34 (1892) 481-502. | JFM

[24] L. Saloff-Coste. Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour 1996) 301-413. Lecture Notes in Math. 1665. Springer, Berlin, 1997. | MR | Zbl

[25] W. Woess. Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge, 2000. | MR | Zbl

[26] V. V. Yurinskiĭ. On a Dirichlet problem with random coefficients. In Stochastic Differential Systems (Proc. IFIP-WG 7/1 Working Conf., Vilnius, 1978) 344-353. Lecture Notes in Control and Information Sci. 25. Springer, Berlin, 1980. | MR | Zbl

[27] V. V. Yurinskiĭ. Averaging of symmetric diffusion in a random medium (in Russian). Sibirsk. Mat. Zh. 27 (1986) 167-180. English transl.: Siberian Math. J. 27 (1986) 603-613. | MR | Zbl

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