Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Variance decay for functionals of the environment viewed by the particle

Jean-Christophe Mourrat

Full-text: Open access

Enhanced PDF (360 KB)

Abstract

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.

Résumé

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

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

Dates
First available in Project Euclid: 4 January 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1294170239

Digital Object Identifier
doi:10.1214/10-AIHP375

Mathematical Reviews number (MathSciNet)
MR2779406

Zentralblatt MATH identifier
1213.60163

Subjects
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]

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

Citation

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. https://projecteuclid.org/euclid.aihp/1294170239


Export citation

References

  • [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.
  • [2] L. Bertini and B. Zegarlinski. Coercive inequalities for Kawasaki dynamics: The product case. Markov Process. Related Fields 5 (1999) 125–162.
    Zentralblatt MATH: 0934.60096
  • [3] D. Boivin. Tail estimates for homogenization theorems in random media. ESAIM Probab. Stat. 13 (2009) 51–69.
    Zentralblatt MATH: 1187.60090
    Digital Object Identifier: doi:10.1051/ps:2007036
  • [4] A. Bourgeat and A. Piatnitski. Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 153–165.
    Zentralblatt MATH: 1058.35023
    Digital Object Identifier: doi:10.1016/S0246-0203(03)00065-7
  • [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.
    Zentralblatt MATH: 1014.60094
    Digital Object Identifier: doi:10.1016/S0246-0203(02)00016-X
  • [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.
    Zentralblatt MATH: 1138.65007
    Digital Object Identifier: doi:10.1016/j.cam.2007.01.043
  • [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.
    Zentralblatt MATH: 0713.60041
    Digital Object Identifier: doi:10.1007/BF01041608
  • [8] J.-P. Deuschel. Algebraic L2 decay of attractive critical processes on the lattice. Ann. Probab. 22 (1994) 264–283.
    Zentralblatt MATH: 0811.60089
    Digital Object Identifier: doi:10.1214/aop/1176988859
    Project Euclid: euclid.aop/1176988859
  • [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.
    Zentralblatt MATH: 1215.35025
    Digital Object Identifier: doi:10.1214/10-AOP571
    Project Euclid: euclid.aop/1300281725
  • [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.
  • [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.
    Zentralblatt MATH: 0588.60058
    Digital Object Identifier: doi:10.1007/BF01210789
    Project Euclid: euclid.cmp/1104114929
  • [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.
  • [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.
    Zentralblatt MATH: 0592.60054
  • [15] R. Künnemann. The diffusion limit for reversible jump processes on ℤd with ergodic random bond conductivities. Comm. Math. Phys. 90 (1983) 27–68.
  • [16] T. M. Liggett. L2 rates of convergence for attractive reversible nearest particle systems: The critical case. Ann. Probab. 19 (1991) 935–959.
    Zentralblatt MATH: 0737.60092
    Digital Object Identifier: doi:10.1214/aop/1176990330
    Project Euclid: euclid.aop/1176990330
  • [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.
    Zentralblatt MATH: 0728.73011
    Digital Object Identifier: doi:10.1007/BF00580850
  • [19] J.-C. Mourrat. Scaling limit of the random walk among random traps on ℤd. Preprint, 2010. Available at arXiv:1001.2459.
    arXiv: 1001.2459
    Mathematical Reviews (MathSciNet): MR2994845
    Digital Object Identifier: doi:10.1214/EJP.v17-2414
  • [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.
  • [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.
  • [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.
    Zentralblatt MATH: 0486.60076
  • [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.
    Zentralblatt MATH: 24.1015.02
    Digital Object Identifier: doi:10.1080/14786449208620364
  • [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.
    Zentralblatt MATH: 0885.60061
  • [25] W. Woess. Random Walks on Infinite Graphs and Groups. Cambridge Tracts in Mathematics 138. Cambridge Univ. Press, Cambridge, 2000.
    Zentralblatt MATH: 0951.60002
  • [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.
  • [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.

Institut Henri Poincaré

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more