no. 3
Un théorème limite central local en environnement aléatoire stationnaire de conductances sur ℤ
Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 3, pp. 467-488

On démontre un théorème limite central local pour les marches aléatoires aux plus proches voisins en environnement aléatoire stationnaire de conductances sur ℤ en s'affranchissant simultanément des deux hypothèses classiques d'uniforme ellipticité et d'indépendance sur les conductances. Outre le théorème limite central, on utilise pour cela des inégalités différentielles discrètes du type « inégalités de Nash » associées à la représentation de Hausdorff des suites complètement décroissantes. La méthode s'adapte aux chaînes de Markov analogues en temps continu.

We prove a local central limit theorem for nearest neighbors random walks in stationary random environment of conductances on ℤ without using any of both classic assumptions of uniform ellipticity and independence on the conductances. Besides the central limit theorem, we use discrete differential Nash-type inequalities associated with the Hausdorff's representation of the completely decreasing sequences. The method is also valid for analogous continuous time Markov chains.

Publié le :
DOI : 10.24033/bsmf.2695
Classification : 60J10, 60K37
Mots-clés : Marches aléatoires, environnement aléatoire stationnaire de conductances, théorème limite central local, inégalités de Nash, représentation de Hausdorff des suites complètement décroissantes, théorèmes ergodiques.
Keywords: Random walks, stationary random environment of conductances, local central limit theorem, Nash's inequalities, Hausdorff's representation of the completely decreasing sequences, ergodic theorems. 
@article{BSMF_2015__143_3_467_0,
     author = {Derrien, Jean-Marc},
     title = {Un th\'eor\`eme limite central local en environnement al\'eatoire stationnaire de conductances sur $\mathbb {Z}$},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {467--488},
     year = {2015},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {143},
     number = {3},
     doi = {10.24033/bsmf.2695},
     mrnumber = {3417730},
     zbl = {1334.60016},
     language = {fr},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2695/}
}
TY  - JOUR
AU  - Derrien, Jean-Marc
TI  - Un théorème limite central local en environnement aléatoire stationnaire de conductances sur $\mathbb {Z}$
JO  - Bulletin de la Société Mathématique de France
PY  - 2015
SP  - 467
EP  - 488
VL  - 143
IS  - 3
PB  - Société mathématique de France
UR  - https://www.numdam.org/articles/10.24033/bsmf.2695/
DO  - 10.24033/bsmf.2695
LA  - fr
ID  - BSMF_2015__143_3_467_0
ER  - 
%0 Journal Article
%A Derrien, Jean-Marc
%T Un théorème limite central local en environnement aléatoire stationnaire de conductances sur $\mathbb {Z}$
%J Bulletin de la Société Mathématique de France
%D 2015
%P 467-488
%V 143
%N 3
%I Société mathématique de France
%U https://www.numdam.org/articles/10.24033/bsmf.2695/
%R 10.24033/bsmf.2695
%G fr
%F BSMF_2015__143_3_467_0
Derrien, Jean-Marc. Un théorème limite central local en environnement aléatoire stationnaire de conductances sur $\mathbb {Z}$. Bulletin de la Société Mathématique de France, Tome 143 (2015) no. 3, pp. 467-488. doi: 10.24033/bsmf.2695

Andres, S.; Barlow, Martin T.; Deuschel, J.-D.; Hambly, B. M. Invariance principle for the random conductance model, Probab. Theory Related Fields, Volume 156 (2013), pp. 535-580 (ISSN: 0178-8051) | MR | Zbl | DOI

Andres, S.; Deuschel, J.-D.; Slowik, M. Invariance principle for the random conductance model in a degenerate ergodic environment (preprint arXiv:1306.2521 ) | MR

Barlow, Martin T. Random walks on supercritical percolation clusters, Ann. Probab., Volume 32 (2004), pp. 3024-3084 (ISSN: 0091-1798) | MR | Zbl | DOI

Barlow, Martin T.; Deuschel, J.-D. Invariance principle for the random conductance model with unbounded conductances, Ann. Probab., Volume 38 (2010), pp. 234-276 (ISSN: 0091-1798) | MR | Zbl | DOI

Barlow, Martin T.; Hambly, B. M. Parabolic Harnack inequality and local limit theorem for percolation clusters, Electron. J. Probab., Volume 14 (2009), pp. 1-27 (ISSN: 1083-6489) | MR | Zbl

Berger, Noam Transience, recurrence and critical behavior for long-range percolation, Comm. Math. Phys., Volume 226 (2002), pp. 531-558 (ISSN: 0010-3616) | MR | Zbl | DOI

Berger, Noam; Biskup, Marek Quenched invariance principle for simple random walk on percolation clusters, Probab. Theory Related Fields, Volume 137 (2007), pp. 83-120 (ISSN: 0178-8051) | MR | Zbl | DOI

Berger, Noam; Biskup, Marek; Hoffman, C. E.; Kozma, G. Anomalous heat-kernel decay for random walk among bounded random conductances, Ann. Inst. Henri Poincaré Probab. Stat., Volume 44 (2008), pp. 374-392 (ISSN: 0246-0203) | MR | Zbl | Numdam | DOI

Biskup, Marek Recent progress on the random conductance model, Probab. Surv., Volume 8 (2011), pp. 294-373 (ISSN: 1549-5787) | MR | Zbl | DOI

Biskup, Marek; Boukhadra, Omar Subdiffusive heat-kernel decay in four-dimensional i.i.d. random conductance models, J. Lond. Math. Soc., Volume 86 (2012), pp. 455-481 (ISSN: 0024-6107) | MR | Zbl | DOI

Biskup, Marek; Prescott, Timothy M. Functional CLT for random walk among bounded random conductances, Electron. J. Probab., Volume 12 (2007), pp. 1323-1348 (ISSN: 1083-6489) | MR | Zbl | DOI

Boivin, Daniel; Depauw, Jérôme Spectral homogenization of reversible random walks on ℤd in a random environment, Stochastic Process. Appl., Volume 104 (2003), pp. 29-56 (ISSN: 0304-4149) | MR | Zbl | DOI

Boivin, Daniel; Derrien, Jean-Marc Geodesics and recurrence of random walks in disordered systems, Electron. Comm. Probab., Volume 7 (2002), pp. 101-115 (ISSN: 1083-589X) | MR | Zbl | DOI

Boukhadra, Omar Heat-kernel estimates for random walk among random conductances with heavy tail, Stochastic Process. Appl., Volume 120 (2010), pp. 182-194 (ISSN: 0304-4149) | MR | Zbl | DOI

Carlen, E. A.; Kusuoka, S.; Stroock, D. W. Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist., Volume 23 (1987), pp. 245-287 (ISSN: 0246-0203) | MR | Zbl | Numdam

Coulhon, Thierry; Saloff-Coste, Laurent Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana, Volume 9 (1993), pp. 293-314 (ISSN: 0213-2230) | MR | Zbl | DOI

Delmotte, Thierry Parabolic Harnack inequality and estimates of Markov chains on graphs, Rev. Mat. Iberoamericana, Volume 15 (1999), pp. 181-232 (ISSN: 0213-2230) | MR | Zbl | DOI

Depauw, Jérôme Théorème ergodique pour cocycle harmonique, applications au milieu aléatoire (preprint arXiv:1309.1566 )

Depauw, Jérôme; Derrien, Jean-Marc Variance limite d'une marche aléatoire réversible en milieu aléatoire sur ℤ, C. R. Math. Acad. Sci. Paris, Volume 347 (2009), pp. 401-406 (ISSN: 1631-073X) | MR | Zbl | DOI

Derrien, Jean-Marc Majoration du noyau de la chaleur en environnement stationnaire de conductances, C. R. Math. Acad. Sci. Paris, Volume 347 (2009), pp. 85-88 (ISSN: 1631-073X) | MR | Zbl | DOI

Doyle, Peter G.; Snell, J. Laurie Random walks and electric networks, Carus Mathematical Monographs, 22, Mathematical Association of America, Washington, DC, 1984, 159 pages (ISBN: 0-88385-024-9) | MR | Zbl | DOI

Feller, William An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, 1971, 669 pages | MR | Zbl

Grimmett, G. R.; Kesten, Harry; Zhang, Y. Random walk on the infinite cluster of the percolation model, Probab. Theory Related Fields, Volume 96 (1993), pp. 33-44 (ISSN: 0178-8051) | MR | Zbl | DOI

Grimmett, G. R.; Stirzaker, D. R. Probability and random processes, The Clarendon Press, Oxford Univ. Press, New York, 1992, 541 pages (ISBN: 0-19-853666-6; 0-19-853665-8) | MR | Zbl

Hausdorff, Felix Summationsmethoden und Momentfolgen. I, Math. Z., Volume 9 (1921), pp. 74-109 (ISSN: 0025-5874) | MR | JFM | DOI

Herglotz, G. Über Potenzreihen mit positivem, reellen Teil im Einheitskreis, Leipziger Berichte, Volume 63 (1911), pp. 501-511 | JFM

Kawazu, Kiyoshi; Kesten, Harry On birth and death processes in symmetric random environment, J. Statist. Phys., Volume 37 (1984), pp. 561-576 (ISSN: 0022-4715) | MR | Zbl | DOI

Kozlov, S. M. The averaging method and walks in inhomogeneous environments, Russ. Math. Surv., Volume 40 (1985), pp. 73-145 (ISSN: 0042-1316) | MR | Zbl | DOI

Lam, H. C. Les théorèmes limites pour des processus stationnaires, Ph. D. Thesis , Université de Tours (2012)

Mathieu, Pierre Quenched invariance principles for random walks with random conductances, J. Stat. Phys., Volume 130 (2008), pp. 1025-1046 (ISSN: 0022-4715) | MR | Zbl | DOI

Mathieu, Pierre; Piatnitski, A. Quenched invariance principles for random walks on percolation clusters, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 463 (2007), pp. 2287-2307 (ISSN: 1364-5021) | MR | Zbl | DOI

Mathieu, Pierre; Remy, Elisabeth Décroissance du noyau de la chaleur et isopérimétrie sur un amas de percolation, C. R. Acad. Sci. Paris Sér. I Math., Volume 332 (2001), pp. 927-931 (ISSN: 0764-4442) | MR | Zbl | DOI

Mathieu, Pierre; Remy, Elisabeth Isoperimetry and heat kernel decay on percolation clusters, Ann. Probab., Volume 32 (2004), pp. 100-128 (ISSN: 0091-1798) | MR | Zbl | DOI

Moser, Jürgen A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., Volume 17 (1964), pp. 101-134 correction dans 20 (1967), 231–236 (ISSN: 0010-3640) | MR | Zbl | DOI

Nash, J. Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., Volume 80 (1958), pp. 931-954 (ISSN: 0002-9327) | MR | Zbl | DOI

Petersen, Karl Ergodic theory, Cambridge Studies in Advanced Math., 2, Cambridge Univ. Press, Cambridge, 1983, 329 pages (ISBN: 0-521-23632-0) | MR | Zbl | DOI

Rudin, Walter Functional analysis, International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991, 424 pages (ISBN: 0-07-054236-8) | MR | Zbl

Sidoravicius, Vladas; Sznitman, Alain-Sol Quenched invariance principles for walks on clusters of percolation or among random conductances, Probab. Theory Related Fields, Volume 129 (2004), pp. 219-244 (ISSN: 0178-8051) | MR | Zbl | DOI

Cité par Sources :