We present here results concerning the asymptotic behavior of isotropic diffusions in random environment that are small perturbations of Brownian motion. When the space dimension is three or more we prove an invariance principle as well as transience. Our methods also apply to questions of homogenization in random media.
Nous présentons ici des résultats sur le comportement asymptotique de diffusions isotropes en milieu aléatoire, qui sont de petites perturbations du mouvement brownien. Lorsque la dimension de l'espace est trois ou plus nous prouvons un principe d'invariance et la transience de la diffusion. Nos méthodes s'appliquent aussi à des problèmes d'homogénéisation en milieu aléatoire.
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Alain-Sol Sznitman 1; Ofer Zeitouni 2, 3
@article{CRMATH_2004__339_6_429_0, author = {Alain-Sol Sznitman and Ofer Zeitouni}, title = {On the diffusive behavior of isotropic diffusions in a random environment}, journal = {Comptes Rendus. Math\'ematique}, pages = {429--434}, publisher = {Elsevier}, volume = {339}, number = {6}, year = {2004}, doi = {10.1016/j.crma.2004.07.012}, language = {en}, }
Alain-Sol Sznitman; Ofer Zeitouni. On the diffusive behavior of isotropic diffusions in a random environment. Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 429-434. doi : 10.1016/j.crma.2004.07.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.012/
[1] Ten Lectures on Random Media, DMV Seminar, vol. 32, Birkhäuser, Basel, 2002
[2] Random walks in asymmetric random environments, Commun. Math. Phys., Volume 142 (1991) no. 2, pp. 345-420
[3] An invariance principle for reversible Markov processes, applications to random motions in random environments, J. Statist. Phys., Volume 55 (1989) no. 3–4, pp. 787-855
[4] Real Analysis and Probability, Wadsworth, Belmont, CA, 1989
[5] A central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Commun. Math. Phys., Volume 104 (1986), pp. 1-19
[6] The method of averaging and walks in inhomogeneous environments, Russian Math. Surveys, Volume 40 (1985) no. 2, pp. 73-145
[7] Lectures on random media, Lecture Notes in Math., vol. 1581, Springer, Berlin, 1994, pp. 242-411
[8] Central limit theorems for tagged particles and for diffusions in random environment, Milieux Aléatoires, Panoramas et Synthèses, vol. 12, Société Mathématique de France, 2001
[9] Boundary value problems with rapidly oscillating random coefficients (J. Fritz; D. Szasz, eds.), Random Fields, Janyos Bolyai Series, North-Holland, Amsterdam, 1981, pp. 835-873
[10] Topics in random walk in random environment, Notes of course at School and Conference on Probability Theory, May 2002, Trieste, ICTP Lecture Series, 2004, pp. 203-266
[11] A.S. Sznitman, O. Zeitouni, An invariance principle for isotropic diffusions in random environment, Preprint
[12] Average of an elliptic boundary problem with random coefficients, Siberian Math. J., Volume 21 (1980), pp. 470-482
[13] Random walks in random environment (J. Picard, ed.), Lectures on Probability Theory and Statistics, Lecture Notes in Math, vol. 1837, Springer, Berlin, 2004, pp. 190-312
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