A homogenization problem for infinite dimensional diffusion processes indexed by having periodic drift coefficients is considered. By an application of the uniform ergodic theorem for the infinite dimensional diffusion processes based on logarithmic Sobolev inequalities, an type homogenization property of the processes with respect to an invariant measure is proved. This is the, so far, best possible analogue in infinite dimensions to a known result in the finite dimensional case (cf. [G. Papanicolaou, S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Seria Coll. Math. Soc. Janos Bolyai, vol. 27, North-Holland, 1979. [4]]).
On considère un problème d'homogénéisation pour des processus de diffusion infini dimensionnels, indéxés par et avec coefficient de transfert périodique. On démontre une propriété d'homogénéisation du type par rapport à une mesure invariante, en utilisant un théorème ergodique uniforme fondé sur les inégalités logarithmiques du type Sobolev. Ce résultat représente le meilleur analogue possible de résultats correspondants en dimension finie (cf. [G. Papanicolaou, S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Seria Coll. Math. Soc. Janos Bolyai, vol. 27, North-Holland, 1979. [4]]).
Accepted:
Published online:
Sergio Albeverio 1; M. Simonetta Bernabei 2; Michael Röckner 3; Minoru W. Yoshida 4
@article{CRMATH_2005__341_11_675_0,
author = {Sergio Albeverio and M. Simonetta Bernabei and Michael R\"ockner and Minoru W. Yoshida},
title = {Homogenization with respect to {Gibbs} measures for periodic drift diffusions on lattices},
journal = {Comptes Rendus. Math\'ematique},
pages = {675--678},
publisher = {Elsevier},
volume = {341},
number = {11},
year = {2005},
doi = {10.1016/j.crma.2005.09.044},
language = {en},
}
TY - JOUR AU - Sergio Albeverio AU - M. Simonetta Bernabei AU - Michael Röckner AU - Minoru W. Yoshida TI - Homogenization with respect to Gibbs measures for periodic drift diffusions on lattices JO - Comptes Rendus. Mathématique PY - 2005 SP - 675 EP - 678 VL - 341 IS - 11 PB - Elsevier DO - 10.1016/j.crma.2005.09.044 LA - en ID - CRMATH_2005__341_11_675_0 ER -
%0 Journal Article %A Sergio Albeverio %A M. Simonetta Bernabei %A Michael Röckner %A Minoru W. Yoshida %T Homogenization with respect to Gibbs measures for periodic drift diffusions on lattices %J Comptes Rendus. Mathématique %D 2005 %P 675-678 %V 341 %N 11 %I Elsevier %R 10.1016/j.crma.2005.09.044 %G en %F CRMATH_2005__341_11_675_0
Sergio Albeverio; M. Simonetta Bernabei; Michael Röckner; Minoru W. Yoshida. Homogenization with respect to Gibbs measures for periodic drift diffusions on lattices. Comptes Rendus. Mathématique, Volume 341 (2005) no. 11, pp. 675-678. doi : 10.1016/j.crma.2005.09.044. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.09.044/
[1] Homogenization of infinite dimensional diffusion processes with periodic drift coefficients, Proceedings of Quantum Information and Complexity, Meijo Univ., 2003 January, World Sci. Publishing, River Edge, NJ, 2004
[2] S. Albeverio, M.S. Bernabei, M. Röckner, M.W. Yoshida, Homogenization of diffusions on the lattice with periodic drift coefficients, Application of logarithmic Sobolev inequality, Preprint, 2005
[3] Diffusions on an infinite dimensional torus, J. Funct. Anal., Volume 42 (1981), pp. 29-63
[4] Boundary value problems with rapidly oscillating random coefficients, Seria Coll. Math. Soc. Janos Bolyai, vol. 27, North-Holland, 1979
[5] Logarithmic Sobolev Inequalities for Gibbs States, Lecture Notes in Math., vol. 1563, Springer-Verlag, Berlin, 1993
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