Comptes Rendus
Probability Theory/Mathematical Physics
Homogenization with respect to Gibbs measures for periodic drift diffusions on lattices
Comptes Rendus. Mathématique, Volume 341 (2005) no. 11, pp. 675-678.

A homogenization problem for infinite dimensional diffusion processes indexed by Zd 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 L1 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 Zd et avec coefficient de transfert périodique. On démontre une propriété d'homogénéisation du type L1 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]]).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2005.09.044
Sergio Albeverio 1; M. Simonetta Bernabei 2; Michael Röckner 3; Minoru W. Yoshida 4

1 Inst. Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany
2 Dipartimento di Matematica e Informatica, Università di Camerino, Via Madonna delle Carceri, 9, 62032 Camerino, Italy
3 Department of Mathematics, Purdue University, Math. Sci. Building, 150N. University Street, West Lafayette, IN 47907-2067, USA
4 The University Electro commun, Department of Systems Engineering, 182-8585 Chofu-shi Tokio, Japan
@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] S. Albeverio; M.S. Bernabei; M. Röckner; M.W. Yoshida 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 Zd with periodic drift coefficients, Application of logarithmic Sobolev inequality, Preprint, 2005

[3] R. Holley; D. Stroock Diffusions on an infinite dimensional torus, J. Funct. Anal., Volume 42 (1981), pp. 29-63

[4] G. Papanicolaou; S. Varadhan Boundary value problems with rapidly oscillating random coefficients, Seria Coll. Math. Soc. Janos Bolyai, vol. 27, North-Holland, 1979

[5] D. Stroock Logarithmic Sobolev Inequalities for Gibbs States, Lecture Notes in Math., vol. 1563, Springer-Verlag, Berlin, 1993

Cited by Sources:

Comments - Policy