no. 4
Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix
Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 4, pp. 981-1001.

This paper deals with homogenization of second order divergence form parabolic operators with locally stationary coefficients. Roughly speaking, locally stationary coefficients have two evolution scales: both an almost constant microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims to give a macroscopic approximation that takes into account the microscopic heterogeneities. This paper follows [Probab. Theory Related Fields (2009)] and improves this latter work by considering possibly degenerate diffusion matrices.

Nous étudions l'homogénéisation d'opérateurs paraboliques du second ordre sous forme divergence à coefficients localement stationnaires. Ces coefficients présentent deux échelles d'évolution: une évolution microscopique presque constante et une évolution macroscopique régulière. La théorie de l'homogénéisation consiste à donner une approximation macroscopique de l'opérateur initial qui tient compte des hétérogénéités microscopiques. Cet article fait suite à [Probab. Theory Related Fields (2009)] et généralise ce dernier en considérant des matrices de diffusion pouvant dégénérer.

DOI: 10.1214/08-AIHP190
Classification: 60F17
Keywords: homogenization, random medium, degenerate diffusion, locally stationary environment 
@article{AIHPB_2009__45_4_981_0,
     author = {Rhodes, R\'emi},
     title = {Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {981--1001},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {4},
     year = {2009},
     doi = {10.1214/08-AIHP190},
     mrnumber = {2572160},
     zbl = {1207.60029},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/08-AIHP190/}
}
TY  - JOUR
AU  - Rhodes, Rémi
TI  - Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2009
SP  - 981
EP  - 1001
VL  - 45
IS  - 4
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/08-AIHP190/
DO  - 10.1214/08-AIHP190
LA  - en
ID  - AIHPB_2009__45_4_981_0
ER  - 
%0 Journal Article
%A Rhodes, Rémi
%T Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2009
%P 981-1001
%V 45
%N 4
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/08-AIHP190/
%R 10.1214/08-AIHP190
%G en
%F AIHPB_2009__45_4_981_0
Rhodes, Rémi. Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix. Annales de l'I.H.P. Probabilités et statistiques, Volume 45 (2009) no. 4, pp. 981-1001. doi : 10.1214/08-AIHP190. http://archive.numdam.org/articles/10.1214/08-AIHP190/

[1] A. Benchérif-Madani and E. Pardoux. Homogenization of a diffusion with locally periodic coefficients. In Séminaire de Probabilités XXXVIII 363-392. Lecture Notes in Math. 1857. Springer, Berlin, 2005. | MR | Zbl

[2] A. Bensoussan, J. L. Lions and G. Papanicolaou. Asymptotic Methods in Periodic Media. North Holland, Amsterdam, 1978. | MR

[3] F. Delarue and R. Rhodes. Stochastic homogenization of quasilinear PDEs with a spatial degeneracy. Asymptot. Anal. 61 (2009) 61-90. | MR | Zbl

[4] M. Fukushima. Dirichlet Forms and Markov Processes. North-Holland, Amsterdam, 1980. | MR | Zbl

[5] M. Hairer and E. Pardoux. Homogenization of periodic linear degenerate PDEs. J. Funct. Anal. 255 2462-2487. | MR | Zbl

[6] J. Jacod and A. N. Shiryaev.Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaft 288. Springer, Berlin, 1987. | MR | Zbl

[7] V. V. Jikov, S. M. Kozlov and O. A. Oleinik.Homogenization of Differential Operators and Integral Functionals. Springer, Berlin, 1994. | MR | Zbl

[8] N. V. Krylov.Controlled Diffusion Processes. Springer, New York, 1980. | MR | Zbl

[9] Z. M. Ma and M. Röckner.Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Universitext. Springer, Berlin, 1992. | MR | Zbl

[10] S. Olla. Homogenization of diffusion processes in Random Fields. Cours de l'école doctorale, Ecole polytechnique, 1994. Available at http://www.ceremade.dauphine.fr/~olla/pubolla.html.

[11] S. Olla and P. Siri. Homogenization of a bond diffusion in a locally ergodic random environment. Stochastic Process. Appl. 109 (2004) 317-326. | MR | Zbl

[12] R. Rhodes. On homogenization of space time dependent random flows. Stochastic Process. Appl. 117 (2007) 1561-1585. | MR | Zbl

[13] R. Rhodes. Diffusion in a locally stationary random environment. Probab. Theory Related Fields 143 (2009) 545-568. | MR | Zbl

[14] D. Stroock. Diffusion semi-groups corresponding to uniformly elliptic divergence form operators. In Séminaires de Probabilités XXII 316-347. Lecture Notes in Math. 1321. Springer, Berlin, 1988. (Section B 35 (1999) 121-141.) | Numdam | MR | Zbl

[15] L. Wu. Forward-Backward martingale decomposition and compactness results for additive functionals of stationary ergodic Markov processes. Ann. Inst. H. Poincaré Probab. Statist. 35 (1999) 121-141. | Numdam | MR | Zbl

Cited by Sources: