Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix
Rhodes, Rémi
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009), p. 981-1001 / Harvested from Numdam
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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.

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.

Publié le : 2009-01-01
DOI : https://doi.org/10.1214/08-AIHP190
Classification:  60F17 
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     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},
     volume = {45},
     year = {2009},
     pages = {981-1001},
     doi = {10.1214/08-AIHP190},
     mrnumber = {2572160},
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     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPB_2009__45_4_981_0}
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Rhodes, Rémi. Homogenization of locally stationary diffusions with possibly degenerate diffusion matrix. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) pp. 981-1001. doi : 10.1214/08-AIHP190. http://gdmltest.u-ga.fr/item/AIHPB_2009__45_4_981_0/

[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 2126985 | Zbl 1067.35009

[2] A. Bensoussan, J. L. Lions and G. Papanicolaou. Asymptotic Methods in Periodic Media. North Holland, Amsterdam, 1978. | MR 503330

[3] F. Delarue and R. Rhodes. Stochastic homogenization of quasilinear PDEs with a spatial degeneracy. Asymptot. Anal. 61 (2009) 61-90. | MR 2499193 | Zbl 1180.35591

[4] M. Fukushima. Dirichlet Forms and Markov Processes. North-Holland, Amsterdam, 1980. | MR 569058 | Zbl 0422.31007

[5] M. Hairer and E. Pardoux. Homogenization of periodic linear degenerate PDEs. J. Funct. Anal. 255 2462-2487. | MR 2473263 | Zbl 1167.60015

[6] J. Jacod and A. N. Shiryaev.Limit Theorems for Stochastic Processes. Grundlehren der Mathematischen Wissenschaft 288. Springer, Berlin, 1987. | MR 959133 | Zbl 0635.60021

[7] V. V. Jikov, S. M. Kozlov and O. A. Oleinik.Homogenization of Differential Operators and Integral Functionals. Springer, Berlin, 1994. | MR 1329546 | Zbl 0838.35001

[8] N. V. Krylov.Controlled Diffusion Processes. Springer, New York, 1980. | MR 601776 | Zbl 0459.93002

[9] Z. M. Ma and M. Röckner.Introduction to the Theory of (Nonsymmetric) Dirichlet Forms. Universitext. Springer, Berlin, 1992. | MR 1214375 | Zbl 0826.31001

[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 2031772 | Zbl 1075.60129

[12] R. Rhodes. On homogenization of space time dependent random flows. Stochastic Process. Appl. 117 (2007) 1561-1585. | MR 2353040 | Zbl 1127.60027

[13] R. Rhodes. Diffusion in a locally stationary random environment. Probab. Theory Related Fields 143 (2009) 545-568. | MR 2475672 | Zbl 1163.60049

[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 960535 | Zbl 0651.47031

[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 1678517 | Zbl 0936.60037