no. 4
On homogenization of space-time dependent and degenerate random flows II
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 673-692.

Nous étudions le comportement asymptotique (homogénéisation) d'une diffusion en milieu aléatoire avec des coefficients dépendant du temps et de l'espace, pour laquelle le coefficient de diffusion peut dégénérer. Dans Stochastic Process. Appl. (2007) (to appear), un principe d'invariance est établi pour le changement d'échelle critique de la diffusion. Ici, une généralisation de cette approche est proposée pour différents changements d'échelle possibles.

We study the long time behavior (homogenization) of a diffusion in random medium with time and space dependent coefficients. The diffusion coefficient may degenerate. In Stochastic Process. Appl. (2007) (to appear), an invariance principle is proved for the critical rescaling of the diffusion. Here, we generalize this approach to diffusions whose space-time scaling differs from the critical one.

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     title = {On homogenization of space-time dependent and degenerate random flows {II}},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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     publisher = {Gauthier-Villars},
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Rhodes, Rémi. On homogenization of space-time dependent and degenerate random flows II. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 673-692. doi : 10.1214/07-AIHP135. http://archive.numdam.org/articles/10.1214/07-AIHP135/

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

[2] M. G. Crandall, H. Ishii and P. L. Lions. User's guide to viscosity solutions of second order Partial Differential Equations. Bull. Amer. Soc. 27 (1992) 1-67. | MR | Zbl

[3] E. B. Davies. One-parameter Semigroups. Academic Press, 1980. | MR | Zbl

[4] Y. Efendiev and A. Pankov. Homogenization of nonlinear random parabolic operators. Adv. Differential Equations 10 (2005) 1235-1260. | MR | Zbl

[5] A. Fannjiang and T. Komorowski. An invariance principle for diffusion in turbulence. Ann. Probab. 27 (1999) 751-781. | MR | Zbl

[6] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin and Hawthorne, New York, 1994. | MR | Zbl

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

[8] C. Landim, S. Olla and H. T. Yau. Convection-diffusion equation with space-time ergodic random flow. Probab. Theory Related Fields 112 (1998) 203-220. | MR | Zbl

[9] A. Lejay. Homogenization of divergence-form operators with lower order terms in random media. Probab. Theory Related Fields 120 (2001) 255-276. | MR | Zbl

[10] T. Komorowski and S. Olla. On homogenization of time-dependent random flows. Probab. Theory Related Fields 121 (2001) 98-116. | MR | Zbl

[11] Z. M. Ma and M. Röckner. Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Springer, Berlin, 1992. | Zbl

[12] J. R. Norris. Long-time behaviour of heat flow: Global estimates and exact asymptotics. Arch. Rational Mech. Anal. 140 (1997) 161-195. | MR | Zbl

[13] K. Oelschläger. Homogenization of a diffusion process in a divergence free random field. Ann. Probab. 16 (1988) 1084-1126. | MR | Zbl

[14] S. Olla. Homogenization of diffusion processes in Random Fields. Cours de l'école doctorale, Ecole polytechnique, 1994.

[15] H. Osada. Homogenization of diffusion processes with random stationary coefficients. Probability Theory and Mathematical Statistics (Tbilisi, 1982) 507-517. Lecture Notes in Math. 1021. Springer, Berlin, 1983. | MR | Zbl

[16] A. Pankov. G-convergence and Homogenization of Nonlinear Partial Differential Operators. Kluwer Publ., 1997. | MR | Zbl

[17] E. Pardoux. BSDE's, weak convergence and homogenization of semilinear PDE's in Nonlinear analysis. In Differential Equations and Control 503-549. F. H. Clarke and R. J. Stern (Eds). Kluwer Acad. Publ., Dordrecht, 1999. | MR | Zbl

[18] R. Rhodes. On homogenization of space-time dependent and degenerate random flows. Stochastic Process. Appl. 17 (2007) 1561-1585. | MR | Zbl

[19] D. W. Stroock and S. R. S. Varadhan. Multidimensional Diffusion Processes. New York, Springer, 1979. | MR | Zbl

[20] N. Svanstedt. Correctors for the homogenization of monotone parabolic operators. J. Nonlinear Math. Phys. 7 (2000) 268-284. | MR | Zbl

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