Cet article traite de l'homogénéisation d'une équation aux dérivées partielles en dimension un d'espace, avec des coefficients aléatoires stationnaires et mélangeants, en présence d'u terme d'ordre zéro fortement oscillant. Nous montrons qu'avec un choix convenable du facteur d'échelle de ce terme d'ordre zéro, les solutions du problème étudié convergent en loi, et nous décrivons le processus limite. On peut noter que la dynamique limite est elle aussi aléatoire.
This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the said zero order terms, the family of solutions of the studied problem converges in law, and describe the limit process. It should be noted that the limit dynamics remain random.
@article{AIHPB_2008__44_3_519_0, author = {Iftimie, Bogdan and Pardoux, \'Etienne and Piatnitski, Andrey}, title = {Homogenization of a singular random one-dimensional PDE}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, volume = {44}, year = {2008}, pages = {519-543}, doi = {10.1214/07-AIHP134}, mrnumber = {2451056}, zbl = {1172.74043}, language = {en}, url = {http://dml.mathdoc.fr/item/AIHPB_2008__44_3_519_0} }
Iftimie, Bogdan; Pardoux, Étienne; Piatnitski, Andrey. Homogenization of a singular random one-dimensional PDE. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) pp. 519-543. doi : 10.1214/07-AIHP134. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_3_519_0/
[1] Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, Vol. 5. North-Holland, Amsterdam, 1978. | MR 503330 | Zbl 0404.35001
, and .[2] Convergence of Probability Measures, Wiley, 1968. | MR 233396 | Zbl 0172.21201
.[3] Probability and Measures, 3d edition. Wiley, 1995. | MR 1324786 | Zbl 0822.60002
.[4] Homogenization of random parabolic operator with large potential. Stochastic Process. Appl. 93 (2001) 57-85. | MR 1819484 | Zbl 1099.35009
, and .[5] Singular homogenization with stationary in time and periodic in space coefficients. J. Funct. Anal. 231 (2006) 1-46. | MR 2190162 | Zbl 1113.35015
, , and .[6] Markov Processes. Characterization and Convergence. Willey, New York, 1986. | MR 838085 | Zbl 0592.60049
and .[7] Dirichlet Forms and Symmetric Markov Processes. De Gruyter, 1994. | MR 1303354 | Zbl 0838.31001
, and .[8] Equations différentielles stochastiques retrogrades réfléchies dans un convexe. Stochastics Stochastic Rep. 57 ( 1996) 111-128. | MR 1407950 | Zbl 0891.60050
and .[9] Brownian Motion and Stochastic Calculus. Springer-Verlag, 1991. | MR 1121940 | Zbl 0734.60060
and .[10] Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, 1990. | MR 1070361 | Zbl 0743.60052
.[11] Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme divergence. Thèse, Université de Provence, 2000.
.[12] Malliavin Calculus and Related Topics, 2nd edition. Probability and Its Applications. Springer-Verlag, Berlin, 1996. | Zbl 0837.60050
.[13] Stochastic calculus with anticipative integrands, Probab. Theory Related Fields 78 (1988) 535-581. | MR 950346 | Zbl 0629.60061
and ,[14] Homogenization of a nonlinear random parabolic PDE. Stochastics Process. Appl. 104 (2003) 1-27. | MR 1956470 | Zbl 1075.35003
and .[15] Continuous Martingales and Brownian Motion. Springer, 1991. | MR 1083357 | Zbl 0731.60002
and .[16] Diffusion semigroups corresponding to uniformly elliptic divergence form operator. In Séminaire de Probabilités XXII. Lectures Notes in Math. 1321 pp. 316-347. Springer, 1988. | Numdam | MR 960535 | Zbl 0651.47031
.