Homogenization of a singular random one-dimensional PDE
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 3, pp. 519-543.

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.

DOI : 10.1214/07-AIHP134
Classification : 74Q10
Mots-clés : stochastic homogenization, random operators 
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     title = {Homogenization of a singular random one-dimensional {PDE}},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
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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) no. 3, pp. 519-543. doi : 10.1214/07-AIHP134. http://archive.numdam.org/articles/10.1214/07-AIHP134/

[1] A. Bensoussan, J.-L. Lions and G. Papanicolaou. Asymptotic Analysis for Periodic Structures. Studies in Mathematics and its Applications, Vol. 5. North-Holland, Amsterdam, 1978. | MR | Zbl

[2] P. Billingsley. Convergence of Probability Measures, Wiley, 1968. | MR | Zbl

[3] P. Billingsley. Probability and Measures, 3d edition. Wiley, 1995. | MR | Zbl

[4] F. Campillo, M. Kleptsyna and A. Piatnitski. Homogenization of random parabolic operator with large potential. Stochastic Process. Appl. 93 (2001) 57-85. | MR | Zbl

[5] M. Diop, B. Iftimie, É. Pardoux and A. Piatnitski. Singular homogenization with stationary in time and periodic in space coefficients. J. Funct. Anal. 231 (2006) 1-46. | MR | Zbl

[6] S. N. Ethier and T. G. Kurtz. Markov Processes. Characterization and Convergence. Willey, New York, 1986. | MR | Zbl

[7] M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. De Gruyter, 1994. | MR | Zbl

[8] A. Gegout-Petit and É. Pardoux. Equations différentielles stochastiques retrogrades réfléchies dans un convexe. Stochastics Stochastic Rep. 57 ( 1996) 111-128. | MR | Zbl

[9] I. Karatzas and S. Shreve. Brownian Motion and Stochastic Calculus. Springer-Verlag, 1991. | MR | Zbl

[10] H. Kunita. Stochastic Flows and Stochastic Differential Equations. Cambridge University Press, 1990. | MR | Zbl

[11] A. Lejay. Méthodes probabilistes pour l'homogénéisation des opérateurs sous forme divergence. Thèse, Université de Provence, 2000.

[12] D. Nualart. Malliavin Calculus and Related Topics, 2nd edition. Probability and Its Applications. Springer-Verlag, Berlin, 1996. | Zbl

[13] D. Nualart and É. Pardoux, Stochastic calculus with anticipative integrands, Probab. Theory Related Fields 78 (1988) 535-581. | MR | Zbl

[14] É. Pardoux and A. Piatnitski. Homogenization of a nonlinear random parabolic PDE. Stochastics Process. Appl. 104 (2003) 1-27. | MR | Zbl

[15] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, 1991. | MR | Zbl

[16] D. W. Stroock. 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 | Zbl

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