Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization

• Autorzy: Lechiheb A. , Nourdin I. , Zheng G. , Haouala E.

Identyfikatory

DOI

10.19195/0208-4147.38.2.2

• Języki publikacji: EN

Abstrakty

EN

This paper deals with the asymptotic behavior of random oscillatory integrals in the presence of long-range dependence. As a byproduct, we solve the corrector problem in random homogenization of onedimensional elliptic equations with highly oscillatory random coefficients displaying long-range dependence, by proving convergence to stochastic integrals with respect to Hermite processes.

Słowa kluczowe

EN

elliptic equation; Hermite process; oscillatory integral; corrector; homogenization

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2018
• Tom: Vol. 38, Fasc. 2
• Strony: 271--286
• Opis fizyczny: Bibliogr. 12 poz.

Twórcy

autor

Lechiheb A.

• Université de Tunis El Manar, Faculté des sciences de Tunis, LR11ES13 Laboratoire d’Analyse stochastique, et applications, 2092, Tunis, Tunisie

autor

Nourdin I.

• Université du Luxembourg, Unité de Recherche en Mathématiques, Maison du Nombre, 6, avenue de la Fonte, L-4364 Esch-sur-Alzette, Grand Duchy of Luxembourg

autor

Zheng G.

• University of Kansas, Mathematics Department, Snow Hall, 1460 Jayhawk Blvd, Lawrence, Kansas 66045, USA

autor

Haouala E.

• Université de Tunis El Manar, Faculté des sciences de Tunis, LR11ES13 Laboratoire d’Analyse stochastique, et applications, 2092, Tunis, Tunisie

Bibliografia

• [1] G. Bal, J. Garnier, S. Motsch, and V. Perrier, Random integrals and correctors in homogenization, Asymptot. Anal. 59 (1-2) (2008), pp. 1-26.
• [2] G. Bal and Y. Gu, Limiting models for equations with large random potential: A review, Commun. Math. Sci. 13 (3) (2015), pp. 729-748.
• [3] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular Variation, Cambridge University Press, Cambridge 1989.
• [4] Y. Gu and G. Bal, Random homogenization and convergence to integrals with respect to the Rosenblatt process, J. Differential Equations 253 (4) (2012), pp. 1069-1087.
• [5] V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer, Berlin 1994.
• [6] O. Kallenberg, Foundations of Modern Probability, second edition, Springer, New York 2002.
• [7] M. Maejima and C. A. Tudor, Wiener integrals with respect to the Hermite process and a non-central limit theorem, Stoch. Anal. Appl. 25 (5) (2007), pp. 1043-1056.
• [8] J. Mourrat and J. Nolen, Scaling limit of the corrector in stochastic homogenization, Ann. Appl. Probab. 27 (2) (2017), pp. 944-959.
• [9] J. Mourrat and F. Otto, Correlation structure of the corrector in stochastic homogenization, Ann. Probab. 44 (5) (2016), pp. 3207-3233.
• [10] G. C. Papanicolaou and S. R. S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, in: Proceedings of the Colloquium on Random Fields (Esztergom, Hungary, 1979), Colloq. Math. Soc. János Bolyai 27, North Holland, 1981, pp. 835-873.
• [11] V. Pipiras and M. S. Taqqu, Integration questions related to fractional Brownian motion, Probab. Theory Related Fields 118 (2) (2000), pp. 251-291.
• [12] M. S. Taqqu, Convergence of integrated processes of arbitrary Hermite rank, Z. Wahrsch. Verw. Gebiete 50 (1) (1979), pp. 53-83.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).