Analytic Methods in Homogenization Theory

  • Tomasz Komorowski
  • Claudio Landim
  • Stefano Olla
Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 345)


Here we present another, more analytic in flavor, approach to the problem of homogenization of diffusions in random environments. We introduce the notions of G-convergence for a class of operators on a separable Hilbert space and Γ-convergence for symmetric quadratic forms. We show how these notions relate to the question of the central limit theorem for diffusions with random coefficients.


  1. Aronson DG (1967) Bounds for the fundamental solution of a parabolic equation. Bull Am Math Soc 73:890–896 MathSciNetzbMATHCrossRefGoogle Scholar
  2. Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures. Studies in mathematics and its applications, vol 5. North-Holland, Amsterdam zbMATHGoogle Scholar
  3. Billingsley P (1999) Convergence of probability measures. Wiley series in probability and mathematical statistics: probability and mathematical statistics, 2nd edn. Wiley, New York. A Wiley-Interscience Publication. zbMATHCrossRefGoogle Scholar
  4. Cioranescu D, Donato P (1999) An introduction to homogenization. Oxford lecture series in mathematics and its applications, vol 17. Clarendon Press, Oxford University Press, New York zbMATHGoogle Scholar
  5. Dal Maso G (1993) An introduction to Γ-convergence. Progress in nonlinear differential equations and their applications, vol 8. Birkhäuser Boston, Boston Google Scholar
  6. De Giorgi E (1975) Sulla convergenza di alcune successioni d’integrali del tipo dell’area. Rend Mat (6) 8:277–294. Collection of articles dedicated to Mauro Picone on the occasion of his ninetieth birthday MathSciNetzbMATHGoogle Scholar
  7. De Giorgi E, Franzoni T (1975) Su un tipo di convergenza variazionale. Atti Accad Naz Lincei, Rend Cl Sci Fis Mat Nat (8) 58(6):842–850 zbMATHGoogle Scholar
  8. Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence. Wiley series in probability and mathematical statistics: probability and mathematical statistics. Wiley, New York zbMATHCrossRefGoogle Scholar
  9. Evans LC (1990) Weak convergence methods for nonlinear partial differential equations. CBMS regional conference series in mathematics, vol 74. Published for the conference board of the mathematical sciences, Washington zbMATHGoogle Scholar
  10. Gilbarg D, Trudinger NS (1983) Elliptic partial differential equations of second order. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol 224, 2nd edn. Springer, Berlin zbMATHCrossRefGoogle Scholar
  11. Moser J (1964) A Harnack inequality for parabolic differential equations. Commun Pure Appl Math 17:101–134 zbMATHCrossRefGoogle Scholar
  12. Murat F (1977) H-convergence. Seminaire d’analyse fonctionnelle et numerique de l’universite d’Alger Google Scholar
  13. Murat F (1978) Compacité par compensation. Ann Sc Norm Super Pisa, Cl Sci (4) 5(3):489–507 MathSciNetzbMATHGoogle Scholar
  14. Murat F, Tartar L (1997) H-convergence. In: Topics in the mathematical modelling of composite materials. Progr nonlinear differential equations appl, vol 31. Birkhäuser Boston, Boston, pp 21–43 CrossRefGoogle Scholar
  15. Ngoan HT (1977a) Convergence of the solutions of boundary value problems for a sequence of elliptic equations. Usp Mat Nauk 32(3(195)):183–184 zbMATHGoogle Scholar
  16. Ngoan HT (1977b) Convergence of the solutions of boundary value problems for a sequence of elliptic systems. Vestn Mosk Univ Ser I Mat Meh 5:83–92 Google Scholar
  17. Papanicolaou GC, Varadhan SRS (1981) Boundary value problems with rapidly oscillating random coefficients. In: Random fields, vols I, II, Esztergom, 1979. Colloq math soc János Bolyai, vol 27. North-Holland, Amsterdam, pp 835–873 Google Scholar
  18. Sbordone C (1975) Su alcune applicazioni di un tipo di convergenza variazionale. Ann Sc Norm Super Pisa, Cl Sci (4) 2(4):617–638 MathSciNetzbMATHGoogle Scholar
  19. Simon L (1979) On G-convergence of elliptic operators. Indiana Univ Math J 28(4):587–594 MathSciNetzbMATHCrossRefGoogle Scholar
  20. Spagnolo S (1967) Sul limite delle soluzioni di problemi di Cauchy relativi all’equazione del calore. Ann Sc Norm Super Pisa, Cl Sci (3) 21:657–699 MathSciNetzbMATHGoogle Scholar
  21. Spagnolo S (1968) Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann Sc Norm Super Pisa, Cl Sci (3) 22(1968):571–597. Errata, Ann Sc Norm Super Pisa, Cl Sci (3) 22:673 MathSciNetzbMATHGoogle Scholar
  22. Stroock DW, Varadhan SRS (1979) Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol 233. Springer, Berlin zbMATHGoogle Scholar
  23. Tartar L (1977) Cours Peccot au collège de France. Paris Google Scholar
  24. Tartar L (1978) Quelques remarques sur homogeneisation. In: Proc of the Japan-France seminar 1976, functional analysis and numerical analysis. Japan Society for the Promotion of Science, Tokyo Google Scholar
  25. Zhikov VV (1983a) Asymptotic behavior and stabilization of solutions of a second-order parabolic equation with lowest terms. Tr Mosk Mat Obŝ 46:69–98 MathSciNetzbMATHGoogle Scholar
  26. Zhikov VV (1983b) G-convergence of elliptic operators. Mat Zametki 33(3):345–356 MathSciNetGoogle Scholar
  27. Zhikov VV, Kozlov SM, Oleĭnik OA, Ngoan HT (1979) Averaging and G-convergence of differential operators. Usp Mat Nauk 34(5(209)):65–133, 256 Google Scholar
  28. Zhikov VV, Kozlov SM, Oleĭnik OA (1981) G-convergence of parabolic operators. Usp Mat Nauk 36(1(217)):11–58, 248 zbMATHGoogle Scholar
  29. Zhikov VV, Kozlov SM, Oleĭnik OA (1994) Homogenization of differential operators and integral functionals. Springer, Berlin. Translated from the Russian by GA Yosifian [GA Iosif’yan] zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Tomasz Komorowski
    • 1
    • 2
  • Claudio Landim
    • 3
    • 4
  • Stefano Olla
    • 5
  1. 1.Institute of MathematicsMaria Curie-Skłodowska UniversityLublinPoland
  2. 2.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  3. 3.Instituto de Matemática (IMPA)Rio de JaneiroBrazil
  4. 4.CNRS UMR 6085Université de RouenSaint-Étienne-du-RouvrayFrance
  5. 5.CEREMADE, CNRS UMR 7534Université Paris-DauphineParisFrance

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