Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aronson DG (1967) Bounds for the fundamental solution of a parabolic equation. Bull Am Math Soc 73:890–896
Bensoussan A, Lions JL, Papanicolaou G (1978) Asymptotic analysis for periodic structures. Studies in mathematics and its applications, vol 5. North-Holland, Amsterdam
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.
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
Dal Maso G (1993) An introduction to Γ-convergence. Progress in nonlinear differential equations and their applications, vol 8. Birkhäuser Boston, Boston
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
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
Ethier SN, Kurtz TG (1986) Markov processes: characterization and convergence. Wiley series in probability and mathematical statistics: probability and mathematical statistics. Wiley, New York
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
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
Moser J (1964) A Harnack inequality for parabolic differential equations. Commun Pure Appl Math 17:101–134
Murat F (1977) H-convergence. Seminaire d’analyse fonctionnelle et numerique de l’universite d’Alger
Murat F (1978) Compacité par compensation. Ann Sc Norm Super Pisa, Cl Sci (4) 5(3):489–507
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
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
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
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
Sbordone C (1975) Su alcune applicazioni di un tipo di convergenza variazionale. Ann Sc Norm Super Pisa, Cl Sci (4) 2(4):617–638
Simon L (1979) On G-convergence of elliptic operators. Indiana Univ Math J 28(4):587–594
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
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
Stroock DW, Varadhan SRS (1979) Multidimensional diffusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental principles of mathematical sciences], vol 233. Springer, Berlin
Tartar L (1977) Cours Peccot au collège de France. Paris
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
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
Zhikov VV (1983b) G-convergence of elliptic operators. Mat Zametki 33(3):345–356
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
Zhikov VV, Kozlov SM, Oleĭnik OA (1981) G-convergence of parabolic operators. Usp Mat Nauk 36(1(217)):11–58, 248
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]
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Komorowski, T., Landim, C., Olla, S. (2012). Analytic Methods in Homogenization Theory. In: Fluctuations in Markov Processes. Grundlehren der mathematischen Wissenschaften, vol 345. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29880-6_14
Download citation
DOI: https://doi.org/10.1007/978-3-642-29880-6_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-29879-0
Online ISBN: 978-3-642-29880-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)