Inventiones mathematicae

, Volume 208, Issue 3, pp 999–1154 | Cite as

The additive structure of elliptic homogenization

  • Scott Armstrong
  • Tuomo Kuusi
  • Jean-Christophe Mourrat


One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in Armstrong et al. (Commun. Math. Phys. 347(2):315–361, 2016): using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

Mathematics Subject Classification

35B27 35B45 



The second author was supported by the Academy of Finland project #258000. We thank Antti Hannukainen (Aalto University) for performing the numerical computations of the corrector and producing Fig. 1.


  1. 1.
    Armstrong, S., Gloria, A., Kuusi, T.: Bounded correctors in almost periodic homogenization. Arch. Ration. Mech. Anal. 222(1), 393–426 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Armstrong, S., Kuusi, T., Mourrat, J.-C.: Mesoscopic higher regularity and subadditivity in elliptic homogenization. Commun. Math. Phys. 347(2), 315–361 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Armstrong, S.N., Mourrat, J.-C.: Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal. 219(1), 255–348 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Armstrong, S.N., Smart, C.K.: Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. Éc. Norm. Supér. (4) 49(2), 423–481 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Avellaneda, M., Lin, F.-H.: Compactness methods in the theory of homogenization. Commun. Pure Appl. Math. 40(6), 803–847 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Avellaneda, M., Lin, F.-H.: Un théorème de Liouville pour des équations elliptiques à coefficients périodiques. C. R. Acad. Sci. Paris Sér. I Math. 309(5), 245–250 (1989)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Billingsley, P.: Convergence of Probability Measures. Wiley Series in Probability and Statistics, 2nd edn. Wiley, New York (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Biskup, M., Salvi, M., Wolff, T.: A central limit theorem for the effective conductance: linear boundary data and small ellipticity contrasts. Commun. Math. Phys. 328(2), 701–731 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Chatterjee, S.: A new method of normal approximation. Ann. Probab. 36(4), 1584–1610 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chatterjee, S.: Fluctuations of eigenvalues and second order Poincaré inequalities. Probab. Theory Relat. Fields 143(1–2), 1–40 (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    Dal Maso, G., Modica, L.: Nonlinear stochastic homogenization. Ann. Mat. Pura Appl. (4) 144, 347–389 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dal Maso, G., Modica, L.: Nonlinear stochastic homogenization and ergodic theory. J. Reine Angew. Math. 368, 28–42 (1986)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Duerinckx, M., Gloria, A., Otto, F.: The structure of fluctuations in stochastic homogenization, preprint. arXiv:1602.01717
  14. 14.
    Fischer, J., Otto, F.: A higher-order large-scale regularity theory for random elliptic operators. Commun. Partial Differ. Equ. 41(7), 1108–1148 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Furlan, M., Mourrat, J.-C.: A tightness criterion for random fields, with application to the Ising model (preprint). arXiv:1502.07335
  16. 16.
    Gloria, A., Neukamm, S., Otto, F.: An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. ESAIM Math. Model. Numer. Anal. 48(2), 325–346 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Gloria, A., Neukamm, S., Otto, F.: Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Invent. Math. 199(2), 455–515 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gloria, A., Neukamm, S., Otto, F.: A regularity theory for random elliptic operators (preprint). arXiv:1409.2678
  19. 19.
    Gloria, A., Nolen, J.: A quantitative central limit theorem for the effective conductance on the discrete torus. Commun. Pure Appl. Math. (in press). arXiv:1410.5734
  20. 20.
    Gloria, A., Otto, F.: An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39(3), 779–856 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Gloria, A., Otto, F.: An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22(1), 1–28 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Gloria, A., Otto, F.: The corrector in stochastic homogenization: near-optimal rates with optimal stochastic integrability (preprint) (2015). arXiv:1510.08290
  23. 23.
    Gloria, A., Otto, F.: The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations (preprint) (2016). arXiv:1510.08290v3
  24. 24.
    Gloria, A., Otto, F.: Quantitative results on the corrector equation in stochastic homogenization. J. Eur. Math. Soc. (in press). arXiv:1409.0801
  25. 25.
    Gu, Y., Mourrat, J.-C.: Scaling limit of fluctuations in stochastic homogenization. Multiscale Model. Simul. 14(1), 452–481 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Gu, Y., Mourrat, J.-C.: On generalized Gaussian free fields and stochastic homogenization (preprint). arXiv:1601.06408
  27. 27.
    Helffer, B., Sjöstrand, J.: On the correlation for Kac-like models in the convex case. J. Stat. Phys. 74(1–2), 349–409 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kozlov, S.M.: Averaging of differential operators with almost periodic rapidly oscillating coefficients. Mat. Sb. (N.S.) 107(149)(2), 199–217, 317 (1978)Google Scholar
  29. 29.
    Marahrens, D., Otto, F.: Annealed estimates on the Green function. Probab. Theory Relat. Fields 163(3–4), 527–573 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Mourrat, J.-C., Nolen, J.: Scaling limit of the corrector in stochastic homogenization. Ann. Appl. Probab. (in press). arXiv:1502.07440
  31. 31.
    Mourrat, J.-C., Otto, F.: Correlation structure of the corrector in stochastic homogenization. Ann. Probab. 44(5), 3207–3233 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Naddaf, A., Spencer, T.: On homogenization and scaling limit of some gradient perturbations of a massless free field. Commun. Math. Phys. 183(1), 55–84 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Naddaf, A, Spencer, T.: Estimates on the variance of some homogenization problems (1998) (unpublished preprint)Google Scholar
  34. 34.
    Nolen, J.: Normal approximation for a random elliptic equation. Probab. Theory Relat. Fields 159(3–4), 661–700 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Nolen, J.: Normal approximation for the net flux through a random conductor. Stoch. Partial Differ. Equ. Anal. Comput. 4(3), 439–476 (2016)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: John, F., Lebowitz, J.L., Szasz, D. (eds.) Random Fields, vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pp. 835–873. North-Holland, Amsterdam (1981)Google Scholar
  37. 37.
    Rossignol, R.: Noise-stability and central limit theorems for effective resistance of random electric networks. Ann. Probab. 44(2), 1053–1106 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sheffield, S.: Gaussian free fields for mathematicians. Probab. Theory Relat. Fields 139(3–4), 521–541 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sjöstrand, J.: Correlation asymptotics and Witten Laplacians. Algebra i Analiz 8(1), 160–191 (1996)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Yurinskiĭ, V.V.: Averaging of symmetric diffusion in a random medium. Sibirsk. Mat. Zh. 27(4), 167–180, 215 (1986)MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Scott Armstrong
    • 1
    • 2
  • Tuomo Kuusi
    • 3
  • Jean-Christophe Mourrat
    • 4
  1. 1.Université Paris-DauphinePSL Research University, CNRS, UMR [7534], CEREMADEParisFrance
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  3. 3.Department of Mathematics and Systems AnalysisAalto UniversityEspooFinland
  4. 4.Ecole normale supérieure de Lyon, CNRSLyonFrance

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