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Probability Theory and Related Fields

, Volume 163, Issue 3–4, pp 527–573 | Cite as

Annealed estimates on the Green function

  • Daniel MarahrensEmail author
  • Felix Otto
Article

Abstract

We consider a random, uniformly elliptic coefficient field \(a(x)\) on the \(d\)-dimensional integer lattice \(\mathbb {Z}^d\). We are interested in the spatial decay of the quenched elliptic Green function \(G(a;x,y)\). Next to stationarity, we assume that the spatial correlation of the coefficient field decays sufficiently fast to the effect that a logarithmic Sobolev inequality holds for the ensemble \(\langle \cdot \rangle \). We prove that all stochastic moments of the first and second mixed derivatives of the Green function, that is, \(\langle |\nabla _x G(x,y)|^p\rangle \) and \(\langle |\nabla _x\nabla _y G(x,y)|^p\rangle \), have the same decay rates in \(|x-y|\gg 1\) as for the constant coefficient Green function, respectively. This result relies on and substantially extends the one by Delmotte and Deuschel (Probab Theory Relat Fields 133:358–390, 2005), which optimally controls second moments for the first derivatives and first moments of the second mixed derivatives of \(G\), that is, \(\langle |\nabla _x G(x,y)|^2\rangle \) and \(\langle |\nabla _x\nabla _y G(x,y)|\rangle \). As an application, we are able to obtain optimal estimates on the random part of the homogenization error even for large ellipticity contrast.

Keywords

Stochastic homogenization Elliptic equations Green function Annealed estimates 

Mathematics Subject Classification

35B27 35J08 39A70 60H25 

References

  1. 1.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic partial differential equations and quasiconformal mappings in the plane, Princeton Mathematical Series 48. Princeton University Press, Princeton (2009)Google Scholar
  2. 2.
    Benjamini, I., Duminil-Copin, H., Kozma, G., Yadin, A.: Disorder, entropy and harmonic functions, Preprint (2011). arXiv:1111.4853
  3. 3.
    Biskup, M., Salvi, M., Wolff, T.: A central limit theorem for the effective conductance: I. Linear boundary data and small ellipticity contrasts, Preprint (2012). arXiv:1210.2371
  4. 4.
    Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23(2), 245–287 (1987)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Conlon, J.C., Giunti, A., Otto, F.: Green’s function for elliptic systems: Delmotte–Deuschel bounds, in preparationGoogle Scholar
  6. 6.
    Conlon, J.C., Naddaf, A.: On homogenization of elliptic equations with random coefficients. Electron. J. Probab. 9(5), 1–58 (2000)MathSciNetGoogle Scholar
  7. 7.
    De Giorgi, E.: Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3(3), 25–43 (1957)MathSciNetGoogle Scholar
  8. 8.
    Delmotte, T.: Inégalité de Harnack elliptique sur les graphes. Colloq. Math. 72(1), 19–37 (1997)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Delmotte, T., Deuschel, J.-D.: On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the \(\nabla \phi \) interface model. Probab. Theory Relat. Fields 133, 358–390 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  10. 10.
    Dolzmann, G., Hungerbühler, N., Müller, S.: Uniqueness and maximal regularity for nonlinear elliptic systems of n-Laplace type with measure valued right hand side. J. Reine Angew. Math. 520, 1–35 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Federbush, P.: Partially alternate derivation of a result by Nelson. J. Math. Phys. 10(1), 50–52 (1969)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Gloria, A.: Fluctuation of Solutions to Linear Elliptic Equations with Noisy Diffusion Coefficients. Comm. PDE 38(2), 304–338 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gloria, A., Marahrens, D.: Improved De Giorgi-Nash-Moser theory in the large, annealed estimates on the Green function and application to fluctuation estimates, in preparationGoogle Scholar
  14. 14.
    Gloria, A., Neukamm, S., Otto, F.: Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics, Max Planck Institute for Mathematics in the Sciences Preprint 3/(2013)Google Scholar
  15. 15.
    Gloria, A., Neukamm, S., Otto, F.: An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. Max Planck Institute for Mathematics in the Sciences Preprint 41/(2013)Google Scholar
  16. 16.
    Gloria, A., Otto, F.: An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39(3), 779–856 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Gross, L.: Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)CrossRefGoogle Scholar
  18. 18.
    Guionnet, A., Zegarlinski, B.: Lecture notes on Logarithmic Sobolev Inequalities. Lecture Notes Math. 1801, 1–134 (2003)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lawler, G. F., Limic, V.: Random walk: a modern introduction, Cambridge Stud. in Adv. Math. 123, CUP, Cambridge (2010)Google Scholar
  20. 20.
    Ledoux, M.: The concentration of measure phenomenon, Math. Surveys and Monographs 89, AMS, Providence, RI (2001)Google Scholar
  21. 21.
    Lieb, E. H., Loss, M.: Analysis, graduate stud. in Math. 14, 2nd edn, AMS, Providence, RI (2001)Google Scholar
  22. 22.
    Naddaf, A., Spencer, T.: Estimates on the variance of some homogenization problems, unpublishedGoogle Scholar
  23. 23.
    Naddaf, A., Spencer, T.: On homogenization and scaling limit of some gradient perturbation of a massless free field. Commun. Math. Phys. 183, 55–84 (1997)zbMATHMathSciNetCrossRefGoogle Scholar
  24. 24.
    Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)Google Scholar
  25. 25.
    Nelson, F.: A quartic interaction in two dimensions, in Mathematical theory of elementary particles. (eds.) Goodman, R., Segal, I. MIT Press, Cambridge, pp. 69–73 (1966)Google Scholar
  26. 26.
    Nelson, F.: The free Markoff field. J. Funct. Anal. 12, 211–227 (1973)zbMATHCrossRefGoogle Scholar
  27. 27.
    Nolen, J., Normal approximation for a random elliptic equation, Preprint (2011). http://math.duke.edu/nolen/preprints/ellipfluctper_rev
  28. 28.
    Riesz, F., Sz.-Nagy, B.: Functional Analysis, Dover Books on Adv. Math. Dover Publ. Inc., New York (1990)Google Scholar
  29. 29.
    Rossignol, R.: Noise-stability and central limit theorems for effective resistance of random electric networks. arXiv:1206.3856 (2012)
  30. 30.
    Stroock, D., Zegarlinski, B.: The logarithmic Sobolev inequality for discrete spin systems on a lattice. Commun. Math. Phys. 149, 175–193 (1992)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany

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