Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Normal approximation for a random elliptic equation

Abstract

We consider solutions of an elliptic partial differential equation in \(\mathbb{R }^d\) with a stationary, random conductivity coefficient that is also periodic with period \(L\). Boundary conditions on a square domain of width \(L\) are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary condition. It is known that in the limit \(L \rightarrow \infty \), this quantity converges to a deterministic constant, almost surely. Our main result is that the law of this random variable is very close to that of a normal random variable, if the domain size \(L\) is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincaré inequality developed recently by Chatterjee.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Bal, G., Garnier, J., Motsch, S., Perrier, V.: Random integrals and correctors in homogenization. Asympt. Anal. 59, 1–26 (2008)

  2. 2.

    Biskup, M., Salvi, M., Wolff, T.: A central limit theorem for the effective conductance: I. linear boundary data and small ellipticity contrasts. Preprint arXiv:1210.2371 (2012)

  3. 3.

    Biskup, M.: Recent progress on the random conductance model. Probab. Surv. 8, 294–373 (2011)

  4. 4.

    Boivin, D.: Tail estimates for homogenization theorems in random media. ESAIM Probab. Stat. 13, 51–69 (2009)

  5. 5.

    Bourgeat, A., Piatnitski, A.: Approximations of effective coefficients in stochastic homogenization. Ann. I.H. Poincaré 40, 153–165 (2004)

  6. 6.

    Bourgeat, A., Piatnitski, A.: Estimates in probability of the residual between the random and the homogenized soutions of one-dimensional second-order operator. Asympt. Anal. 21, 303–315 (1999)

  7. 7.

    Chatterjee, S.: Fluctuations of eigenvalues and second order Poincaré inequalities. Prob. Theory Relat. Fields 143, 1–40 (2009)

  8. 8.

    Conlon, J., Naddaf, A.: Green’s functions for elliptic and parabolic equations with random coefficients. N. Y. J. Math. 6, 153–225 (2000)

  9. 9.

    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)

  10. 10.

    Giaquinta, M., Modica, G.: Regularity results for some classes of higher order non linear elliptic systems. J. Reine Angew. Math. 311/312, 145–169 (1979)

  11. 11.

    Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Springer, New York (2001)

  12. 12.

    Gloria, A., Neukamm, S., Otto F.: Convergence rate for the approximation of effective coefficients by periodization in stochastic homogenization of discrete elliptic equations. In preparation (2011)

  13. 13.

    Gloria, A., Otto, F.: An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39, 779–856 (2011)

  14. 14.

    Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of differential operators and integral functionals. Springer, New York (1994)

  15. 15.

    Komorowski, T., Ryzhik, L.: A sharp bound on the \(L^2\) norm of the solution of a random elliptic difference equation. Commun. Math. Sci. 9, 607–622 (2011)

  16. 16.

    Kozlov, S.M.: The averaging of random operators. Math. USSR Sb. 109, 188–202 (1979)

  17. 17.

    Meyers, N.: An \(L^p\)-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa Classe Sci. 17, 189–206 (1963)

  18. 18.

    Naddaf, A., Spencer, T.: Estimates on the variance of some homogenization problems. Unpublished manuscript (1998)

  19. 19.

    Owhadi, H.: Approximation of the effective conductivity of ergodic media by periodization. Probab. Theory Relat. Fields 125, 225–258 (2003)

  20. 20.

    Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In: Random Fields Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, 27, pp. 835–873. North Holland, New York (1981)

  21. 21.

    Rossignol, R.: Noise-stability and central limit theorems for effective resistance of random electric networks. Preprint arXiv:1206.3856 (2012)

  22. 22.

    Steele, J.M.: An Efron–Stein inequality for nonsymmetric statistics. Ann. Stat. 14, 753–758 (1986)

  23. 23.

    Torquato, S.: Random Heterogeneous Materials. Springer, New York (2001)

  24. 24.

    Wehr, J.: A lower bound on the variance of conductance in random resistor networks. J. Stat. Phys. 86(5,6), 1359–1365 (1997)

  25. 25.

    Wehr, J., Aizenman, M.: Fluctuations of extensive functions of quenched random couplings. J. Stat. Phys. 60, 287–306 (1990)

  26. 26.

    Yurinskii, V.V.: Averaging of symmetric diffusion in a random medium. Sib. Math. J. 4, 603–613 (1986)

Download references

Acknowledgments

I am grateful to Felix Otto whose insight and helpful comments led to improvement of the main argument. I also thank Jan Wehr, Sourav Chatterjee, and Jonathan Mattingly for stimulating discussion in the early stages of this work. The anonymous referees also provided very helpful comments. The author’s research is partially funded by grant DMS-1007572 from the US National Science Foundation.

Author information

Correspondence to James Nolen.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Nolen, J. Normal approximation for a random elliptic equation. Probab. Theory Relat. Fields 159, 661–700 (2014). https://doi.org/10.1007/s00440-013-0517-9

Download citation

Mathematics Subject Classification

  • 35B27
  • 35J15
  • 60F05
  • 60H25