The Annals of Probability

Correlation structure of the corrector in stochastic homogenization

Jean-Christophe Mourrat and Felix Otto

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Recently, the quantification of errors in the stochastic homogenization of divergence-form operators has witnessed important progress. Our aim now is to go beyond error bounds, and give precise descriptions of the effect of the randomness, in the large-scale limit. This paper is a first step in this direction. Our main result is to identify the correlation structure of the corrector, in dimension $3$ and higher. This correlation structure is similar to, but different from that of a Gaussian free field.

Article information

Source
Ann. Probab., Volume 44, Number 5 (2016), 3207-3233.

Dates
Received: February 2014
Revised: January 2015
First available in Project Euclid: 21 September 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1474462096

Digital Object Identifier
doi:10.1214/15-AOP1045

Mathematical Reviews number (MathSciNet)
MR3551195

Zentralblatt MATH identifier
1353.35040

Subjects
Primary: 35B27: Homogenization; equations in media with periodic structure [See also 74Qxx, 76M50] 35J15: Second-order elliptic equations 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 82D30: Random media, disordered materials (including liquid crystals and spin glasses)

Keywords
Homogenization random media two-point correlation function

Citation

Mourrat, Jean-Christophe; Otto, Felix. Correlation structure of the corrector in stochastic homogenization. Ann. Probab. 44 (2016), no. 5, 3207--3233. doi:10.1214/15-AOP1045. https://projecteuclid.org/euclid.aop/1474462096


Export citation

References

  • [1] Armstrong, S. N. and Smart, C. K. (2014). Quantitative stochastic homogenization of elliptic equations in nondivergence form. Arch. Ration. Mech. Anal. 214 867–911.
  • [2] Bal, G. (2008). Central limits and homogenization in random media. Multiscale Model. Simul. 7 677–702.
  • [3] Bal, G. (2010). Homogenization with large spatial random potential. Multiscale Model. Simul. 8 1484–1510.
  • [4] Bal, G. (2011). Convergence to homogenized or stochastic partial differential equations. Appl. Math. Res. Express. AMRX 2 215–241.
  • [5] Bal, G., Garnier, J., Gu, Y. and Jing, W. (2012). Corrector theory for elliptic equations with long-range correlated random potential. Asymptot. Anal. 77 123–145.
  • [6] Bal, G., Garnier, J., Motsch, S. and Perrier, V. (2008). Random integrals and correctors in homogenization. Asymptot. Anal. 59 1–26.
  • [7] Bal, G. and Gu, Y. (2015). Limiting models for equations with large random potential: A review. Commun. Math. Sci. 13 729–748.
  • [8] Bal, G. and Jing, W. (2011). Corrector theory for elliptic equations in random media with singular Green’s function. Application to random boundaries. Commun. Math. Sci. 9 383–411.
  • [9] Berger, N. and Biskup, M. (2007). Quenched invariance principle for simple random walk on percolation clusters. Probab. Theory Related Fields 137 83–120.
  • [10] Biskup, M. and Spohn, H. (2011). Scaling limit for a class of gradient fields with nonconvex potentials. Ann. Probab. 39 224–251.
  • [11] Boivin, D. (2009). Tail estimates for homogenization theorems in random media. ESAIM Probab. Stat. 13 51–69.
  • [12] Bourgeat, A. and Piatnitski, A. (2004). Approximations of effective coefficients in stochastic homogenization. Ann. Inst. Henri Poincaré Probab. Stat. 40 153–165.
  • [13] Caffarelli, L. A. and Souganidis, P. E. (2010). Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media. Invent. Math. 180 301–360.
  • [14] Caputo, P. and Ioffe, D. (2003). Finite volume approximation of the effective diffusion matrix: The case of independent bond disorder. Ann. Inst. Henri Poincaré Probab. Stat. 39 505–525.
  • [15] Conlon, J. G. and Fahim, A. (2015). Strong convergence to the homogenized limit of parabolic equations with random coefficients. Trans. Amer. Math. Soc. 367 3041–3093.
  • [16] Conlon, J. G. and Spencer, T. (2014). Strong convergence to the homogenized limit of elliptic equations with random coefficients. Trans. Amer. Math. Soc. 366 1257–1288.
  • [17] Figari, R., Orlandi, E. and Papanicolaou, G. (1982). Mean field and Gaussian approximation for partial differential equations with random coefficients. SIAM J. Appl. Math. 42 1069–1077.
  • [18] Funaki, T. (2005). Stochastic interface models. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1869 103–274. Springer, Berlin.
  • [19] Giacomin, G., Olla, S. and Spohn, H. (2001). Equilibrium fluctuations for $\nabla\phi$ interface model. Ann. Probab. 29 1138–1172.
  • [20] Gloria, A. and Mourrat, J.-C. (2012). Spectral measure and approximation of homogenized coefficients. Probab. Theory Related Fields 154 287–326.
  • [21] Gloria, A., Neukamm, S. and Otto, F. (2014). An optimal quantitative two-scale expansion in stochastic homogenization of discrete elliptic equations. ESAIM Math. Model. Numer. Anal. 48 325–346.
  • [22] Gloria, A., Neukamm, S. and Otto, F. (2015). Quantification of ergodicity in stochastic homogenization: Optimal bounds via spectral gap on Glauber dynamics. Invent. Math. 199 455–515.
  • [23] Gloria, A. and Otto, F. (2011). An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 779–856.
  • [24] Gloria, A. and Otto, F. (2012). An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 1–28.
  • [25] Gu, Y. and Bal, G. (2012). Random homogenization and convergence to integrals with respect to the Rosenblatt process. J. Differential Equations 253 1069–1087.
  • [26] Gu, Y. and Bal, G. (2015). Fluctuations of parabolic equations with large random potentials. Stoch. Partial Differ. Equ. Anal. Comput. 3 1–51.
  • [27] Helffer, B. and Sjöstrand, J. (1994). On the correlation for Kac-like models in the convex case. J. Stat. Phys. 74 349–409.
  • [28] Künnemann, R. (1983). The diffusion limit for reversible jump processes on $\textbf{Z}^{d}$ with ergodic random bond conductivities. Comm. Math. Phys. 90 27–68.
  • [29] Marahrens, D. and Otto, F. (2016). Annealed estimates on the Green function. Probab. Theory Related Fields 163 527–573.
  • [30] Miller, J. (2011). Fluctuations for the Ginzburg–Landau $\nabla\phi$ interface model on a bounded domain. Comm. Math. Phys. 308 591–639.
  • [31] Mourrat, J.-C. (2011). Variance decay for functionals of the environment viewed by the particle. Ann. Inst. Henri Poincaré Probab. Stat. 47 294–327.
  • [32] Mourrat, J.-C. (2014). Kantorovich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients. Probab. Theory Related Fields 160 279–314.
  • [33] Mourrat, J.-C. (2015). First-order expansion of homogenized coefficients under Bernoulli perturbations. J. Math. Pures Appl. (9) 103 68–101.
  • [34] Mourrat, J.-C. and Gu, Y. (2015). Scaling limit of fluctuations in stochastic homogenization. Unpublished manuscript. Available at arXiv:1503.00578.
  • [35] Mourrat, J.-C. and Nolen, J. (2015). Scaling limit of the corrector in stochastic homogenization. Unpublished manuscript. Available at arXiv:1502.07440.
  • [36] Naddaf, A. and Spencer, T. (1997). On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys. 183 55–84.
  • [37] Naddaf, A. and Spencer, T. (1998). Estimates on the variance of some homogenization problems. Unpublished manuscript.
  • [38] Papanicolaou, G. C. and Varadhan, S. R. S. (1981). Boundary value problems with rapidly oscillating random coefficients. In Random Fields, Vol. I, II (Esztergom, 1979). Colloquia Mathematica Societatis János Bolyai 27 835–873. North-Holland, Amsterdam.
  • [39] Sjöstrand, J. (1996). Correlation asymptotics and Witten Laplacians. Algebra i Analiz 8 160–191.
  • [40] Yurinskiĭ, V. V. (1986). Averaging of symmetric diffusion in a random medium. Sibirsk. Mat. Zh. 27 167–180, 215.