Skip to main content

High order correctors and two-scale expansions in stochastic homogenization

Abstract

In this paper, we study high order correctors in stochastic homogenization. We consider elliptic equations in divergence form on \(\mathbb {Z}^d\), with the random coefficients constructed from i.i.d. random variables. We prove moment bounds on the high order correctors and their gradients under dimensional constraints. It implies the existence of stationary correctors and stationary gradients in high dimensions. As an application, we prove a two-scale expansion of the solutions to the random PDE, which identifies the first and higher order random fluctuations in a strong sense.

This is a preview of subscription content, access via your institution.

References

  1. Armstrong, S.N., Smart, C.K.: Quantitative stochastic homogenization of convex integral functionals. Annales scientifiques de l’Ecole normale supérieure 48, 423–481 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  2. Armstrong, S.N., Kuusi, T., Mourrat, J.-C.: Mesoscopic higher regularity and subadditivity in elliptic homogenization. Comm. Math. Phys. 347, 315–361 (2016)

  3. Armstrong, S.N., Kuusi, T., Mourrat, J.-C.: Scaling limits of energies and correctors (2016, arXiv preprint). arXiv:1603.03388

  4. Armstrong, S.N., Kuusi, T., Mourrat, J.-C.: The additive structure of elliptic homogenization (2016, preprint). arXiv:1602.00512

  5. Armstrong, S.N., Mourrat, J.-C.: Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal. 219, 255–348 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bella, P., Fehrman, B., Fischer, J., Otto, F.: Stochastic homogenization of linear elliptic equations: high-order error estimates in weak norms via second-order correctors (2016, preprint). arXiv:1609.01528

  7. 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, 701–731 (2014)

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  9. Caffarelli, L., Souganidis, P.: Rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media. Invent. Math. 180, 301–360 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Conlon, J., Naddaf, A.: On homogenization of elliptic equations with random coefficients. Electron. J. Probab. 5, 1–58 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Conlon, J., Spencer, T.: Strong convergence to the homogenized limit of elliptic equations with random coefficients. Trans. Am. Math. Soc. 366, 1257–1288 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  12. Deuschel, J.-D., Delmotte, T.: 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)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gérard-Varet, D., Masmoudi, N.: Homogenization in polygonal domains. J. Eur. Math. Soc. (JEMS) 13, 1477–1503 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gérard-Varet, D., Masmoudi, N.: Homogenization and boundary layers. Acta Math. 209, 1–46 (2012)

  15. Duerinckx, M., Gloria, A., Otto, F.: The structure of fluctuations in stochastic homogenization (2016, preprint). arXiv:1602.01717

  16. Fischer, J., Otto, F.: A higher-order large-scale regularity theory for random elliptic operators (2015). arXiv:1503.07578

  17. Gloria, A.: Fluctuation of solutions to linear elliptic equations with noisy diffusion coefficients. Commun. Partial Differ. Equ. 38, 304–338 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  18. Gloria, A., Marahrens, D.: Annealed estimates on the Green functions and uncertainty quantification, Ann. Inst. H. Poincar Anal. Non Linaire. 33, 1153–1197 (2016)

  19. Gloria, A., Neukamm, S., Otto, F.: Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on glauber dynamics. Invent. Math. 199, 455–515 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  20. 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, 325–346 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  21. Gloria, A., Neukamm, S., Otto, F.: A regularity theory for random elliptic operators (2014, preprint). arXiv:1409.2678

  22. Gloria, A., Nolen, J.: A quantitative central limit theorem for the effective conductance on the discrete torus. Commun. Pure Appl. Math. 69, 2304–2348 (2016)

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

    Article  MATH  MathSciNet  Google Scholar 

  24. Gloria, A., Otto, F.: An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22, 1–28 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Gloria, A., Otto, F.: Quantitative results on the corrector equation in stochastic homogenization. J. Eur. Math. Soc. (2015, to appear). arXiv:1409.0801

  26. Gloria, A., Otto, F.: The corrector in stochastic homogenization: Near-optimal rates with optimal stochastic integrability (2015, preprint). arXiv:1510.08290

  27. Gu, Y.: A central limit theorem for fluctuations in 1d stochastic homogenization. Stoch PDE: Anal Comp. 4, 713–745 (2016)

  28. Gu, Y., Mourrat, J.-C.: Pointwise two-scale expansion for parabolic equations with random coefficients. Probab. Theory Relat. Fields. 166, 585–618 (2016)

  29. Gu, Y., Mourrat, J.-C.: Scaling limit of fluctuations in stochastic homogenization. Multiscale Model. Simul. 14, 452–481 (2016)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kozlov, S.M.: Averaging of random operators. Matematicheskii Sbornik 151, 188–202 (1979)

    MathSciNet  Google Scholar 

  31. Kunnemann, R.: The diffusion limit for reversible jump processes on \({\mathbb{Z}}^d\) with ergodic random bond conductivities. Commun. Math. Phys. 90, 27–68 (1983)

    Article  MathSciNet  Google Scholar 

  32. Marahrens, D., Otto, F.: Annealed estimates on the Green’s function. Probab. Theory Relat. Fields 163, 527–573 (2015)

    Article  MATH  MathSciNet  Google Scholar 

  33. Mourrat, J.-C.: Variance decay for functionals of the environment viewed by the particle. Ann. Inst. H. Poincaré Probab. Statist 47, 294–327 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  34. Mourrat, J.-C.: Kantorovich distance in the martingale clt and quantitative homogenization of parabolic equations with random coefficients. Probab. Theory Relat. Fields 160, 279–314 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  35. Mourrat, J.-C., Nolen, J.: Scaling limit of the corrector in stochastic homogenization. Ann. Appl. Probab. (2016, to appear). arXiv:1502.07440

  36. Mourrat, J.-C., Otto, F.: Correlation structure of the corrector in stochastic homogenization. Ann. Probab. 44, 3207–3233 (2016)

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

  38. Nolen, J.: Normal approximation for a random elliptic equation. Probab. Theory Relat. Fields 159, 661–700 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  39. Nolen, J.: Normal approximation for the net flux through a random conductor. SPDE: Anal. Comput. 4, 439–476 (2016)

    MATH  MathSciNet  Google Scholar 

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

  41. Rossignol, R.: Noise-stability and central limit theorems for effective resistance of random electric networks. Ann. Probab. 44, 1053–1106 (2016)

  42. Yurinskii, V.: Averaging of symmetric diffusion in random medium. Sib. Math. J. 27, 603–613 (1986)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the anonymous referee for several helpful suggestions. The author’s research is partially funded by Grant DMS-1613301 from the US National Science Foundation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Yu Gu.

Appendices

Appendix A: The existence of stationary correctors: removing the massive term

The goal in this section is to show that for the corrector equation (2.8) defined when \(d\geqslant 2n-1\)

$$\begin{aligned} (\lambda +\nabla ^*a(y)\nabla ) \psi _n^\lambda (y)=F(y)=\left\{ \begin{array}{l} \nabla _i^* (a_i(y)\psi _{n-1}(y)),\\ a_i(y) \nabla _i\psi _{n-1}(y)-\langle a_i(y)\nabla _i \psi _{n-1}(y)\rangle ,\\ a_{i}(y)\psi _{n-2}(y)-\langle a_{i}(y)\psi _{n-2}(y)\rangle , \end{array} \right. \end{aligned}$$
(A.1)

the uniform estimates \(\Vert \psi _n^\lambda \Vert _p\lesssim 1\) and \(\Vert \nabla \psi _n^\lambda \Vert _p\lesssim 1\) implies the existence of a stationary corrector and a stationary gradient, respectively.

In Sect. 2.3.1, we proved that when \(d\geqslant 2n-1\), there exists a stationary random field \(\Psi \) such that \(\Psi (0)\in L^2(\Omega )\) and

$$\begin{aligned} F=\nabla ^* \Psi . \end{aligned}$$

The rest is standard, and we present it for the convenience of the reader.

We first introduce some notations. Let \(\omega =(\omega _e)_{e\in \mathbb {B}}\in \Omega \) denote the sample point, and for \(x\in \mathbb {Z}^d\), we define the shift operator \(\tau _x\) on \(\Omega \) by \((\tau _x \omega )_e=\omega _{x+e}\), where \(x+e:=(x+\underline{e},x+\bar{e})\) is the edge obtained by shifting e by x. Since \(\{\omega _e\}_{e\in \mathbb {B}}\) are i.i.d., \(\{\tau _x\}_{x\in \mathbb {Z}^d}\) is a group of measure-preserving transformations. We can define the operator

$$\begin{aligned} T_x f(\zeta )=f(\tau _x\zeta ) \end{aligned}$$

for any measurable function f on \(\Omega \), and the generators of \(T_x\), denoted by \(\{D_i\}_{i=1}^d\), are defined by \(D_if:=T_{e_i}f-f\). The adjoint \(D_i^*\) is defined by \(D_i^* f:=T_{-e_i}f-f\). We denote the gradient on \(\Omega \) by \(D=(D_1,\ldots ,D_d)\) and the divergence \(D^*F:=\sum _{i=1}^d D_i^*F_i\) for \(F:\Omega \rightarrow \mathbb {R}^d\).

Lemma A.1

Let \(\phi _\lambda \) solve \((\lambda +\nabla ^* a\nabla )\phi _\lambda =\nabla ^*\Psi \) with \(\Psi \) a stationary random field on \(\mathbb {Z}^d\) such that \(\Psi (0)\in L^2(\Omega )\). For the equation

$$\begin{aligned} \nabla ^*a\nabla \phi =\nabla ^*\Psi , \end{aligned}$$
(A.2)
  1. (i)

    there exists a random field \(\phi \) solving (A.2) such that \(\nabla \phi \) is stationary and \(\nabla \phi (0)\in L^2(\Omega )\).

  2. (ii)

    if \(\langle |\phi _\lambda |^2\rangle \lesssim 1\), then there exists a stationary random field \(\phi \) solving (A.2) such that \(\phi (0)\in L^2(\Omega )\).

Proof

Part (i) comes from [31, Theorem 3]. For Part (ii), let \(\tilde{a}(\omega )=(\omega _{e_1},\ldots ,\omega _{e_d})\) and \(\tilde{\Psi }\in L^2(\Omega )\) so that \(\Psi (x)=\tilde{\Psi }(\tau _x\omega )\), we lift the equation to the probability space

$$\begin{aligned} (\lambda +D^*\tilde{a} D)\tilde{\phi }_\lambda =D^*\tilde{\Psi }, \end{aligned}$$

so it is clear that \(\phi _\lambda (x)=\tilde{\phi }_\lambda (\tau _x\omega )\). Since \(\langle |\tilde{\phi }_\lambda |^2\rangle \lesssim 1\), we can extract a subsequence \(\tilde{\phi }_\lambda \rightarrow \tilde{\phi }\) weakly in \(L^2\), which implies \(D\tilde{\phi }_\lambda \rightarrow D\tilde{\phi }\) weakly in \(L^2(\Omega )\).

For any \(G\in L^2(\Omega )\), we have

$$\begin{aligned} \lambda \langle \tilde{\phi }_\lambda G\rangle +\langle \tilde{a}D\tilde{\phi }_\lambda DG\rangle =\langle D^*\tilde{\Psi }G\rangle , \end{aligned}$$

and sending \(\lambda \rightarrow 0\) leads to

$$\begin{aligned} \langle \tilde{a}D\tilde{\phi }DG\rangle =\langle D^*\tilde{\Psi }G\rangle , \end{aligned}$$

so \(D^*\tilde{a}D\tilde{\phi }=D^*\tilde{\Psi }\). Now we define \(\phi (x)=\tilde{\phi }(\tau _x\omega )\), and it is clear that

$$\begin{aligned} \nabla ^*a\nabla \phi =\nabla ^*\Psi . \end{aligned}$$

The proof is complete. \(\square \)

Appendix B: Finite difference approximation

The following are some classical results and we present it for the convenience of the reader. We recall the classical Green’s function estimates:

$$\begin{aligned} \mathcal {G}^\alpha (x,y)\lesssim \frac{e^{-c\sqrt{\alpha }|x-y|}}{|x-y|_*^{d-2}} \end{aligned}$$
(B.1)

for some \(c>0\), with \(\mathcal {G}^\alpha \) the Green’s function of \(\alpha +\nabla ^*\nabla \) on \(\mathbb {Z}^d\). Let \(\mathcal {G}_\varepsilon ^\alpha (x,y)\) be the Green’s function of \(\alpha +\nabla _\varepsilon ^*\nabla _\varepsilon \) on \(\varepsilon \mathbb {Z}^d\), it is clear that

$$\begin{aligned} \mathcal {G}_\varepsilon ^\alpha (x,y)=\varepsilon ^2\mathcal {G}^{\varepsilon ^2 \alpha }\left( \frac{x}{\varepsilon },\frac{y}{\varepsilon }\right) . \end{aligned}$$

Lemma B.1

Let \(u_\varepsilon \) solve

$$\begin{aligned} (\alpha +\nabla _\varepsilon ^*\nabla _\varepsilon )u_\varepsilon =f_\varepsilon , \end{aligned}$$

on \(\varepsilon \mathbb {Z}^d\), with \(f_\varepsilon \) satisfying (i) there exists \(\lambda >0\) s.t. \(|f_\varepsilon (x)|\lesssim e^{-\lambda |x|}\), (ii) all discrete derivatives of \(f_\varepsilon \) satisfies (i), then for \(v_\varepsilon =u_\varepsilon \) or any derivative of \(u_\varepsilon \), we have

$$\begin{aligned} |v_\varepsilon (x)|\lesssim e^{-c|x|} \end{aligned}$$

for constant \(c>0\) independent of \(\varepsilon >0,x\in \varepsilon \mathbb {Z}^d\).

Proof

It suffices to consider the case \(v_\varepsilon =u_\varepsilon \). For derivatives, e.g., \(v_\varepsilon =\nabla _{\varepsilon ,i} u_\varepsilon \), using the fact that \(\nabla _{\varepsilon ,i}\) commutes with \(\nabla _{\varepsilon ,j}\) and \(\nabla _{\varepsilon ,j}^*\), we have

$$\begin{aligned} (\alpha +\nabla _\varepsilon ^*\nabla _\varepsilon )v_\varepsilon =\nabla _{\varepsilon ,i}f_\varepsilon , \end{aligned}$$

so we only need to apply the result when \(v_\varepsilon =u_\varepsilon \).

For \(u_\varepsilon (x)\), we use the Green’s function representation

$$\begin{aligned} u_\varepsilon (x)=\sum _{y\in \varepsilon \mathbb {Z}^d} \mathcal {G}^\alpha _\varepsilon (x,y)f(y)=\sum _{y\in \varepsilon \mathbb {Z}^d}\varepsilon ^2 \mathcal {G}^{\varepsilon ^2\alpha }\left( \frac{x}{\varepsilon },\frac{y}{\varepsilon }\right) f(y), \end{aligned}$$

and by (B.1), we have

$$\begin{aligned} \varepsilon ^2 \mathcal {G}^{\varepsilon ^2\alpha }\left( \frac{x}{\varepsilon },\frac{y}{\varepsilon }\right) \lesssim \varepsilon ^2 \frac{e^{-\rho |x-y|}}{|\frac{x-y}{\varepsilon }|_*^{d-2}} \end{aligned}$$

for some constant \(\rho >0\). Thus

$$\begin{aligned} |u_\varepsilon (x)|\lesssim \varepsilon ^d\sum _{y\in \varepsilon \mathbb {Z}^d, y\ne x} \frac{e^{-\rho |x-y|}}{|x-y|^{d-2}}e^{-\lambda |y|}+\varepsilon ^2e^{-\lambda |x|}. \end{aligned}$$

For the first term on the r.h.s. of the above expression, we only need to decompose the summation into \(\sum _{|y|<|x|/2}\) and \(\sum _{|y|\geqslant |x|/2}\) to complete the proof. \(\square \)

Lemma B.2

Let \(u_\varepsilon \) solve

$$\begin{aligned} (\alpha +\nabla _\varepsilon ^* \nabla _\varepsilon )u_\varepsilon =f_\varepsilon \end{aligned}$$

on \(\varepsilon \mathbb {Z}^d\), with \(f_\varepsilon \) satisfying (i) there exists \(\lambda >0\) s.t. \(|f_\varepsilon (x)|\lesssim e^{-\lambda |x|}\), (ii) there exists a continuous function \(\bar{f}:\mathbb {R}^d\rightarrow \mathbb {R}\) such that \(\Vert f_\varepsilon -\bar{f}\Vert _{2,\varepsilon }\rightarrow 0\) as \(\varepsilon \rightarrow 0\), then for \(\bar{u}\) solving

$$\begin{aligned} (\alpha -\Delta )\bar{u}=\bar{f} \end{aligned}$$

on \(\mathbb {R}^d\), we have \(\Vert u_\varepsilon -\bar{u}\Vert _{2,\varepsilon }\rightarrow 0\) as \(\varepsilon \rightarrow 0\).

Proof

We first note that \(|\bar{f}(x)|\lesssim e^{-\lambda |x|}\) for \(x\in \mathbb {R}^d\). The rest of the proof is decomposed into three steps.

Step 1. We write

$$\begin{aligned} u_\varepsilon (x)=\sum _{y\in \varepsilon \mathbb {Z}^d} \varepsilon ^2 \mathcal {G}^{\varepsilon ^2\alpha }\left( \frac{x}{\varepsilon },\frac{y}{\varepsilon }\right) f_\varepsilon (y), \end{aligned}$$

and define

$$\begin{aligned} \bar{u}_\varepsilon (x):=\sum _{y\in \varepsilon \mathbb {Z}^d} \varepsilon ^2 \mathcal {G}^{\varepsilon ^2\alpha }\left( \frac{x}{\varepsilon },\frac{y}{\varepsilon }\right) \bar{f}(y) \end{aligned}$$

on \(\varepsilon \mathbb {Z}^d\). The goal is to show \(\Vert u_\varepsilon -\bar{u}_\varepsilon \Vert _{2,\varepsilon }\rightarrow 0\) as \(\varepsilon \rightarrow 0\). By (B.1), we deduce

$$\begin{aligned} |u_\varepsilon (x)-\bar{u}_\varepsilon (x)|\lesssim & {} \sum _{y\in \varepsilon \mathbb {Z}^d} \varepsilon ^2 \frac{e^{-\rho |x-y|}}{|\frac{x-y}{\varepsilon }|_*^{d-2}}|f_\varepsilon (y)-\bar{f}(y)|\\\lesssim & {} \varepsilon ^d \sum _{y\in \varepsilon \mathbb {Z}^d,y\ne x} \frac{e^{-\rho |x-y|}}{|x-y|^{d-2}}|f_\varepsilon (y)-\bar{f}(y)|+\varepsilon ^2|f_\varepsilon (x)-\bar{f}(x)|, \end{aligned}$$

thus

$$\begin{aligned}&\varepsilon ^d\sum _{x\in \varepsilon \mathbb {Z}^d} |u_\varepsilon (x)-\bar{u}_\varepsilon (x)|^2 \\&\quad \lesssim \varepsilon ^d \sum _{x\in \varepsilon \mathbb {Z}^d} \left( \sum _{y\in \varepsilon \mathbb {Z}^d,y\ne x} \varepsilon ^d\frac{e^{-\rho |x-y|}}{|x-y|^{d-2}} \sum _{y\in \varepsilon \mathbb {Z}^d,y\ne x}\varepsilon ^d\frac{e^{-\rho |x-y|}}{|x-y|^{d-2}}|f_\varepsilon (y)-\bar{f}(y)|^2\right) \\&\qquad +\varepsilon ^d\varepsilon ^4\sum _{x\in \varepsilon \mathbb {Z}^d}|f_\varepsilon (x)-\bar{f}(x)|^2:=I_1+I_2 \end{aligned}$$

It is clear by assumption that \(I_2\rightarrow 0\), and

$$\begin{aligned} I_1\lesssim & {} \varepsilon ^d \sum _{x\in \varepsilon \mathbb {Z}^d}\sum _{y\in \varepsilon \mathbb {Z}^d,y\ne x}\varepsilon ^d\frac{e^{-\rho |x-y|}}{|x-y|^{d-2}}|f_\varepsilon (y)-\bar{f}(y)|^2\\\lesssim & {} \varepsilon ^d \sum _{y\in \varepsilon \mathbb {Z}^d}|f_\varepsilon (y)-\bar{f}(y)|^2\rightarrow 0 \end{aligned}$$

as \(\varepsilon \rightarrow 0\).

Step 2. Define

$$\begin{aligned} \tilde{u}_\varepsilon (x)=\varepsilon ^d\sum _{y\in \varepsilon \mathbb {Z}^d,y\ne x} \mathscr {G}^\alpha (x,y)\bar{f}(y) \end{aligned}$$

on \(\varepsilon \mathbb {Z}^d\), with \(\mathscr {G}^\alpha \) the continuous Green’s function of \(\alpha -\Delta \) in \(\mathbb {R}^d\). The goal is to show \(\Vert \bar{u}_\varepsilon -\tilde{u}_\varepsilon \Vert _{2,\varepsilon }\rightarrow 0\) as \(\varepsilon \rightarrow 0\). We first have

$$\begin{aligned} |\bar{u}_\varepsilon (x)-\tilde{u}_\varepsilon (x)|\lesssim \sum _{y\in \varepsilon \mathbb {Z}^d,y\ne x} \left| \varepsilon ^2 \mathcal {G}^{\varepsilon ^2\alpha }\left( \frac{x}{\varepsilon },\frac{y}{\varepsilon }\right) -\varepsilon ^2 \mathscr {G}^{\varepsilon ^2\alpha }\left( \frac{x}{\varepsilon },\frac{y}{\varepsilon }\right) \right| |\bar{f}(y)|+\varepsilon ^2|\bar{f}(x)|. \end{aligned}$$

where we used the scaling property \(\mathscr {G}^{\varepsilon ^2\alpha }(x/\varepsilon ,y/\varepsilon )=\varepsilon ^{d-2}\mathscr {G}^\alpha (x,y)\). By [11, Lemma 3.1],

$$\begin{aligned} |\mathcal {G}^\lambda (x,y)-\mathscr {G}^\lambda (x,y)|\lesssim \frac{e^{-c \sqrt{\lambda }|x-y|}}{|x-y|_*^{d-1}} \end{aligned}$$

for some \(c>0\), so

$$\begin{aligned} |\bar{u}_\varepsilon (x)-\tilde{u}_\varepsilon (x)|\lesssim \sum _{y\in \varepsilon \mathbb {Z}^d,y\ne x} \varepsilon ^{d+1} \frac{e^{-c\sqrt{\alpha }|x-y|}}{|x-y|^{d-1}}|\bar{f}(y)|+\varepsilon ^2|\bar{f}(x)| \lesssim \varepsilon e^{-c|x|} \end{aligned}$$

for some \(c>0\), which implies \(\Vert \bar{u}_\varepsilon -\tilde{u}_\varepsilon \Vert _{2,\varepsilon }\rightarrow 0\).

Step 3. We first extend \(\tilde{u}_\varepsilon (x)\) from \(\varepsilon \mathbb {Z}^d\) to \(\mathbb {R}^d\) such that \(\tilde{u}_\varepsilon (x)=\tilde{u}_\varepsilon ([x]_\varepsilon )\) with \([x]_\varepsilon \) the \(\varepsilon \)-integer part of x. Then we consider

$$\begin{aligned} \Vert \tilde{u}_\varepsilon -\bar{u}\Vert _{2,\varepsilon }^2=\varepsilon ^d\sum _{x\in \varepsilon \mathbb {Z}^d} |\tilde{u}_\varepsilon (x)-\bar{u}(x)|^2 =\int _{\mathbb {R}^d} |\tilde{u}_\varepsilon (x)-\bar{u}([x]_\varepsilon )|^2dx. \end{aligned}$$

It is clear that \(\int _{\mathbb {R}^d} |\bar{u}([x]_\varepsilon )-\bar{u}(x)|^2dx\rightarrow 0\) as \(\varepsilon \rightarrow 0\), so we only need to show

$$\begin{aligned} \int _{\mathbb {R}^d} |\tilde{u}_\varepsilon (x)-\bar{u}(x)|^2dx\rightarrow 0. \end{aligned}$$

Since \(\bar{u}(x)=\int _{\mathbb {R}^d} \mathscr {G}^\alpha (x,y)\bar{f}(y)dy\), we have \(\tilde{u}_\varepsilon (x)\rightarrow \bar{u}(x)\) for \(x\in \mathbb {R}^d\). Now by Lemma B.1, \(|\tilde{u}_\varepsilon (x)|+|\bar{u}(x)|\lesssim e^{-c|x|}\) for some \(c>0\), so by dominated convergence theorem, the proof is complete. \(\square \)

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gu, Y. High order correctors and two-scale expansions in stochastic homogenization. Probab. Theory Relat. Fields 169, 1221–1259 (2017). https://doi.org/10.1007/s00440-016-0750-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00440-016-0750-0

Keywords

  • Quantitative homogenization
  • High order corrector
  • Two-scale expansion
  • Random fluctuation

Mathematics Subject Classification

  • 35B27
  • 35K05
  • 34E05
  • 35R60