1 Introduction and Statement of Rigorous Result

1.1 Uniformly Elliptic Coefficient Fields

The basic objects of this paper are \(\lambda \)-uniformly elliptic tensor fields \(a=a(x)\) (that are not necessarily symmetric) in d-dimensional space, by which we mean that for all points x

$$\begin{aligned} \xi \cdot a(x)\xi \ge \lambda |\xi |^2\quad \text{ and }\quad \xi \cdot a(x)\xi \ge |a(x)\xi |^2\quad \text{ for } \text{ all }\;x,\xi \in {\mathbb {R}}^d. \end{aligned}$$

Note that the second condition implies \(|a(x)\xi |\) \(\le |\xi |\). These conditions are not equivalent unless a(x) is symmetric; however, the form of the bounds in (1) is the one preserved under homogenization [58, Definition 6]. Such a tensor field a gives rise to the heterogeneous elliptic operator \(-\nabla \cdot a \nabla \) acting on functions u.Footnote 1

Homogenization means assimilating the effective, i.e., large scale, behavior of a heterogeneous medium to a homogeneous one, as described by the constant tensor \(a_{\textrm{hom}}\). By this, one means that the difference of the solution operators \((-\nabla \cdot a\nabla )^{-1}\) \(-(-\nabla \cdot a_{\textrm{hom}}\nabla )^{-1}\) converges to zero when applied to functions f varying only on scales \(L\uparrow \infty \). Homogenization is known to take place in a number of situations, see [58] for a general notion, e. g. when the coefficient field a is periodic or when it is sampled from a stationary and ergodic ensemble \(\langle \cdot \rangle \). While we are interested in the latter, it is convenient to introduce the representative volume element (RVE) method in the context of the former.

1.2 The RVE Method

Unless \(d=1\), there is no explicit formula that allows to compute in practice \(a_{\textrm{hom}}\) for a general ensemble \(\langle \cdot \rangle \). Early work treated specific ensembles that admit asymptotic explicit formulas in limiting regimes, like spherical inclusions covering a low volume fraction in [49, p. 365]. Explicit upper and lower bounds on \(a_{\textrm{hom}}\) in terms of features of the ensemble \(\langle \cdot \rangle \) play a major role in the engineering literature, see for instance [50, 59]. On the contrary, the RVE method is a computational method to obtain convergent approximations to \(a_{\textrm{hom}}\) for a general the ensemble \(\langle \cdot \rangle \). As the name “volume element” indicates, it is based on samples a of \(\langle \cdot \rangle \) in a (computational) domain, typically a cube of side-length L. It consists in inverting \(-\nabla \cdot a\nabla \) for d (representative) boundary conditions. The question of the appropriate size L of the RVE evolved from a philosophical one in [36] (large enough to be statistically typical and so that boundary effects are dominated by bulk effects) towards a more practical one in [17] (just large enough so that the statistical properties relevant for the physical quantity \(a_\textrm{hom}\) are captured). The convergence of the method has been extensively investigated by numerical experiments in the engineering literature, see some references below. In this paper, we rigorously analyze some aspects of the convergence for a certain class of ensembles \(\langle \cdot \rangle \).

We now introduce the RVE method. Suppose (momentarily) that the coefficient field a is L-periodic, meaning that \(a(x+Lk)=a(x)\) for all x and \(k\in {\mathbb {Z}}^d\). Given a Cartesian coordinate direction \(i=1,\ldots ,d\) and denoting by \(e_i\) the unit vector in the i-th direction, we define (up to additive constants) \(\phi _i^{(1)}\) as the L-periodic solution of

$$\begin{aligned} -\nabla \cdot a(\nabla \phi _i^{(1)}+e_i)=0. \end{aligned}$$

The function \(\phi _i^{(1)}\) is called first-order corrector, because it additively corrects the affine coordinate function \(x_i\) in such a way that the resulting function \(x\mapsto x_i+\phi _i^{(1)}(x)\) is a-harmonic, by which we understand that it vanishes under application of \(-\nabla \cdot a\nabla \).

Let us momentarily adopt the language of a conducting medium: On the microscopic level, multiplication with the tensor field a converts the electric field into the electric flux. On the macroscopic level, it is \(a_{\textrm{hom}}\) that relates the averaged field to the averaged flux. In view of (2), \(\nabla \phi _i^{(1)}+e_i\) can be considered as an electric field in the absence of charges, arising from the electric potential \(-(\phi _i^{(1)}+x_i)\). In view of the periodicity of \(\phi _i^{(1)}\), the large-scale average of \(\nabla \phi _i^{(1)}+e_i\) is just \(e_i\). Now \(a(\nabla \phi _i^{(1)}+e_i)\) is the corresponding flux. It is periodic, so its large-scale average is given by its average on the periodic cell


Observe that the notation \({{\bar{a}}}\) without reference to the period L is legitimate since (3) is equivalent to \({{\bar{a}}} e_i\) . A well-known feature of homogenization is that \({{\bar{a}}}\) inherits the bounds (1) from a, as can be derived with the help of the dual problem (20). In the periodic case, (3) in fact coincides with the homogenized coefficient \(a_{\textrm{hom}}\).

On the contrary, in the random case which we introduce now, (3) provides only a fluctuating approximation to the deterministic \(a_{\textrm{hom}}\). Homogenization is known to take place when a is sampled from a stationary and ergodic ensemble \(\langle \cdot \rangle \), see [38, 53]. By the latter, we mean a probability measureFootnote 2 on the space of tensor fields a satisfying (1); we use the symbol \(\langle \cdot \rangle \) to address both the ensemble and to denote its expectation operator. Stationarity is the crucial structural assumption and means that the shifted random field \(x\mapsto a(z+x)\) has the same (joint) distribution as a for any shift vector \(z\in {\mathbb {R}}^d\). Ergodicity is a qualitative assumptionFootnote 3 that encodes the decorrelation of the values of a over large distances.

1.3 Two Strategies of Periodizing

In order to apply the RVE method in form of (3), considered as an approximation for \(a_{\textrm{hom}}\), one needs to produce samples a of L-periodic coefficient fields connected to the given ensemble \(\langle \cdot \rangle \). The goal of this paper is to compare two strategies to procure such L-periodic samples. The first strategy relies on “periodizing the realizations” in its most naive form—we shall actually consider a seemingly less naive form of it, see Sect. 4—and goes as follows: Taking a coefficient field a in \({\mathbb {R}}^d\), we restrict it to the box \([0,L)^d\) and then periodically extend it. This defines a map \(a\mapsto a_L\). We then take \(\overline{a_L}\), cf. (3), as an approximation for \(a_{\textrm{hom}}\). One unfavorable aspect of this strategy is obvious: The push-forward of \(\langle \cdot \rangle \) under this map \(a\mapsto a_L\) is no longer stationary—an imagined glance at a typical realization would reveal d families of parallel artificial hypersurfaces.

Related variants of this strategy consist in still restricting a to \([0,L)^d\) but then imposing Dirichlet or Neumann boundary conditions instead of periodic boundary conditions. (Neumann conditions will actually be analyzed in Sect. 4.) Both boundary conditions for the random (vector) field \(\nabla \phi ^{(1)}=\nabla \phi ^{(1)}(a,x)\) destroy its stationarity in the sense of shift-covariance: It is no longer true that for any shift-vector \(z\in {\mathbb {R}}^d\) we have \(\nabla \phi ^{(1)}(a,z+x)\) \(=\nabla \phi ^{(1)}(a(z+\cdot ),x)\).

The second strategy relies on “periodizing the ensemble” and is more subtle: Given an ensemble \(\langle \cdot \rangle \), one constructs a “related” stationary ensemble \(\langle \cdot \rangle _L\) of L-periodic fields, samples a from \(\langle \cdot \rangle _L\) and takes \({{\bar{a}}}\) as an approximation. The quality of this second method was numerically explored in [35] for random non-overlapping inclusions and (next to the first strategy) in [40] for random Voronoi tessellationsFootnote 4; in both cases, the periodization is obvious. Requirements on the periodization of ensembles were formulated in [55, Section 4], a general construction idea was given in [28, Remark 5]. In this paper, we advocate thinking of the map \(\langle \cdot \rangle \leadsto \langle \cdot \rangle _L\) as conditioning on periodicity and will construct it for a specific but relevant class of \(\langle \cdot \rangle \) given in Assumption 1.

The second strategy obviously capitalizes on the knowledge of the ensemble \(\langle \cdot \rangle \) and not just of a single realization (a “snapshot”), in the sense of “known unknowns” as opposed to “unknown unknowns”. This is in contrast to the numerical analysis on inferring \(a_{\textrm{hom}}\) in [51], or on constructing effective boundary conditions in [47, 48] from a snapshot.

1.4 Fluctuations and Bias

In this paper, we are interested in comparing these two strategies in terms of their bias (also called systematic error): How much do the two expected values \(\langle \overline{a_L}\rangle \) and \(\langle {{\bar{a}}}\rangle _L\) deviate from \(a_{\textrm{hom}}\), which by qualitative theory is their common limit for \(L\uparrow \infty \) (see [15] for \(\langle \overline{a_L}\rangle \) and Corollary 1iii) for \(\langle {\overline{a}}\rangle _L\)). We shall heuristically argue that

$$\begin{aligned} \langle \overline{a_L}\rangle -a_{\textrm{hom}}=O(L^{-1}), \end{aligned}$$

see Sect. 4, while proving

$$\begin{aligned} \langle {{\bar{a}}}\rangle _L-a_{\textrm{hom}}=O(L^{-d}), \end{aligned}$$

see Theorems 1 and 2. Here L should be thought of as the (non-dimensional) ratio between the actual period L and a suitably defined correlation length of \(\langle \cdot \rangle \) set to unity. The quantification of the convergence in L is clearly of practical interest: After a discretization that resolves the correlation length, the number of unknowns of the linear algebra problem (2) scales with \(L^d\) for \(L\gg 1\). Numerical experiments confirm the \(O(L^{-d})\) scaling [41, Figure 5] and the substantially worse behavior (4) for the other strategy [56]. In this regard, result (4) is not unexpected either from a theoretical or a numerical perspective, cf. [24, (3.4)], [15, 23], respectively. Nevertheless, to the best of our knowledge, we provide here the first formal argument in favor of such a behavior.

We note that fluctuations (which are at the origin of the random part of the error), as for instance measured in terms of the square root of the variance, are in many situations proven to be of the order (see, e.g., [28, Theorem 2])

$$\begin{aligned} \langle |{{\bar{a}}}-\langle \bar{a}\rangle _L|^2\rangle _L^\frac{1}{2}=O(L^{-\frac{d}{2}}), \end{aligned}$$

see, e.g., [41, Figure 6] for a numerical validation, and the same is expected to hold for the other strategy ([60, Fig.3] and [24, (3.3)])

$$\begin{aligned} \langle |\overline{a_L}-\langle \overline{a_L}\rangle |^2\rangle ^\frac{1}{2}=O(L^{-\frac{d}{2}}). \end{aligned}$$

Hence, the variance scales like the inverse of the volume \(L^d\) of the periodic cell \([0,L)^d\), as if we were averaging over \([0,L)^d\) some field of unit range of dependence instead of the long-range correlated \(a(\nabla \phi _i+e_i)\). In view of this identical fluctuation scaling for both strategies, the different bias scaling is significant in the most relevant dimension \(d=3\), which we mostly focus in this paper: For the first strategy, the bias dominates, so that taking the empirical mean of \(\overline{a_L}\) over many realizations a does not substantially reduce the total error. It does so in the second scenario, which suggests to use variance reduction methods, like analyzed in [25, 45].

Theoretical results on the random error in RVE, at least for the second strategy like in (6), are by now abundant, starting from [30, Theorem 2.1] for a discrete medium with i. i. d. coefficient, over [27, Theorem 1] for a class of continuum media based on the Poisson point process, to the leading-order identification of the variance in in [19, Theorem 2]. The last result arises from the characterization of leading-order variances in stochastic homogenization in general, starting from [52, Theorem 2.1] for correctors, and is in the spirit of the general approach laid out in [34]. These estimates of variances and fluctuations in homogenization rely on a functional calculus (Malliavin derivatives, Spectral Gap inequalities). There is an alternative approach based on a finite range assumption (and its relaxation via mixing conditions) that was shown to yield optimal results in [3, 31], and culminated in the monograph [4]. In this paper, we make use of the first approach.

Theoretical results on the systematic error in RVE, again for the second strategy as in (5), seem to have been restricted to the case of a discrete medium with i. i. d. coefficients, see [28, Proposition 3], where the construction of \(\langle \cdot \rangle _L\) is obvious. The argument for [28, Proposition 3] is based on a (necessarily non-stationary) coupling of \(\langle \cdot \rangle \) and \(\langle \cdot \rangle _L\) and introduces a massive term into the corrector equation in order to screen the resulting boundary layer, which leads to a logarithmically worse estimate than (5). Our analysis avoids this coupling and suggests that such a logarithmic correction is artificial. (Incidentally, the phenomenon that the bias decays to an order that is twice the order of the fluctuation decay occurs also in the analysis of the homogenization error \((-\nabla \cdot a\nabla )^{-1}f-(-\nabla \cdot a_{\text {hom}}\nabla )^{-1}f\) itself: While the variance can be characterized to order \(O(L^{-d})\), where \(L\gg 1\) now is the ratio between the scale of f and the correlation length, see [20, Theorem 1], the expectation seems to be characterized to order \(O(L^{-2d})\), see [14, 18, 43].)

The first strategy is appealing since it only requires a snapshot, which could come from an actual material image, whereas the second one requires knowledge of the underlying ensemble, which has to be estimated or imposed as a model. Several methods to overcome the effect of boundary layers on the first strategy have been proposed: Motivated by the treatment of periodic coefficient fields of unknown period and the ensuing resonance error, oversampling [37] and filtering [11] strategies were proposed. Until recently, in case of random media, however, because of the slow decay of the boundary layer, they were not expected to perform better than \(O(L^{-1})\), see [24, (3.4)]. This motivated [26] to screen the boundary effects by a massive term to the corrector equation (2), cf. (51). However, results in preparation [8] suggest that, in the case of an isotropic ensemble, oversampling strategies may give rise to an improved rate that is at most of the order of the random error \(O(L^{-\frac{d}{2}})\). Screening strategies based on semi-group [1, 51] or wave-equation [2] versions of the corrector equation have also been analyzed. Screening by a massive term, in conjunction with extrapolation in the massive parameter, has been proven to reduce the systematic error to \(O(L^{-d})\) [28, Thm. 2]. Based on screening and extrapolation, [51, Prop. 1.1 & Th. 1.2] formulated a numerical algorithm that extracts \(a_{\textrm{hom}}\) from a snapshot a up to the optimal total error \(O(L^{-\frac{d}{2}})\) with only \(O(L^d)\) operations.

1.5 Assumptions and Formulation of Rigorous Result

We now introduce a class of ensembles \(\langle \cdot \rangle \) of \(\lambda \)-uniform coefficient fields a that can be easily periodized. Loosely speaking, the natural way to periodize a general stationary ensemble \(\langle \cdot \rangle \) of coefficient fields a on \({\mathbb {R}}^d\) is to condition on a being L-periodic. Clearly, this conditioning is highly singular, and we thus shall restrict ourselves to stationary and centered Gaussian ensembles \(\langle \cdot \rangle \). Since a realization g of such a Gaussian ensemble is obviously not (\(\lambda \)-uniformly) elliptic, we will work with a (nonlinear) map A and consider the pointwise transformation \(a(x)=A(g(x))\). More precisely, we will identify \(\langle \cdot \rangle \) with its push-forward under

$$\begin{aligned} g\mapsto a:=\big (x\mapsto A(g(x))\big ). \end{aligned}$$

Centered Gaussian ensembles on some (infinite-dimensional) Banach space X are characterized through their covariance, which is a semi-definite bounded bilinear form on \(X^*\), defining a Hilbert space (known as the Cameron–Martin space) \(H\subset X\). When H is a Hilbert space of Hölder continuous functions on \({\mathbb {R}}^d\), this operator is best represented by its kernel \(c(x,y)=\langle g(x) g(y)\rangle \). Stationarity of \(\langle \cdot \rangle \) then amounts to \(c=c(x-y)\); the positive-semidefinite character of the bilinear form translates into non-negativity of the Fourier transform \({\mathcal F}c(q)\ge 0\) for all wave vectors \(q\in {\mathbb {R}}^d\). We now argue that periodization by conditioning can be characterized as follows: \(\langle \cdot \rangle _L\) is the stationary centered Gaussian measure with L-periodic covariance \(c_L\) with Fourier coefficients given by restricting the Fourier transform \({{\mathcal {F}}}c(q)\) to the dual lattice \(q\in \frac{2\pi }{L}{\mathbb {Z}}^d\), defining the k-th Fourier mode of \(c_L\) as:

$$\begin{aligned} \frac{1}{\sqrt{L^d}}\int _{[0,L)^d}\textrm{d}x\, e^{-i\frac{2\pi k}{L}\cdot x}c_L(x):=\frac{1}{\sqrt{L^d}}{{\mathcal {F}}}c\left( \frac{2\pi k}{L}\right) \;\;\text{ for } \text{ all }\;k\in {\mathbb {Z}}^d. \end{aligned}$$

Since loosely speaking, the contributions to \(\langle \cdot \rangle \) from every wave vector \(q\in {\mathbb {R}}^d\) are independent, this restriction indeed corresponds to conditioning. This definition also highlights that information is lost when passing from \(\langle \cdot \rangle \) to \(\langle \cdot \rangle _L\). In terms of real space, the passage from c to \(c_L\) amounts to periodization of the covariance function:

$$\begin{aligned} c_L(x)=\sum _{k\in {\mathbb {Z}}^d}c(x+Lk). \end{aligned}$$

As for the whole space ensemble, we identify \(\langle \cdot \rangle _L\) with its push-forward under (7).

We now collect the technical assumptions on \(\langle \cdot \rangle \), that is, on the covariance function c and the map A. Loosely speaking, we need that A is regular and that c is regular with integrable decay, both up to second derivatives. A subclass of these ensembles, namely those of Matérn form \({\mathcal F}c(q)=(1+|q|^2)^{-\frac{d}{2}-\nu }\), is, for instance, used as a prior for elastic microstructures, where the smoothness parameter \(\nu \) is estimated from real material images, and the effective moduli \(a_{\textrm{hom}}\) are computed by RVE via periodization of ensembles, see [42, (2.10) and Fig. 9].

Assumption 1

Let \(\langle \cdot \rangle \) be a stationary, ergodic and centered Gaussian ensemble of scalarFootnote 5 fields g on \({\mathbb {R}}^d\), as determined by the covariance function \(c(x):=\langle g(x) g(0)\rangle \). We assume that there exists an \(\alpha >0\) such that

$$\begin{aligned} \sup _{x\in {\mathbb {R}}^d}(1+|x|^2)^{\frac{d+2}{2}+\alpha }|\nabla ^2c(x)|<\infty . \end{aligned}$$

We identify \(\langle \cdot \rangle \) with its push-forward under the map (7), where \(A:{\mathbb {R}}\rightarrow {\mathbb {R}}^{d\times d}\) is such that the coefficient field a is \(\lambda \)-uniformly elliptic, see (1). We assume that

$$\begin{aligned} \sup _{g\in {\mathbb {R}}}|A'(g)|+|A''(g)|<\infty . \end{aligned}$$

We now comment on some direct consequences of Assumption 1. On the one hand, since we implicitly assume that \(\lim _{|x|\uparrow \infty }c(x)=0\) by ergodicity, (10) implies by integration \(\sup _{x\in {\mathbb {R}}^d}(1+|x|^2)^{\frac{d}{2}+\alpha }|c(x)|<\infty \) and thus because of \(\alpha >0\)

$$\begin{aligned} \sup _{q\in {\mathbb {R}}^d}{{\mathcal {F}}}c(q)\lesssim \int _{{\mathbb {R}}^d} \textrm{d}x|c(x)|<\infty , \end{aligned}$$

where, all along the paper, \(\lesssim \) means \(\le \) up to a multiplicative constant that only depends on d, \(\lambda \), and the constants implicit in (10) and (11) of Assumption 1. Further subscripts will indicate an additional dependence. Via (8), (12) yields that the Cameron–Martin norm of \(\langle \cdot \rangle _L\) dominates the \(L^2([0,L)^d)\)-norm. This implies that \(\langle \cdot \rangle _L\) endowed with the Hilbert structure of \(L^2([0,L)^d)\) has a uniform spectral gap in L, see [12, Poincaré inequality (5.5.2)]. By (11), this transmits to the ensemble of a’s, see (7), and will be used for the stochastic estimates.

On the other hand, (10) ensures that the realizations of a belongs to \(C^{0,\alpha }_{\textrm{loc}}\) for any \(\alpha <1\) [unrelated to the one in (10)], namely

$$\begin{aligned} \sup _{x}\langle \Vert a\Vert _{C^{0,\alpha }(B_1(x))}^p\rangle<\infty \quad \text{ for } \text{ all }\;\alpha<1,\;p<\infty , \end{aligned}$$

which will allow us to appeal to Schauder theory for local regularity. For the reader’s convenience, we repeat the standard Kolmogorov argument for (13). The assumption (10) implies \(\sup _{x}|\nabla ^2 c(x)|<\infty \) and thus \(\sup _{x,y}|y-x|^{-2}\langle (g(y)\) \(-g(x))^2\rangle \) \(=\sup _{z}|z|^{-2}(c(0)\) \(-c(z))\) \(<\infty \). Since g is Gaussian, this extends to arbitrary moments: \(\sup _{x,y}|y-x|^{-1}\) \(\langle |g(y)-g(x)|^p\rangle ^{\frac{1}{p}}<\infty \). Estimating the Hölder semi-norm \([g]_{\alpha ,B_1}^p\) by the Besov norm \(\int _{B_1} \textrm{d}z|z|^{-d-p\alpha }\int _{B_1} \textrm{d}x|g(x+z)\) \(-g(x)|^p\), one derives \(\sup _{x}\langle [g]_{\alpha ,B_1(x)}^p\rangle <\infty \) for any \(\alpha <1\) [unrelated to the one in (10)] and \(p<\infty \). By (11), this transmits to the coefficient field a in form of (13).

Since because of (10) we also have \(\sup _{L\ge 1}\vert \nabla ^2 c_L(x)\vert <\infty \), (13) extends to \(\langle \cdot \rangle _L\):

$$\begin{aligned} \sup _{L\ge 1}\sup _x\langle \Vert a\Vert ^p_{C^{0,\alpha }(B_1(x))}\rangle _L<\infty \quad \text {for any }\alpha<1, p<\infty . \end{aligned}$$

Equipped with the definition of the periodized ensembles \(\langle \cdot \rangle _L\), we can state our main result.

Theorem 1

Let \(d>2\) and A be symmetric. Under Assumption 1 on \(\langle \cdot \rangle \), for all L, and with \(\langle \cdot \rangle _L\) defined with (8) we have for the expectation \(\langle {{\bar{a}}}\rangle _L\) of \({{\bar{a}}}\) defined in (3)

$$\begin{aligned} \limsup _{L\uparrow \infty }L^{d}|\langle {{\bar{a}}}\rangle _L-a_\textrm{hom}|<\infty . \end{aligned}$$

Let us motivate the scaling (15). For fixed \(i=1,\ldots ,d\) we consider the flux

$$\begin{aligned} q:=a(\nabla \phi ^{(1)}_i+e_i), \end{aligned}$$

which is a random (vector) field, meaning that \(q=q(g,x)\). We note that by uniqueness for (2), q is stationary, where we recall that it means that for every shift vector \(z\in {\mathbb {R}}^d\), we have \(q(g,z+x)=q(g(z+\cdot ),x)\) for all points x and (periodic) fields g. Hence by stationarity of \(\langle \cdot \rangle _L\), we may write \(\langle {{\bar{a}}}\rangle _L\) \(=\langle q(0)\rangle _L\). Clearly, q(0), as arising from the solution of the PDE (2), depends via \(a=A(g)\) on the value of g in any point y, no matter how distant from 0.

Let us assume for a moment that q were more local, meaning that q(0) depends on g only through its restriction \(g_{|B_R}\) for some radius \(R<\infty \). Let us also assume for simplicity that \(\langle \cdot \rangle \) has unit range, which amounts to assume that c is supported in \(B_1\), a sharpening of (10). We then claim that

$$\begin{aligned} \langle q(0)\rangle _L\quad \text{ is } \text{ independent } \text{ of }\;L\ge 2R+2. \end{aligned}$$

Indeed, by the locality assumption and (centered) Gaussianity, the distribution of the value \(q(0)=q(g,0)\) is determined by \({c_L}_{|B_{2R}}\). In view of (9) and by the finite range assumption, \({c_L}_{|B_{2R}}=c_{|B_{2R}}\) for \(L\ge 1+2R+1\).

As mentioned, our flux q(g, 0) does depend on g(y) even for \(R=|y|\gg 1\). This dependence is described by the mixed derivative \(\nabla \nabla G(a,0,y)\) of the Green function G(axy) for \(-\nabla \cdot a\nabla \), see Sect. 2.2. Stochastic estimates show that, at least on an annealedFootnote 6 level, the decay of this variable-coefficient Green’s function is no worse than of its constant-coefficient counterpart so that \(R^d|\nabla \nabla G(a,0,y)|\lesssim 1\). Loosely speaking, it is this exponent d that shows up in (15).

In Sect. 2, we will refine Theorem 1 by characterizing the leading-order error term in Theorem 2.

2 Theorem 1: Refinement and Main Ideas

The two ingredients for Theorem 1 are a suitable representation formula for \(\langle {{\bar{a}}}\rangle _L\), see Sect. 2.1, and its asymptotics through stochastic homogenization, here on the level of the mixed derivatives of the Green function, see Sect. 2.2. We need the second-order version of stochastic homogenization because of an inversion symmetry. We refine Theorem 1 in Sect. 2.3 by identifying the leading-order error term, see Theorem 2. In Sect. 2.4, we will argue that the leading-order error typically does not vanish, by exploring the regime of small ellipticity contrast. In Sect. 2.5, we discuss the structure of the leading-order error term in the case of an isotropic ensemble.

2.1 Representation Formula

We start with an informal, but detailed, derivation of the representation formula, see (26), which might be the most conceptual piece of our work.

Let us fix two vectors \(\xi \) and \(\xi ^*\) and focus on the component \(\xi ^*\cdot {{\bar{a}}}\xi \); we denote by \(\phi ^{(1)}\) the solution of (2) with \(e_i\) replaced by \(\xi \), where by linearity and uniqueness (up to additive constants) we have \(\phi ^{(1)} = \sum _i\xi _i\phi ^{(1)}_i\). By stationarity of \(\nabla \phi ^{(1)}\) and \(\langle \cdot \rangle _L\), we have

$$\begin{aligned} \langle \xi ^*\cdot {{\bar{a}}} \xi \rangle _L=\langle F\rangle _L\quad \text{ where }\quad F:=(\xi ^*\cdot a(\nabla \phi ^{(1)}+\xi ))(0). \end{aligned}$$

Instead of directly estimating \(\langle \xi ^*\cdot {{\bar{a}}}\xi \rangle _L -\xi ^*\cdot a_{\textrm{hom}}\xi \), we will estimate its derivative w.r.t. L, that is \(\frac{\textrm{d}}{\textrm{d}L}\langle \xi ^*\cdot \bar{a}\xi \rangle _L\). The reason is that by general Gaussian calculus (in form of the Price formula) applied to the ensemble \(\langle \cdot \rangle _L\) of (periodic) fields g that depends on a parameter L, we have for any \(F=F(g)\)

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}\langle F\rangle _L =\frac{1}{2}\int _{{\mathbb {R}}^d}\textrm{d}x \int _{{\mathbb {R}}^d}\textrm{d}y\Big \langle \frac{\partial ^2 F}{\partial g(-x)\partial g(-y)}\Big \rangle _L\frac{\partial c_L}{\partial L}(x-y), \end{aligned}$$

where the two minus signs in the denominator are for later convenience. Here \(\frac{\partial ^2 F}{\partial g(x)\partial g(y)}\) denotes the kernel representing the second Fréchet derivative of F, seen as a bilinear form on the space of functions on \({\mathbb {R}}^d\). As a derivative w.r.t. the noise g, it can be seen as a Malliavin derivative. We refer the reader to [16] for a rigorous proof of (18).

We define F by (17). By the change of variables \(z\rightsquigarrow x-y\), which capitalizes on the translation invariance of the covariance, and (more directly) by the stationarity of \(\langle \cdot \rangle _L\) in conjunction with the stationarity of \(\nabla \phi ^{(1)}\) that leads to \(\left\langle \frac{\partial ^2 F}{\partial g(-x)\partial g(z-x)}\right\rangle _L\) \(=\left\langle \xi ^*\cdot \frac{\partial ^2 a(\nabla \phi ^{(1)}+\xi )(x)}{\partial g(0)\partial g(z)} \right\rangle _L\), we obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}\langle \xi ^*\cdot {{\bar{a}}}\xi \rangle _L =\frac{1}{2}\int _{{\mathbb {R}}^d}\textrm{d}z \Big \langle \int _{{\mathbb {R}}^d}\xi ^*\cdot \frac{\partial ^2 a(\nabla \phi ^{(1)}+\xi )}{\partial g(0)\partial g(z)}\Big \rangle _L \frac{\partial c_L}{\partial L}(z). \end{aligned}$$

With help of the corrector for the (pointwise) dual coefficient field \(a^*\) in direction \(\xi ^*\) (while we work with the assumption \(A^*=A\) and thus have \(a^*=a\), keeping the primal and dual medium apart reveals more of the structure), i.e., the periodic solution \({\phi ^*}^{(1)}\) of

$$\begin{aligned} \nabla \cdot a^*(\nabla {\phi ^*}^{(1)}+\xi ^*)=0, \end{aligned}$$

the inner integral can be rewritten more symmetrically as

$$\begin{aligned} \int _{{\mathbb {R}}^d}\xi ^*\cdot \frac{\partial ^2 a(\nabla \phi ^{(1)}+\xi )}{\partial g(0)\partial g(z)}&=\int _{{\mathbb {R}}^d}(\nabla {\phi ^*}^{(1)}+\xi ^*)\cdot \frac{\partial ^2 a(\nabla \phi ^{(1)}+\xi )}{\partial g(0)\partial g(z)}\nonumber \\&=\int _{{\mathbb {R}}^d}(\nabla {\phi ^*}^{(1)}+\xi ^*)\cdot \left[ \frac{\partial ^2 }{\partial g(0)\partial g(z)},a\right] (\nabla \phi ^{(1)}+\xi );\nonumber \end{aligned}$$

indeed, the first identity (formally) follows from applying \(\frac{\partial ^2}{\partial g(0)\partial g(z)}\) to (2) and then testing with \({\phi ^*}^{(1)}\), whereas the second identify follows from testing (20) with \(\frac{\partial ^2\phi ^{(1)}}{\partial g(0)\partial g(z)}\). Resolving the commutator \(\left[ \frac{\partial ^2 }{\partial g(0)\partial g(z)},a\right] \) by Leibniz’ rule we obtain

$$\begin{aligned} \int _{{\mathbb {R}}^d}\xi ^*\cdot \frac{\partial ^2 a(\nabla \phi ^{(1)}+\xi )}{\partial g(0)\partial g(z)}= & {} 2\int _{{\mathbb {R}}^d}(\nabla {\phi ^*}^{(1)}+\xi ^*)\cdot \frac{\partial ^2 a }{\partial g(0)\partial g(z)}(\nabla \phi ^{(1)}+\xi ) \nonumber \\{} & {} +\int _{{\mathbb {R}}^d}(\nabla {\phi ^*}^{(1)}+\xi ^*)\cdot \frac{\partial a }{\partial g(0)}\nabla \frac{\partial \phi ^{(1)}}{\partial g(z)} \nonumber \\{} & {} +\int _{{\mathbb {R}}^d}(\nabla {\phi ^*}^{(1)}+\xi ^*)\cdot \frac{\partial a }{\partial g(z)}\nabla \frac{\partial \phi ^{(1)}}{\partial g(0)}. \end{aligned}$$

Denoting \(a':=A'(g)\) and \(a'':=A''(g)\), we remark that by (7) we have \(\frac{\partial a(x)}{\partial g(z)}\) \(=a'(z)\delta (x-z)\). Applying operator \(\frac{\partial }{\partial g(z)}\) on (2), we thus obtain the representation

$$\begin{aligned} \frac{\partial \nabla \phi ^{(1)}(x)}{\partial g(z)}=-\nabla \nabla G(x,z)a'(z)(\nabla \phi +e)(z) \end{aligned}$$

in terms of the mixed derivatives of the non-periodic Green function (since we are only interested in the mixed gradient of the Green function, the dimension \(d=2\) poses no problems here) \(G=G(a,x,y)\) associated with the operator \(-\nabla \cdot a\nabla \). Hence the above turns into

$$\begin{aligned} \int _{{\mathbb {R}}^d}\xi ^*\cdot \frac{\partial ^2 a(\nabla \phi ^{(1)}+\xi )}{\partial g(0)\partial g(z)}&=\delta (z)\big ((\nabla {\phi ^*}^{(1)}+\xi ^*)\cdot a''(\nabla \phi ^{(1)}+\xi )\big )(0) \\&\quad -\big (a'(\nabla {\phi ^*}^{(1)}+\xi ^*)\big )(0) \cdot \nabla \nabla G(0,z) \big (a' (\nabla \phi ^{(1)}+\xi )\big )(z)\\&\quad -\big (a'(\nabla {\phi ^*}^{(1)}+\xi ^*)\big )(z) \cdot \nabla \nabla G(z,0) \big (a' (\nabla \phi ^{(1)}+\xi )\big )(0). \end{aligned}$$

Applying \(\langle \cdot \rangle _L\), we obtain by stationarity

$$\begin{aligned}{} & {} \big \langle \int _{{\mathbb {R}}^d}\xi ^*\cdot \frac{\partial ^2 a(\nabla \phi ^{(1)}+\xi )}{\partial g(0)\partial g(z)}\big \rangle _L\nonumber \\{} & {} \quad =\delta (z)\big \langle (\nabla {\phi ^*}^{(1)}+\xi ^*)\cdot a''(\nabla \phi ^{(1)}+\xi )\big \rangle _L\nonumber \\{} & {} \qquad -\big \langle \big (a'(\nabla {\phi ^*}^{(1)}+\xi ^*)\big )(0) \cdot \nabla \nabla G(0,z) \big (a' (\nabla \phi ^{(1)}+\xi )\big )(z)\big \rangle _L\nonumber \\{} & {} \qquad -\big \langle \big (a'(\nabla {\phi ^*}^{(1)}+\xi ^*)\big )(0)\cdot \nabla \nabla G(0,-z) \big (a' (\nabla \phi ^{(1)}+\xi )\big )(-z)\big \rangle _L. \end{aligned}$$

Inserting this into (19), and noting that since \(\frac{\partial c_L}{\partial L}\) is even (as derivative of a covariance function), the two last terms have the same contribution, we obtain

$$\begin{aligned} \begin{aligned}&\frac{\textrm{d}}{\textrm{d}L}\langle \xi ^*\cdot {{\bar{a}}}\xi \rangle _L \\&\quad =-\int _{{\mathbb {R}}^d}\textrm{d}z\big \langle \big (a'(\nabla {\phi ^*}^{(1)}+\xi ^*)\big )(0) \cdot \nabla \nabla G(0,z) \big (a' (\nabla \phi ^{(1)}+\xi )\big )(z)\big \rangle _L\frac{\partial c_L}{\partial L}(z)\\&\qquad +\frac{1}{2}\big \langle (\nabla {\phi ^*}^{(1)}+\xi ^*)\cdot a''(\nabla \phi ^{(1)}+\xi )\big \rangle _L \frac{\partial c_L}{\partial L}(0). \end{aligned} \end{aligned}$$

We now insert (9) in form of

$$\begin{aligned} \frac{\partial c_L}{\partial L}(z){\mathop {=}\limits ^{(9)}} \sum _{k\in {\mathbb {Z}}^d}k\cdot \nabla c(z+Lk). \end{aligned}$$

This relation highlights that the z-integral in (24) is not absolutely convergent for \(|z|\uparrow \infty \), not even borderline: While \(\nabla \nabla G(0,z)\) decays as \(|z|^{-d}\), a glance at (25) reveals that \(\frac{\partial c_L}{\partial L}(z)\) grows as |z|. Part of the rigorous work is devoted to justify this formal derivation of (24) by replacing the operator \(-\nabla \cdot a\nabla \) by \(\frac{1}{T}-\nabla \cdot a\nabla \), see Proposition 1.

In order to access the cancellations, we will perform a re-summation. Assuming for simplicity for this exposition that \(\langle \cdot \rangle \) has unit range of dependence, so that c is supported in the unit ball, we have that \(c_L(z=0)\) does not depend on \(L\ge 2\). Hence, the second r.h.s. term in (24) does not contribute. By L-periodicity of the correctors, (24) can be re-summed to

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}\langle \xi ^*\cdot {{\bar{a}}}\xi \rangle _L= & {} \int _{{\mathbb {R}}^d}\textrm{d}z \big \langle \big (a'(\nabla {\phi ^*}^{(1)}+\xi ^*)\big )(0) \nonumber \\{} & {} \cdot \big (\sum _{k\in {\mathbb {Z}}^d}k_n\nabla \nabla G(0,z+Lk)\big ) \big (a'(\nabla \phi ^{(1)}+\xi )\big )(z)\big \rangle _L \partial _nc(z), \nonumber \\ \end{aligned}$$

where from now on we use Einstein’s convention of summation over repeated indices, here \(n\in \{1,\ldots ,d\}\). Formula (26) is our final representation. Clearly, the sum over k is still not absolutely convergent. However, as we shall see in the next subsection, it converges after homogenization.

2.2 Approximation by Second-Order Homogenization

In this subsection, we turn to the asymptotics of the representation (26) for \(L\uparrow \infty \). In particular, we shall argue why first-order homogenization is not sufficient and give an efficient introduction into second-order correctors.

As there is no contribution from \(k=0\), and since by our finite range assumption (for the sake of this discussion), z is constrained to the unit ball, the argument \(z+Lk\) of the Green function satisfies \(|z+Lk|\gtrsim L\). Hence, we may appeal to homogenization to replace G(xy) by \({\overline{G}}(x-y)\), where \({{\bar{G}}}\) denotes the fundamental solution of \(-\nabla \cdot \bar{a}\nabla \). This appears like periodic homogenization as long as L is fixed, but in fact amounts to stochastic homogenization since we are interested in \(L\uparrow \infty \). Since we are interested in its gradient, we need to replace G by the two-scale expansion of \({\overline{G}}\). (See below for more details on the two-scale expansion.) Since we are interested in the mixed gradient, the two-scale expansion acts on both variables. Hence in a first Ansatz, we approximate

$$\begin{aligned} \nabla \nabla G(0,x)\approx -\partial _{ij}{\overline{G}}(x) (e_i+\nabla \phi _i^{(1)})(0)\otimes (e_j+\nabla {\phi _j^*}^{(1)})(x), \end{aligned}$$

where \({\phi _j^*}^{(1)}\) denotes the solution of (20) with \(\xi ^*\) replaced by \(e_j\). To leading order, this yields by the periodicity of correctors

$$\begin{aligned} \nabla \nabla G(0,z+Lk)\approx -\partial _{ij}{\overline{G}}(Lk)\;(e_i+\nabla \phi _i^{(1)})(0)\otimes (e_j+\nabla {\phi _j^*}^{(1)})(z). \end{aligned}$$

Applying \(\sum _{k\in {\mathbb {Z}}^d}k_n\) to the r.h.s., we see that it vanishes by parity w.r.t. inversion \(k\leadsto -k\). This is an indication that the first-order two-scale expansion (27) is not sufficient and that we have to go to a second-order expansion, which we shall describe now.

We need to replace the first-order version of the two-scale expansion of \({\overline{G}}\) by its second-order version. We recall the two-scale expansion in its first-order version: Given an \(\bar{a}\)-harmonic function \({{\bar{u}}}\), one considers \(u=(1+\phi ^{(1)}_i\partial _i){{\bar{u}}}\) as a good approximation to an a-harmonic function. Indeed, it follows from (2) that when \({{\bar{u}}}\) is a first-order polynomial, u is exactly a-harmonic. In fact, this is a characterization of the first-order correctors \(\phi _i^{(1)}\). Second-order correctors \(\phi ^{(2)}_{ij}\) can be characterized in a similar way: For every \({{\bar{a}}}\)-harmonic second-order polynomial \({{\bar{u}}}\), we impose that u \(=(1+\phi _i^{(1)}\partial _i+\phi _{ij}^{(2)}\partial _{ij}){{\bar{u}}}\) is a-harmonic.Footnote 7 It is clear from this characterization that \(\phi _{ij}^{(2)}\) depends on the choice of the additive constant in \(\phi _i^{(1)}\), which we now fix through


Since for our second-order polynomial \({{\bar{u}}}\) we have

$$\begin{aligned} \nabla u=\partial _i{{\bar{u}}}(e_i+\nabla \phi _i^{(1)})+ \partial _{ij}{{\bar{u}}}(\phi ^{(1)}_ie_j+\nabla \phi _{ij}^{(2)}), \end{aligned}$$

so that \(\nabla \cdot a\nabla u=0\) turns into \(\nabla \partial _i\bar{u}\cdot a(e_i+\nabla \phi _i^{(1)})\) \(+\partial _{ij}{{\bar{u}}}\nabla \cdot a(\phi ^{(1)}_ie_j\) \(+\nabla \phi _{ij}^{(2)})\) \(=0\), and using that \(\nabla \cdot {{\bar{a}}}\nabla {{\bar{u}}}=0\), we obtain the following standard PDE characterization of \(\phi _{ij}^{(2)}\):

$$\begin{aligned} -\nabla \cdot a(\nabla \phi _{ij}^{(2)}+\phi ^{(1)}_ie_j) =e_j\cdot (a(\nabla \phi _i^{(1)}+e_i)-{{\bar{a}}}e_i). \end{aligned}$$

Note that (31) is uniquely solvable (up to additive constants) for a periodic \(\phi _{ij}^{(2)}\) because the r.h.s. of (31) has vanishing average in view of (3). The definition of \({\phi _{ij}^*}^{(2)}\) for the dual medium \(a^*\) is analogous.

In view of (30), we thus replace (27) by

$$\begin{aligned} \begin{aligned} \nabla \nabla G(0,x)&\approx -\partial _{ij}{\overline{G}}(x) (e_i+\nabla \phi _i^{(1)})(0)\otimes (e_j+\nabla {\phi _j^*}^{(1)})(x)\\&\quad -\partial _{ijm}{\overline{G}}(x) (\phi _i^{(1)}e_m+\nabla \phi _{im}^{(2)})(0)\otimes (e_j+\nabla {\phi _j^*}^{(1)})(x)\\&\quad +\partial _{ijm}{\overline{G}}(x) (e_i+\nabla \phi _i^{(1)})(0)\otimes ({\phi _j^*}^{(1)}e_m+\nabla \phi _{jm}^{*(2)})(x). \end{aligned} \end{aligned}$$

It is here that the assumption of symmetry of A is convenient: Otherwise, the instance of \({\overline{G}}\) in the first r.h.s. term of (32) would have to be replaced by \({\overline{G}}+{\overline{G}}^{(2)}\) where \({\overline{G}}^{(2)}\) is the \((1-d)\)-homogeneous solution of \(\nabla \cdot (\bar{a}\nabla {\overline{G}}^{(2)}+\bar{a}^{(2)}_m\nabla \partial _m{\overline{G}})=0\), where \({{\bar{a}}}^{(2)}\) is the second-order homogenized coefficient, see (66). Since \({\overline{G}}^{(2)}\), as a dipole, is odd w.r.t. point inversion, its contribution does not vanish as for \({\overline{G}}\), c. f. (28). For the analogue of (28), we now turn to the first-order Taylor expansion (recall \(k\not =0\))

$$\begin{aligned} \nabla \nabla G(0,z+Lk)&\!\approx -\big (\partial _{ij}{\overline{G}}(Lk)\!+\! z_m\partial _{ijm}{\overline{G}}(Lk)\!\big ) (e_i\!+\!\nabla \phi _i^{(1)})(0)\!\otimes \! (e_j+\nabla {\phi _j^*}^{(1)})(z)\\&\quad -\partial _{ijm}{\overline{G}}(Lk) (\phi _i^{(1)}e_m+\nabla \phi _{im}^{(2)})(0)\otimes (e_j+\nabla {\phi _j^*}^{(1)})(z)\\&\quad +\partial _{ijm}{\overline{G}}(Lk) (e_i+\nabla \phi _i^{(1)})(0)\otimes ({\phi _j^*}^{(1)}e_m+\nabla \phi _{jm}^{*(2)})(z). \end{aligned}$$

By the inversion symmetry of \({\overline{G}}\) and the \(-d-1\)-homogeneity of \(\partial _{ijm}{\overline{G}}\), this implies

$$\begin{aligned} \begin{aligned} \sum _{k\in {\mathbb {Z}}^d}k_n\nabla \nabla G(0,z+Lk)&\approx L^{-d-1}\sum _{k\in {\mathbb {Z}}^d}k_n\partial _{ijm}{\overline{G}}(k)\Big (-z_m (e_i+\nabla \phi _i^{(1)})(0)\\&\quad \otimes (e_j+\nabla {\phi _j^*}^{(1)})(z)-(\phi _i^{(1)}e_m+\nabla \phi _{im}^{(2)})(0) \\&\quad \otimes (e_j+\nabla {\phi _j^*}^{(1)})(z)+(e_i+\nabla \phi _i^{(1)})(0)\otimes ({\phi _j^*}^{(1)}e_m\\&\quad +\nabla \phi _{jm}^{*(2)})(z)\Big ). \end{aligned} \end{aligned}$$

In view of \({{\bar{a}}}\approx a_{\textrm{hom}}\), we finally replace \({\overline{G}}\), which is still random, by the deterministic \(G_\textrm{hom}\) that may be pulled out of \(\langle \cdot \rangle _L\) when inserting (33) into (26). Hence, we obtain the approximation

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}\langle \xi ^*\cdot {{\bar{a}}}\xi \rangle _L \approx L^{-d-1}\Gamma _{\textrm{hom},ijmn}\int _{{\mathbb {R}}^d}\textrm{d}z\,\xi ^*\cdot {{\mathcal {Q}}}_{Lijm}(z)\xi \,\partial _nc(z), \end{aligned}$$

where the five-tensor field \({{\mathcal {Q}}}_L\) is defined through a combination of three covariances of quadratic expressions in correctors, see Definition 1, and where the four-tensor \(\Gamma _{\textrm{hom}}\) is formally given by the (borderline) divergent lattice sum \(\sum _{k\in {\mathbb {Z}}^d}k_n\partial _{ijm} G_{T,\textrm{hom}}(k)\), which in line with the remark at the end of Sect. 2.1 we replace by

$$\begin{aligned} \Gamma _{\textrm{hom}}=\lim _{T\uparrow \infty }\Gamma _{\textrm{hom},T}\;\;\text{ where }\;\; \Gamma _{\textrm{hom},Tijmn}:=\sum _{k\in {\mathbb {Z}}^d}k_n\partial _{ijm} G_{T,\textrm{hom}}(k), \end{aligned}$$

with \(G_{T,\textrm{hom}}\) denoting the fundamental solution of \(\frac{1}{T}-\nabla \cdot a_{\textrm{hom}}\nabla \).

2.3 Refinement of Rigorous Result

We start with the full definition of the tensor field \({\mathcal Q}_L\) appearing in (34).

Definition 1

Recall the definitions (2) and (31) of first- and second-order correctors \(\phi _i^{(1)}\) and \(\phi _{ij}^{(2)}\), and their versions \({\phi _i^*}^{(1)}\) and \(\phi _{ij}^{*(2)}\) with a replaced by \(a^*\). For given vectors \(\xi \) and \(\xi ^*\), we continue to write \(\phi ^{(1)}=\xi _i\phi _i\) and \({\phi ^*}^{(1)}=\xi ^*_i{\phi ^*}^{(1)}_i\). Consider the random tensor fields

$$\begin{aligned} \xi ^*\cdot Q^{(1)}_{ij}(z)\xi:= & {} \big ((\xi ^*+\nabla {\phi ^*}^{(1)})\cdot a'(e_i+\nabla \phi _i^{(1)})\big )(0) \nonumber \\{} & {} \big ((e_j+\nabla {\phi _j^*}^{(1)})\cdot a' (\xi +\nabla \phi ^{(1)})\big )(z), \end{aligned}$$
$$\begin{aligned} \xi ^*\cdot Q^{(2)}_{ijm}(z)\xi:= & {} -\big ((\xi ^*+\nabla {\phi ^*}^{(1)})\cdot a' (\phi ^{(1)}_ie_m+\nabla \phi _{im}^{(2)})\big )(0)\nonumber \\{} & {} \big ((e_j+\nabla {\phi _j^*}^{(1)})\cdot a' (\xi +\nabla \phi ^{(1)})\big )(z) \nonumber \\{} & {} +\big ((\xi ^*+\nabla {\phi ^*}^{(1)})\cdot a'(e_i+\nabla \phi _i^{(1)})\big )(0)\nonumber \\{} & {} \big ((\phi ^{*(1)}_je_m+\nabla \phi _{jm}^{*(2)}) \cdot a'(\xi +\nabla \phi ^{(1)})\big )(z). \end{aligned}$$

For any L, we consider the ensemble \(\langle \cdot \rangle _L\) from Definition (9) and define

$$\begin{aligned} {{\mathcal {Q}}}_{Lijm}(z) :=-z_m\langle Q_{ij}^{(1)}(z)\rangle _L+\langle Q_{ijm}^{(2)}(z)\rangle _L. \end{aligned}$$

Here comes the more precise version of Theorem 1, which consists in making (34) rigorous:

Theorem 2

Let \(d>2\) and A be symmetric. Suppose \(\langle \cdot \rangle \) satisfies Assumption 1 and let \(a_{\textrm{hom}}\) denote the homogenized coefficient. For all L, let \(\langle \cdot \rangle _L\) defined with (8), \({{\bar{a}}}\) be defined by (3), \(\Gamma _{\textrm{hom},T}\) defined by (35), and \({{\mathcal {Q}}}_L\) be as in Definition 1. Then, the following limits exist:

$$\begin{aligned} \Gamma _{\textrm{hom},ijmn}&:=\lim _{T\uparrow \infty }\Gamma _{\textrm{hom},Tijmn},\\ {{\mathcal {Q}}}_{ijm}(z)&:=\lim _{L\uparrow \infty }{\mathcal Q}_{Lijm}(z)\quad \text{ for } \text{ any } z\in {\mathbb {R}}^d, \end{aligned}$$

and the latter only depends on \(\langle \cdot \rangle \) (and not the lattice). Moreover, we have

$$\begin{aligned} \lim _{L\uparrow \infty }L^{d+1}\frac{d\langle {{\bar{a}}}\rangle _L}{\textrm{d}L} =\Gamma _{\textrm{hom},ijmn}\int _{{\mathbb {R}}^d}\textrm{d}z{\mathcal Q}_{ijm}(z)\partial _nc(z). \end{aligned}$$

With the tools of this paper, the asymptotics of \(\frac{d\langle {{\bar{a}}}\rangle _L}{\textrm{d}L}\) could be characterized up to order \(O(L^{-d-\frac{d}{2}})\). Let us comment on the representation of the leading error term arising from (39), namely

$$\begin{aligned} d \lim _{L\uparrow \infty } L^{d}\big (a_{\textrm{hom}}-\langle \bar{a}\rangle _L\big ) =\Gamma _{\textrm{hom},ijmn}\int _{{\mathbb {R}}^d}\textrm{d}z{{\mathcal {Q}}}_{ijm}(z)\partial _nc(z). \end{aligned}$$

This representation separates a first factor \(\Gamma _{\textrm{hom}}\), which only depends on the type of the periodic lattice (here cubic) and the homogenized coefficient \(a_{\textrm{hom}}\), from a second factor that only depends on the whole-space ensemble \(\langle \cdot \rangle \), via its covariance function c and covariances involving its first- and second-order correctors.

Let us address the coordinate-free interpretation of \({\mathcal Q}_L\) (and its limit \({{\mathcal {Q}}}\)), i.e., its transformation behavior. We note that \(\xi \), and likewise \(\xi ^*\), should be seen as a linear form (rather than a vector), since it gives rise to a coordinate function: namely affine coordinates via \(\xi \cdot x\) and harmonic coordinates via \(\phi (x)+\xi \cdot x\). A glance at the first r.h.s. term in (38) shows that the indices i and j label the first-order correctors and thus take in linear forms; this is even more obvious for the index m that takes in a linear form in the z-variable. The second, and likewise the third, r.h.s. term in (38) is of the same nature since the second-order corrector naturally takes in a (homogeneous) second-order polynomial, which can be identified with linear combinations of (symmetric) tensor products of linear coordinates. Hence, in the language of differential geometry \({{\mathcal {Q}}}_L(z)\) is a five-contravariant tensor field—as it takes in the five linear forms.

The four-tensor \(\Gamma _{\textrm{hom},T}\) (and its limit \(\Gamma _\textrm{hom}\)) allows for a coordinate-free interpretation: \(\Gamma _{\textrm{hom},T}\) takes in three vectors (namely the directions of the derivatives of \(G_{\textrm{hom}}\)) and renders a vector; as a form it is thus three-covariant and one-contravariant, and in the traditional notation of differential geometry one would write \(\Gamma _{\textrm{hom},Tijm}^n\), highlighting that contraction in (39) with the three-contravariant tensor field \(\xi ^*\cdot {\mathcal {Q}}^{ijm}\xi \) (with \(\xi \), \(\xi ^*\) fixed) is natural. In view of calculus, \(\Gamma _{\textrm{hom},T}\) is invariant under permutation of the covariant indices. There is an isomorphic way of seeing \(\Gamma _{\textrm{hom},T}\) that allows for an electrostatic interpretation: \(\Gamma _{\textrm{hom},T}\) in fact takes in an endomorphismFootnote 8 and renders a (symmetric) bilinear form. Indeed, for some endomorphism B of \({\mathbb {R}}^d\) consider the lattice \(B{\mathbb {Z}}^d\), and the accordingly periodized version of \(G_{T,\textrm{hom}}\), that is \(G_{T,\textrm{hom},B}\) \(:=\sum _{k\in {\mathbb {Z}}^d}G_{T,\textrm{hom}}(x+Bk)\). We then have

$$\begin{aligned} \Gamma _{\textrm{hom},Tijm}^n v^i v^j u^m \xi _n =\frac{\textrm{d}}{\textrm{d}t}_{|t=0}v\cdot \nabla ^2G_{T,\textrm{hom},\textrm{id}+t u\otimes \xi }(x=0)v. \end{aligned}$$

Hence, \(\Gamma _{\textrm{hom},Tijm}^{n}\) describes, on the level of the second derivatives, how (the regular part of) the fundamental solution (infinitesimally) depends on the lattice w.r.t. which one periodizes it.

2.4 Small Contrast Regime and Non-degeneracy

In this subsection, we (formally) identify the leading order (42) of the r.h.s. of (40) in the small-contrast regime. We then argue that this leading-order error term typically does not vanish, even in the high-symmetry case of an isotropic ensemble.

We start with the derivation of (42): To leading order in a small ellipticity contrast \(1-\lambda \), the quantity \(\nabla \phi _i^{(1)}\) may be neglected w.r.t. \(e_i\); likewise \(\phi _i^{(1)}e_m+\nabla \phi _{im}^{(2)}\) may be neglected w.r.t. \(e_i\). Hence to leading order, (38) reduces to

$$\begin{aligned} \xi ^*\cdot {{\mathcal {Q}}}_{ijm}(z)\xi \approx -z_m\big \langle \xi ^*\cdot a'(0)e_i\; e_j\cdot a'(z)\xi \big \rangle . \end{aligned}$$

Restricting to the case of scalar A for convenience, the expression further simplifies to

$$\begin{aligned} {{\mathcal {Q}}}_{ijm}(z) \approx -z_m\,\langle a'(0)a'(z)\rangle \,e_i\otimes e_j. \end{aligned}$$

Restricting ourselves w. l. o. g. to ensembles \(\langle \cdot \rangle \) with \(c(0)=\langle g^2(0)\rangle =\langle g^2(z)\rangle =1\), we see that \(\langle a'(0)a'(z)\rangle \) depends on the Gaussian ensemble \(\langle \cdot \rangle \) only through c(z). We thus write \(\langle a'(0)a'(z)\rangle \) \(={{\mathcal {A}}}'(c(z))\) for some function \({{\mathcal {A}}}\), so that by the chain rule

$$\begin{aligned} {{\mathcal {Q}}}_{ijm}(z)\partial _nc(z)\approx -z_m \,\partial _n{\mathcal A}(c(z))\,e_i\otimes e_j. \end{aligned}$$

Normalizing \({{\mathcal {A}}}\) such that \({{\mathcal {A}}}(0)=0\), we obtain by integration by parts

$$\begin{aligned} \int _{{\mathbb {R}}^d}\textrm{d}z {{\mathcal {Q}}}_{ijm}(z)\partial _nc(z) \approx \delta _{mn}\, \,\int _{{\mathbb {R}}^d}\textrm{d}z{\mathcal A}(c(z))\,e_i\otimes e_j. \end{aligned}$$

Hence, the r.h.s. of (40) is given by

$$\begin{aligned} \Big (\lim _{T\uparrow \infty }\sum _{k\in {\mathbb {Z}}^d}k_m\partial _m\nabla ^2 G_{T,\textrm{hom}}(k)\Big ) \int _{{\mathbb {R}}^d}\textrm{d}z{{\mathcal {A}}}(c(z)) \end{aligned}$$

to leading order in the contrast.

It remains to argue that the two factors in (42) typically do not vanish. The second factor in (42) does not vanish in the typical case of \(A'>0\) and \(c\ge 0\). Indeed, by definition of \({{\mathcal {A}}}\), we then have \({{\mathcal {A}}}'> 0\) and thus \({\mathcal A}(c)>0\) for \(c>0\), so that \(\int _{{\mathbb {R}}^d}\textrm{d}z{\mathcal A}(c(z))>0\) because of \(c(0)=1\).

For the first factor in (42), we restrict ourselves to an isotropic ensemble, namely the case where c is radially symmetric, in addition to A being scalar. In line with this, we show that the trace of the first factor in (42) does not vanish:

$$\begin{aligned} \lim _{T\uparrow \infty }\sum _{k\in {\mathbb {Z}}^d}k_m\partial _m\Delta G_{T,\textrm{hom}}(k)\not =0. \end{aligned}$$

For our isotropic ensemble, the contravariant two-form a is invariant in law under orthogonal transformations, and so is \(a_\textrm{hom}\), which thus is a multiple of the identity, so that \(\triangle \) is a multiple of \(\nabla \cdot a_{\textrm{hom}}\nabla \). Hence by definition of \(G_{T,\textrm{hom}}\), (43) follows from

$$\begin{aligned} \lim _{T\uparrow \infty }\frac{1}{T}\sum _{k\in {\mathbb {Z}}^d}k_m\partial _m G_{T,\textrm{hom}}(k) \not =0. \end{aligned}$$

By scaling, we have \(G_{T,\textrm{hom}}(k)\) \(=\frac{1}{\sqrt{T}^{d-2}} G_{1,\textrm{hom}}(\frac{k}{\sqrt{T}})\). Hence we see that the sum in (44) can be interpreted as a Riemann sum that in the limit \(T\uparrow \infty \) converges to the integral

$$\begin{aligned} \int _{{\mathbb {R}}^d} dk k_m\partial _m G_{1,\textrm{hom}}(k) =-d\int _{{\mathbb {R}}^d} dk G_{1,\textrm{hom}}(k)=-d, \end{aligned}$$

where the identity follows from integrating the defining equation \(G_{1,\textrm{hom}}- \nabla \cdot a_{\textrm{hom}}\nabla G_{1,\textrm{hom}}=\delta \) over \({\mathbb {R}}^d\). In particular, we find that \(\langle {{\bar{a}}}\rangle _L>a_{\textrm{hom}}\) for L large enough, which is consistent with numerical simulations in [40, Fig. 7 & 8], [56, Tab. 3] and [41, Tab. 5.2], where however types of ensembles are considered that are different from our class.

2.5 Isotropic Ensembles

In this subsection, we address the case of an isotropic ensemble. The main step is to characterize the structure of \(\Gamma _\textrm{hom}\), see (50), which amounts to an elementary exercise in representation theory.

We recall that by an isotropic ensemble we mean that c is radially symmetric and that A is scalar. As a consequence, the law of the scalar a under \(\langle \cdot \rangle _L\) is invariant under a change of variables by the octahedral group, and its law under \(\langle \cdot \rangle \) is invariant under the full orthogonal group. As a consequence, both \(\langle {{\bar{a}}}\rangle _L\) and \(a_{\textrm{hom}}\) are multiples of the identity. As a consequence \(G_{T,\textrm{hom}}\) is radially symmetric. Hence by definition (35), the 3-covariant and 1-contravariant tensor \(\Gamma _{\textrm{hom},T}\), like its limit \(\Gamma _{\textrm{hom}}\), is invariant under the octahedral group. Furthermore, it is obviously invariant under the permutation of its first three (covariant) derivatives.

We now derive the (quite restricted) form \(\Gamma _{\textrm{hom}}\) takes as a consequence of these symmetries. We recall that the four-linear form \(\Gamma _{\textrm{hom}}=\Gamma _{\textrm{hom}}(v,v',u,\xi )\) takes in three vectors v, \(v'\), u and the form \(\xi \). Choosing the standard basis \(\{e_m\}_m\) and its dual basis \(\{e^n\}_n\), by linearity and invariance under the octahedral group, it is enough to characterize the two bilinear forms \(\Gamma _{\textrm{hom}}(v,v',e_1,e^1)\) and \(\Gamma _{\textrm{hom}}(v,v',e_2,e^1)\). The first form is invariant under the octahedral subgroup that fixes \(e_1\), which contains in particular reflections \(x_i\leadsto -x_i\) for \(i\not =1\). Since the form is symmetric and thus diagonalizable, this first implies that \(e_1\) is an eigenvector, and then that \(\{e_1\}^\perp \) is an eigenspace. Hence, the bilinear form can be written as:

$$\begin{aligned} \Gamma _{\textrm{hom}}(v,v',e_1,e^1)=\mu _{\perp }v\cdot v'+\mu _{||}(v\cdot e_1)(v'\cdot e_1) \end{aligned}$$

for some constants \(\mu _{\perp }\) and \(\mu _{||}\). For the second bilinear form \(\Gamma _{\textrm{hom}}(v,v',e_2,e^1)\), the same argument yields that it has block diagonal form w.r.t.  the span of \(\{e_1,e_2\}\) and its orthogonal complement. In particular, we have

$$\begin{aligned} \Gamma _{\textrm{hom}}(v,v',e_2,e^1)=c v\cdot v'\quad \text{ for }\;v\cdot e_1=v\cdot e_2=0 \end{aligned}$$

for some constant c, which we may recover through \(c=\Gamma _\textrm{hom}(e_3,e_3,e_2,e^1)\). By invariance under the octahedral transformation \(x_2\leadsto -x_2\), this expression vanishes, so that in fact

$$\begin{aligned} \Gamma _{\textrm{hom}}(v,v',e_2,e^1)=0\quad \text{ for }\;v\cdot e_1=v\cdot e_2=0. \end{aligned}$$

For the same reason, we have

$$\begin{aligned} \Gamma _{\textrm{hom}}(e_2,e_2,e_2,e^1)=0. \end{aligned}$$

By the permutation symmetry in the first three arguments, we obtain from (45)

$$\begin{aligned} \Gamma _{\textrm{hom}}(e_1,e_1,e_2,e^1)=0\;\;\text{ and }\;\;\Gamma _\textrm{hom}(e_1,e_2,e_2,e^1) =\Gamma _\textrm{hom}(e_2,e_1,e_2,e^1)=\mu _{\perp }. \end{aligned}$$

Statements (46), (48), and (47) combine to

$$\begin{aligned} \Gamma _{\textrm{hom}}(v,v',e_2,e^1)= \mu _{\perp }\big ((v\cdot e_1)(v'\cdot e_2)+(v'\cdot e_1)(v\cdot e_2)\big ). \end{aligned}$$

A short computation shows that the combination of this with (45) yields

$$\begin{aligned} \Gamma _{\textrm{hom}}(v,v',u,\xi )= & {} \xi .\Big (\mu _{\perp }\big ((v\cdot v')u+(v\cdot u) v'+(v'\cdot u)v\big ) \nonumber \\{} & {} +(\mu _{||}-2\mu _{\perp })T(v,v',u)\big ), \end{aligned}$$

where we have introduced the trilinear map

$$\begin{aligned} T_i(v,v',u)=v_i{v'}_iu_i\quad \text{(no } \text{ summation) }, \end{aligned}$$

which is invariant under permutations and octahedral transformations, but not under all orthogonal transformations. In terms of indices, we may rewrite (49) as

$$\begin{aligned} \Gamma _{\textrm{hom},ijmn}=\mu _{\perp }\big (\delta _{ij}\delta _{mn}+\delta _{im}\delta _{jn} +\delta _{in}\delta _{jm}\big ) +(\mu _{||}-2\mu _{\perp })\delta _{ijmn}. \end{aligned}$$

Hence in the isotropic case, \(\Gamma _{\textrm{hom}}\) is determined by just two numbers.

We now turn to the second factor on the r.h.s. of (40). As discussed after Definition 1, \(\xi ^*\cdot {\mathcal Q}_{ijm}(z)\xi \) is a five-covariant tensor field, so that \(\int _{{\mathbb {R}}^d}\textrm{d}z\xi ^*\cdot {\mathcal Q}_{ijm}(z)\xi \partial _nc(z)\) is a five-covariant and one-contravariant tensor. In our case of an isotropic ensemble, \({{\mathcal {Q}}}\) is invariant under the entire orthogonal group (not just the discrete octahedral group) as a consequence of \(L\uparrow \infty \). Since the l.h.s. of (40) is a multiple of the identity, it is enough to consider the trace of \(\int _{{\mathbb {R}}^d}\textrm{d}z\xi ^*\cdot {\mathcal Q}_{ijm}(z)\xi \partial _nc(z)\) in \(\xi ,\xi ^*\):

$$\begin{aligned} Q_{ijmn}:=\int _{{\mathbb {R}}^d}\textrm{d}z\big (e_1\cdot {\mathcal Q}_{ijm}(z)e_1+\cdots +e_d\cdot {\mathcal Q}_{ijm}(z)e_d\big )\partial _nc(z), \end{aligned}$$

which is a three-covariant and one-contravariant tensor, still invariant under the (full) orthogonal group. Since in (40), it is contracted with a tensor, namely \(\Gamma _{\textrm{hom}}\), that is symmetric under permutation of ijm, we may pass to the orthogonal projection \(Q^{\textrm{sym}}\) of Q onto this subspace, which preserves invariance under the orthogonal group. Hence as for \(\Gamma _{\textrm{hom}}\), we obtain that \(Q^{\textrm{sym}}\) must be of the form (50). However, while the first three terms in (50) are invariant under the entire orthogonal group, the last is not. Hence, \(Q^{\textrm{sym}}\) must be of the more restricted form

$$\begin{aligned} Q_{ijmn}^{\textrm{sym}}=\nu _{\perp } \big (\delta _{ij}\delta _{mn}+\delta _{im}\delta _{jn}+\delta _{in}\delta _{jm}\big ) \end{aligned}$$

for some constant \(\nu _{\perp }\). Hence for an isotropic ensemble, the relevant information of the entire six-tensor \(\int _{{\mathbb {R}}^d}\textrm{d}z\xi ^*\cdot {\mathcal Q}_{ijm}(z)\xi \partial _nc(z)\) is the single number \(\nu _{\perp }\).

3 Structure of the Proof of Theorem 2

In this section, we formulate the main intermediate results that lead to Theorem 2: In Sect. 3.1, we introduce the massive approximation in order to rigorously derive the analogue of the representation formula (44) from Sect. 2.1, see Proposition 1. In Sect. 3.2 we argue, following Sect. 2.1, that a re-summation allows for removing the massive approximation in the representation formula, see Proposition 2. It relies on second-order homogenization, as introduced in Sect. 2.2. In Sect. 5.1, we sketch how to pass from the representation given by Proposition 2 to the asymptotics stated in Theorem 2. This essentially relies on corrector estimates and the estimate of the homogenization error, see Sects. 3.3 and 3.4. In Sect. 3.3, we formulate the uniform stochastic estimates on first- and second-order correctors needed to capture the asymptotics \(L\uparrow \infty \), see Proposition 3. In Sect. 3.4, we formulate the stochastic second-order estimate of the homogenization error, applied to the Green function, see Proposition 4.

3.1 Massive Approximation

As became apparent in Sect. 2.1, there is divergence in the sum over the periodic cells, see (24). We avoid it by replacing the operator \(-\nabla \cdot a\nabla \) by \(\frac{1}{T}-\nabla \cdot a\nabla \) where \(T<\infty \) will eventually tend to infinity. This has the desired effect that the corresponding Green’s function \(G_T(a,x,y)\) and its derivatives now decay exponentially in \(\frac{|y-x|}{\sqrt{T}}\), which can be seen for instance from the homogenization result in Proposition 5. The language of “massive” approximation arises from field theory where such a zero-order term is often introduced to suppress an infrared divergence, like here. Assimilating \(m^2\) to the inverse of a time scale T, however, makes the connection to stochastic processes, since \(\frac{1}{T}-\nabla \cdot a\nabla \) is the generator of a diffusion-desorption process where T is the time scale of desorption, and ultimately to parabolic intuition. As a collateral of the massive approximation, we have to replace the definitions (2) and (3) by


with analogous definitions for the transposed medium \(a^*\).

We collect in the following some estimates on the massive quantities that are useful in the proofs of this section.

From Schauder’s theory, \(\phi _T^{(1)}\) belongs to \(C^{1,\alpha }_{\textrm{loc}}({\mathbb {R}}^d)\) and

$$\begin{aligned}{} & {} \Vert (\phi _T^{(1)},\nabla \phi _T^{(1)})\Vert _{C^{0,\alpha }([0,L)^d)}\le C(L^{\alpha }[a]_{\alpha }) \\{} & {} \quad \text {and}\quad \Vert (\phi _T^{(1)}-\phi ^{(1)},\nabla \phi _T^{(1)}-\nabla \phi ^{(1)})\Vert _{C^{0,\alpha }([0,L)^d)}\le C(L^{\alpha }[a]_{\alpha })T^{-1}, \end{aligned}$$

where we recall that \([a]_\alpha \) denotes the Hölder semi-norm of a. Knowing that C grows at most polynomially in its argument \([a]_{\alpha }\), we deduce from (14) that the estimates above can be converted into, for any \(p<\infty \)

$$\begin{aligned}{} & {} \langle \Vert (\phi _T^{(1)},\nabla \phi _T^{(1)})\Vert ^p_{C^{0,\alpha }([0,L)^d)}\rangle _L\lesssim _{p,L}1 \nonumber \\{} & {} \quad \text {and}\quad \langle \Vert (\phi _T^{(1)}-\phi ^{(1)},\nabla \phi _T^{(1)}-\nabla \phi ^{(1)})\Vert ^p_{C^{0,\alpha }([0,L)^d)}\rangle _L \lesssim _{p,L}T^{-1}. \end{aligned}$$

Analogously, we obtain at the level of the massive Green functions \(G_T\) and \({{\bar{G}}}_T\) of the operators \(\tfrac{1}{T}-\nabla \cdot a\nabla \) and \(\tfrac{1}{T}-\nabla \cdot {{\bar{a}}}\nabla \), respectively:

$$\begin{aligned} \langle \vert \nabla \nabla G_T(x,y)-\nabla \nabla G(x,y)\vert ^p\rangle _L\underset{T\uparrow \infty }{\rightarrow }0\quad \text {for any }x\ne y, \end{aligned}$$

as well as

$$\begin{aligned} \langle \vert (\nabla ^3{{\bar{G}}}_T(x),\nabla ^2{{\bar{G}}}_T(x))-(\nabla ^3\bar{G}(x),\nabla ^2\bar{G}(x))\vert ^p\rangle _L\underset{T\uparrow \infty }{\rightarrow }\ 0 \quad \text { for any }x\ne 0. \end{aligned}$$

Finally, we have the following moment bounds on the massive Green function \(G_T\),

$$\begin{aligned} \langle \vert \nabla \nabla \!\!\!{} & {} G_T(x,y)\vert ^p\rangle ^{\frac{1}{p}}_L\lesssim _{p,L}\vert x-y\vert ^{-d}\exp \left( -\frac{\vert x-y\vert }{C\sqrt{T}}\right) \nonumber \\{} & {} \quad \text {provided }T\ge L^2 \text { and } \frac{L}{2}\le \vert x-y\vert <\infty , \end{aligned}$$

that we deduce from Proposition 5 and the bound on the constant-coefficient Green function \({\bar{G}}_T\) and its derivatives

$$\begin{aligned} \vert \nabla ^2 {\bar{G}}_T(x)\vert +\vert x\vert \vert \nabla ^3{\bar{G}}_T(x)\vert \lesssim \vert x\vert ^{-d}\exp \big (-\frac{\vert x-y\vert }{C\sqrt{T}}\big )\quad \text {for any }x\ne 0, \end{aligned}$$

that are uniform in \(T\uparrow \infty \).

We now can state the massive version of formula (24). Its rigorous proof will be established in [16].

Proposition 1

It holds

$$\begin{aligned} \begin{aligned} \frac{\textrm{d}}{\textrm{d}L} \langle \xi ^*\cdot {{\bar{a}}}_T\xi \rangle _L=&-\int _{{\mathbb {R}}^d}\textrm{d}z \big \langle \big (a' (\nabla \phi _T^{*(1)}+\xi ^*)\big )(0)\nabla \nabla G_T(0,z) \big (a'(\nabla \phi ^{(1)}_T+\xi )\big )(z)\big \rangle _L \frac{\partial c_L}{\partial L}(z)\\&+\frac{1}{2}\big \langle (\nabla \phi _T^{*(1)}+\xi ^*)\cdot a'' (\nabla \phi ^{(1)}_T+\xi )\big \rangle _L\frac{\partial c_L}{\partial L}(0), \end{aligned} \end{aligned}$$

where we recall that \(\phi ^{(1)}_T=\sum _i \xi _i \phi ^{(1)}_{Ti}\).

The z-integral on the r.h.s. of (57) converges absolutely for \(|z|\uparrow \infty \) since the exponential decay of \(\nabla \nabla G_T(0,z)\) dominates the linear growth of \(\frac{\partial c_L}{\partial L}(z)\), cf. (25). The singularity at \(z=0\) is to be interpreted by duality, using that the other factors are continuous in z.

3.2 Re-summation

Following Sect. 2.2, we now appeal to second-order homogenization, which allows for a re-summation. As a by-product of the re-summation, we may pass to the limit \(T\uparrow \infty \) in (57). The difficulty with passing to the limit \(T\uparrow \infty \) lies in the \(\{|z|\ge L\}\)-part of the integral in (57). We thus fix a smooth cutoff function \(\eta \) for \(B_\frac{1}{2}\) in \(B_1\), rescaled according to

$$\begin{aligned} \eta _L(z)=\eta ({\textstyle \frac{z}{L}}), \end{aligned}$$

and we split the z-integral into the benign near-field part \(\int _{{\mathbb {R}}^d}\textrm{d}z \eta _L(z)\) and the delicate far-field part \(\int _{{\mathbb {R}}^d}\textrm{d}z(1-\eta _L)(z)\). On the far-field part, we appeal to the two-scale expansion (32). Hence, we have to monitor the homogenization error

$$\begin{aligned} \begin{aligned} {{{\mathcal {E}}}(x,y)}&:= \nabla \nabla G(x,y)+\partial _{ij}{\overline{G}}(x-y) (e_i+\nabla \phi _i^{(1)})(x)\otimes (e_j+\nabla {\phi _j^*}^{(1)})(y)\\&\quad +\partial _{ijm}{\overline{G}}(x-y) (\phi _i^{(1)}e_m+\nabla \phi _{im}^{(2)})(x)\otimes (e_j+\nabla {\phi _j^*}^{(1)})(y)\\&\quad -\partial _{ijm}{\overline{G}}(x-y) (e_i+\nabla \phi _i^{(1)})(x)\otimes ({\phi _j^*}^{(1)}e_m+\nabla \phi _{jm}^{*(2)})(y), \end{aligned} \end{aligned}$$

where we recall that \({\overline{G}}\) denotes the fundamental solution for the constant-coefficient operator \(-\nabla \cdot {{\bar{a}}}\nabla \).

The translation invariance of \({\overline{G}}\) together with the periodicity of \(\phi ^{(1)}\) and \(\phi ^{(2)}\) allows for a re-summation. As in Sect. 2.2, we feed in a zeroth- and first-order Taylor expansion of \({\overline{G}}\). This gives rise to the analogue of (35), namely

$$\begin{aligned} {\overline{\Gamma }}_{ijmn}=\lim _{T\uparrow \infty }{\overline{\Gamma }}_{Tijmn}\;\;\text{ where }\;\; {\overline{\Gamma }}_{Tijmn}:=\sum _{k\in {\mathbb {Z}}^d}k_n\partial _{ijm}{\overline{G}}_{T}(k), \end{aligned}$$

where we recall that \({\overline{G}}_T\) denotes the fundamental solution of \(\frac{1}{T}-\nabla \cdot {{\bar{a}}}\nabla \). The existence of this limit, which is borderline summable, is established in Step 2 of the proof of Proposition 2. The Taylor expansion generates the additional error terms

$$\begin{aligned} \epsilon _{Lijn}^{(1)}(z)&:=\sum _{k\in {\mathbb {Z}}^d}k_n\big (((1-\eta _L)\partial _{ij}{\overline{G}})(z+Lk) -\partial _{ij}{\overline{G}}(Lk)-z_m\partial _{ijm}{\overline{G}}(Lk)\big ), \end{aligned}$$
$$\begin{aligned} \epsilon _{Lijmn}^{(2)}(z)&: =\sum _{k\in {\mathbb {Z}}^d}k_n\big (((1-\eta _L)\partial _{ijm}{\overline{G}})(z+Lk) -\partial _{ijm}{\overline{G}}(Lk)\big ). \end{aligned}$$

Thanks to this re-summation, the subtlety of the \(T\uparrow \infty \) is limited to the not absolutely convergent sum in (59). The sums in (60) and (61) are absolutely convergent since both summands decay as \(|k|^{-(d+1)}\) for \(|k|\gg \frac{|z|}{L}\), see (120) and (121) for a more quantitative discussion. Equipped with these definitions, we are now able to express the limit \(T\uparrow \infty \) of (57):

Proposition 2

Let \({{\bar{\Gamma }}}\) be as in (59), \(\epsilon ^{(1)}\) and \(\epsilon ^{(2)}\) as in (60) and (61), and \({{\mathcal {E}}}\) as in (58). Let \(Q^{(1)}\) and \(Q^{(2)}\) be defined as in (36) and (37). Then, we have

$$\begin{aligned}{} & {} {\frac{\textrm{d}}{\textrm{d}L}\langle \xi ^*\cdot {{\bar{a}}}\xi \rangle _L} \nonumber \\{} & {} \quad = L^{-(d+1)}\int _{{\mathbb {R}}^d}\textrm{d}z \big \langle {\overline{\Gamma }}_{ijmn}\big (\xi ^*\cdot Q^{(2)}_{ijm}(z)\xi -z_m \xi ^*\cdot Q^{(1)}_{ij}(z)\xi \big )\big \rangle _L \partial _nc(z) \nonumber \\{} & {} \qquad +\int _{{\mathbb {R}}^d}\textrm{d}z \big \langle \epsilon ^{(2)}_{Lijmn}(z)\xi ^*\cdot Q^{(2)}_{ijm}(z)\xi +\epsilon ^{(1)}_{Lijn}(z)\xi ^*\cdot Q^{(1)}_{ij}(z)\xi \big )\big \rangle _L \partial _nc(z) \nonumber \\{} & {} \qquad -\int _{{\mathbb {R}}^d}\textrm{d}z(1-\eta _L)(z) \big \langle \big (a'(\nabla \phi ^{*(1)}+\xi ^*)\big )(0) {\mathcal E}(0,z) \big (a'(\nabla \phi ^{(1)}+\xi )\big )(z)\big \rangle _L \frac{\partial c_L}{\partial L}(z) \nonumber \\{} & {} \qquad -\int _{{\mathbb {R}}^d}\textrm{d}z\eta _L(z) \big \langle \big (a'(\nabla \phi ^{*(1)}+\xi ^*)\big )(0)\nabla \nabla G(0,z) \big (a'(\nabla \phi ^{(1)}+\xi )\big )(z)\big \rangle _L \frac{\partial c_L}{\partial L}(z) \nonumber \\{} & {} \qquad +\frac{1}{2}\big \langle (\nabla \phi ^{*(1)}+\xi ^*)\cdot a'' (\nabla \phi ^{(1)}+\xi )\big \rangle _L\frac{\partial c_L}{\partial L}(0). \end{aligned}$$

Periodic homogenization theory suffices to establish Proposition 2 and in particular to ensure that all five expressions on the r.h.s. of (62) are well-defined, including the third one. Indeed, it helps to momentarily think of having rescaled length by the fixed L. This puts us into the context of a 1-periodic coefficient field a, which in addition is Hölder continuous. By periodic homogenization, we prove in Proposition 5 (in a more general case for the massive quantity \({\mathcal {E}}_T\))

$$\begin{aligned} \sup _{x,y}|y-x|^{d+2}\langle |{\mathcal E}(x,y)|^p\rangle ^{\frac{1}{p}}_L<\infty \quad \text {for any } p<\infty . \end{aligned}$$

This estimate yields the absolute convergence of the third term on the r.h.s. of (62), since the decay (63) over-compensates the linear growth of \(\frac{\partial c_L}{\partial L}\).

In order to pass from the representation in Proposition 2 to the asymptotics in Theorem 2, we have to show that the first r.h.s. term of (62), up to the factor \(L^{d+1}\), converges to the r.h.s. term of (39), and that the remaining terms are \(o(L^{-(d+1)})\). The proof relies on stochastic estimates on the correctors \(\phi ^{(1)}_i\) and \(\phi ^{(2)}_{ij}\) in order to control moments of \(Q^{(1)}_{ij}\) and \(Q^{(2)}_{ijm}\) together with moment estimates on the homogenization error \({\mathcal {E}}\). This is the purpose of the two next section. The proof of Theorem 2 is carried out in Sect. 5.1.

3.3 Stochastic Corrector Estimates Up to Second Order

As just discussed, the proof of Theorem 2 will rely on estimates of not only the first-order corrector \(\phi ^{(1)}_i\), but also its second-order version \(\phi ^{(2)}_{ij}\), see part i) of Proposition 3. Since the period L of the ensemble \(\langle \cdot \rangle _L\) tends to infinity, these have to rely on stochastic (and not periodic) homogenization. This is the reason for the restriction to \(d>2\) (which is just a more telling way of saying \(d\ge 3\) since it is rather \(d=2\) that is borderline): For \(d=2\), the first-order corrector in the whole-space ensemble \(\langle \cdot \rangle \) is not stationary, so that one looses (pointwise) control even of a centered second-order corrector. Only for \(d>2\) one has the middle item in (69), see for instance [32]. For the (limiting) whole-space ensemble \(\langle \cdot \rangle \), such higher-order corrector estimates have first been established in [33] (however suboptimal in odd dimensions) and [7, Theorem 3.1] (see [20, Proposition 2.2] for a treatment of any order). These works, like ours, rely on Malliavin calculus and a suitable spectral gap estimate, as is available under Assumption 1. (Incidentally, the quantitative theory based on finite-range assumptions as started in [5] has also been extended to get stochastic estimates on \(\phi ^{(2)}\) in [48].) Unfortunately, we cannot simply quote [7] since we need the estimate for the periodized ensembles \(\langle \cdot \rangle _L\) (uniform for \(L\uparrow \infty \), of course).

For Proposition 4, we need to also estimate the flux correctors, both first order and second order, which we shall recall now. (We also refer to [20, Section 2] for a compact introduction into all higher-order correctors.) It follows from (2) and (3) that \(a(\nabla \phi ^{(1)}_i+e_i)-{{\bar{a}}} e_i\) is divergence-free, periodic, and of zero average. Hence it allows for, in the language of \(d=2\), a periodic stream function, or in the language of \(d=3\), a periodic vector potential. For general d, it can be represented in terms of a periodic tensor field \(\sigma _i\) with

$$\begin{aligned} a(\nabla \phi ^{(1)}_i+e_i)={{\bar{a}}} e_i+\nabla \cdot \sigma _i^{(1)}\quad \text{ and }\quad \sigma _{imn}^{(1)}=-\sigma _{inm}^{(1)}, \end{aligned}$$

where for a (skew symmetric) tensor field \(\sigma \), we write \((\nabla \cdot \sigma )_m\) \(:=\partial _n\sigma _{mn}\), as an instance of an exterior derivative. Observe that (64) does not determine \(\sigma _{i}^{(1)}\). Indeed, \(\sigma _i^{(1)}\), which can be interpreted as an alternating \((d-2)\)-form, is only determined up to a \((d-3)\)-form. For estimates like in Proposition 3, we choose a suitable (and simple) gauge, that is

$$\begin{aligned} -\Delta \sigma ^{(1)}_{imn}=\partial _m(e_n\cdot a(e_i+\nabla \phi ^{(1)}_i))-\partial _n(e_m\cdot a(e_i+\nabla \phi ^{(1)}_i)). \end{aligned}$$

Note also that (31) can be reformulated in divergence form

$$\begin{aligned} \nabla \cdot a(\nabla \phi _{ij}^{(2)}+\phi ^{(1)}_ie_j) =(\nabla \cdot \sigma ^{(1)}_i)e_j. \end{aligned}$$

This shows that there is a second-order analogue of (64): For every coordinate direction i, let the matrix \({{\bar{a}}}^{(2)}\) be defined through


for any \(j=1,\ldots ,d\), and the periodic tensor field \(\sigma ^{(2)}_{ij}\) through

$$\begin{aligned} \begin{array}{c} a(\nabla \phi ^{(2)}_{ij}+\phi ^{(1)}_ie_j)={{\bar{a}}}_i^{(2)} e_j+\sigma _i^{(1)}e_j +\nabla \cdot \sigma _{ij}^{(2)}\quad \text{ and }\quad \sigma _{ijmn}^{(2)}=-\sigma _{ijnm}^{(2)}. \end{array} \end{aligned}$$

The merits of the flux correctors \(\sigma _i^{(1)}\) and \(\sigma _{ij}^{(2)}\) will become clear in Sect. 3.4. In fact, in that context it will be convenient to have yet one more object, namely the periodic solution \(\omega _i\) of

$$\begin{aligned} -\triangle \omega _i=\phi _i^{(1)}. \end{aligned}$$

Proposition 3

Let \(d>2\) and \(\langle \cdot \rangle \) satisfy Assumptions 1; let \(\langle \cdot \rangle _L\) be defined with (8). Let \(p<\infty \) be arbitrary.

(i) We have

$$\begin{aligned} \langle |\nabla \phi ^{(1)}_i|^p\rangle _L^\frac{1}{p} +\langle |\phi ^{(1)}_i|^p\rangle _L^\frac{1}{p} +\langle |\nabla \phi ^{(2)}_{ij}|^p\rangle _L^\frac{1}{p} \lesssim _p 1. \end{aligned}$$

(ii) The random tensor fields \(\sigma ^{(1)}_i\) and \(\sigma ^{(2)}_{ij}\) can be constructed such that

$$\begin{aligned} \langle |\sigma ^{(1)}_i|^p\rangle _L^\frac{1}{p} +\langle |\nabla \sigma ^{(2)}_{ij}|^p\rangle _L^\frac{1}{p} \lesssim _p 1. \end{aligned}$$

(iii) We have for any deterministic periodic vector field h and function \(\eta \)

$$\begin{aligned}{} & {} \Big \langle \Big |\int _{[0,L)^d} h\cdot (q_i-\langle q_i\rangle ,\nabla \phi ^{(1)}_i)\Big |^p\Big \rangle _L^\frac{1}{p} \lesssim _p\Big (\int _{[0,L)^d}|h|^2\Big )^\frac{1}{2} \nonumber \\ {}{} & {} \quad \text { and }\quad \Big \langle \Big \vert \int _{[0,L)^d} \eta \phi ^{(1)}_i\Big \vert ^p\Big \rangle _L^{\frac{1}{p}}\lesssim _p \Big (\int _{[0,L)^d} \vert \eta \vert ^{\frac{2d}{d+2}}\Big )^{\frac{d+2}{2d}}, \end{aligned}$$

where we recall the definition of the flux \(q_i:=a(\nabla \phi ^{(1)}_i+e_i)\).

(iv) We have for all z

$$\begin{aligned} \langle |\phi _{ij}^{(2)}(z)-\phi _{ij}^{(2)}(0)|^p\rangle _L^\frac{1}{p} +\langle |\sigma _{ij}^{(2)}(z)-\sigma _{ij}^{(2)}(0)|^p\rangle _L^\frac{1}{p} \lesssim _p\mu _d^{(2)}(|z|), \end{aligned}$$


$$\begin{aligned} \mu _d^{(2)}(r):=\left\{ \begin{array}{ccccc} r&{}\text{ for }&{}2> r&{}&{}\\ r^\frac{1}{2}&{}\text{ for }&{} 2\le r&{}\text{ and }&{}d=3\\ \ln ^\frac{1}{2}r&{}\text{ for }&{}2\le r&{}\text{ and }&{}d=4\\ 1&{}\text{ for }&{}2\le r&{}\text{ and }&{}d>4 \end{array}\right\} . \end{aligned}$$

(v) We have for all z

$$\begin{aligned} \langle |\nabla \omega _{i}(z)-\nabla \omega _i(0)|^p\rangle _L^\frac{1}{p}\lesssim _p\mu _d^{(2)}(|z|). \end{aligned}$$

While part i) of Proposition 3 is explicitly used in Sect. 5.1, the usage of the other parts is more indirect: Part ii) is used in Corollary 1, part iii) is used to estimate the second-order homogenization error in Lemma 3; and part iv) and v) are used to apply this to the Green function, see Proposition 4.

The proof of Proposition 3 essentially follows the strategy of [39, Section 4] and extends it from first-order to second-order correctors; the passage from \(\langle \cdot \rangle \) to \(\langle \cdot \rangle _L\) is only a minor change. Another additional feature is the second estimate of (70) that we deduce as follows: Given a deterministic periodic function \(\eta \) (where w. l. o. g we may assume that \(\int _{[0,L)^d}\eta =0\) since \(\int _{[0,L)^d}\phi ^{(1)}=0\)), we consider the solution of \(-\triangle \zeta =\eta \) and set \(h=\nabla \zeta \) to the effect of \(\int _{[0,L)^d}\eta \phi _i^{(1)}=\int _{[0,L)^d}h\cdot \nabla \phi _i^{(1)}\). By maximal regularity for the Laplacian and Sobolev’s estimate, we have \((\int _{[0,L)^d}|h|^2)^\frac{1}{2}\lesssim (\int _{[0,L)^d}|\eta |^\frac{2d}{d+2})^\frac{d+2}{2d}\) so that the second estimate follows from the first. In this paper, we will only establish the most important ingredient for Proposition 3, namely the characterization of stochastic cancellations of the gradient of the correctors in Lemma 1. While (74) reproduces [39, Proposition 4.1], the new element is its second-order counterpart (75). The first item of (71) is a consequence of (75), adapting [39, Proposition 4.1, Part 1, Step 5]. The second item of (71) follows from the analogue of (75) on the level of the second-order flux (67), adapting [39, Proposition 4.1, Part 2].

Lemma 1

Let \(d>2\) and \(\langle \cdot \rangle \) satisfy Assumptions 1; let \(\langle \cdot \rangle _L\) be defined with (8). For any deterministic periodic vector field h and any \(p<\infty \), we have

$$\begin{aligned} \Big \langle \Big |\int _{[0,L)^d}h\cdot \nabla \phi ^{(1)}_{i}\Big |^p\Big \rangle _L^\frac{1}{p}&\lesssim _p\Big (\int _{[0,L)^d}|h|^2\Big )^\frac{1}{2}, \end{aligned}$$
$$\begin{aligned} \Big \langle \Big |\int _{[0,L)^d}h\cdot \nabla \phi ^{(2)}_{ij}\Big |^p\Big \rangle _L^\frac{1}{p}&\lesssim _p\Big (\int _{[0,L)^d}|x|^2_L|h|^2\Big )^\frac{1}{2}, \end{aligned}$$

where \(\vert x\vert _L:=\inf _{k\in {\mathbb {Z}}^d}\vert x+kL\vert \).

The choice of the origin of the weight in (75) is of course arbitrary. We note that (75) also holds with the weighted \(L^2\)-norm \(\big (\int _{[0,L)^d}|x|^2_L|h|^2\big )^\frac{1}{2}\) replaced by the \(L^q\)-norm of the same scaling, namely \(\big (\int _{[0,L)^d}|h|^q\big )^\frac{1}{q}\) with \(q=\frac{2d}{d+2}\). However when passing from (75) to (71), we essentially choose \(h=\nabla {{\bar{G}}}(\cdot -z)-\nabla {{\bar{G}}}\), and in the critical dimension \(d=4\), we thus would have \(|f|^q=O(|x|^{-4})\) for \(1\ll |x|\ll |z|\) and thus would obtain a power \(\frac{3}{4}\) on the logarithm \(\ln |z|\) instead of the optimal power \(\frac{1}{2}\).

In establishing (75), we use the same approach as [39, Proposition 4.1] for (74), namely we identify and estimate the Malliavin derivative of the l.h.s. and then appeal to the spectral gap estimate. However, while, for the first-order result (74), a buckling is required, it is not necessary for its second-order counterpart (75). One can avoid it by appealing to the quenched Calderón–Zygmund estimate, see [20] and [39, Proposition 7.1 ii)], albeit in the weighted form of Lemma 2:

Lemma 2

Let \(d>2\) and let \(\langle \cdot \rangle _L\) be an ensemble of \(\lambda \)-uniformly elliptic coefficient fields that are L-periodic. Let the random periodic fields f and u be related by

$$\begin{aligned} \nabla \cdot (a\nabla u+f)=0. \end{aligned}$$

Let \(1<p<p'<\infty \) and \(1<q<\infty \). Suppose that w is arbitrary L-periodic function in Muckenhoupt class \(A_q\),Footnote 9 then the weighted annealed Calderón–Zygmund estimates hold, i.e.,

$$\begin{aligned} \bigg (\int _{[-\small \frac{L}{2},\small \frac{L}{2})^d} \text {d}x w\big<|\nabla u|^{p}\big>^{\frac{q}{p}}_L \bigg )^{1/q} \lesssim _{p,p',q} \bigg (\int _{[-\small \frac{L}{2},\small \frac{L}{2})^d} \text {d}x w\big <|f|^{p'}\big >^{\frac{q}{p'}}_L \bigg )^{1/q}, \end{aligned}$$

where the implicit multiplicative constant depends in addition on the Muckenhoupt norm of w. In particular, for \(w=|x|_L^2\), we obtain

$$\begin{aligned} \bigg (\int _{[-\small \frac{L}{2},\small \frac{L}{2})^d} \text {d}x |x|_L^2\big<|\nabla u|^{p}\big>^{\frac{2}{p}}_L \bigg )^{1/2} \lesssim _{p,p'} \bigg (\int _{[-\small \frac{L}{2},\small \frac{L}{2})^d} \text {d}x |x|_L^2\big <|f|^{p'}\big >^{\frac{2}{p'}}_L \bigg )^{1/2}. \end{aligned}$$

An inspection of the proof of [39, Proposition 7.1 ii)] shows that the argument extends to the case with a weight in the corresponding Muckenhoupt class. Indeed, the only essential new ingredient is that this weighted annealed estimate holds for the constant coefficient operator, i.e., the analogue of [39, Lemma 7.4]. This in turn follows from [54, Theorem 5, p.219] or [46, Theorem 7.1]. Alternatively, one can derive the weighted estimate from the unweighted one and the dualized Lipschitz estimate Lemma 5, following the strategy of [29, Corollary 5]. We finally mention [22, Theorem 4.4] where such annealed regularity estimates are stated and will be proven in [21].

The limit \(L\uparrow \infty \) for the first r.h.s. term in (62) relies on the following purely qualitative consequence of Proposition 3.

Corollary 1

Let \(d>2\) and \(\langle \cdot \rangle \) satisfy Assumptions 1; let \(\langle \cdot \rangle _L\) be defined with (8).

(i) For \(i=1,\cdots ,d\) there exists a unique stationary random field \(\phi _i^{(1)}\) with

$$\begin{aligned} \langle |\phi _i^{(1)}|^p+|\nabla \phi _i^{(1)}|^p\rangle {\mathop {\sim }\limits ^{<}}_p 1\quad \text {for any }p<\infty , \end{aligned}$$

which decays in the sense of

$$\begin{aligned} \Big \langle \Big |\int _{{\mathbb {R}}^d}\eta \phi _i^{(1)}\Big |^p\Big \rangle ^\frac{1}{p}\lesssim _p \Big (\int _{{\mathbb {R}}^d}|\eta |^\frac{2d}{d+2}\Big )^\frac{d+2}{2d} \quad \text { for } \text { all } \text { deterministic } \text { functions }\;\eta , \end{aligned}$$

and which satisfies

$$\begin{aligned} \nabla \cdot a(\nabla \phi _i^{(1)}+e_i)=0\quad \text{ a. } \text{ s. }. \end{aligned}$$

(ii) For \(i,j=1,\cdots ,d\) there exists a unique random field \(\phi ^{(2)}_{ij}\) such that \(\nabla \phi _{ij}^{(2)}\) is stationary and satisfies \(\langle |\nabla \phi _{ij}^{(2)}|^p\rangle \lesssim _p 1\), such that \(\phi ^{(2)}_{ij}\) has moderate growthFootnote 10 in the senseFootnote 11 of

$$\begin{aligned} \langle \vert \phi ^{(2)}_{ij}(z)\vert ^p\rangle ^{\frac{1}{p}}\lesssim _p\mu ^{(2)}_d(\vert z\vert ) \quad \text{ for } \text{ all }\;z, \end{aligned}$$

and which satisfies

$$\begin{aligned} -\nabla \cdot a(\nabla \phi _{ij}^{(2)}+\phi _i^{(1)}e_j) =e_j\cdot (a(\nabla \phi ^{(1)}_i+e_i)-a_{\textrm{hom}}e_i) \quad \text{ a. } \text{ s. }. \end{aligned}$$

(iii) We have

$$\begin{aligned}{} & {} \lim _{L\uparrow \infty }\langle \vert {\overline{a}}-a_{\text {hom}}\vert \rangle _L=0, \quad \lim _{L\uparrow \infty }\langle Q^{(1)}_{ij}(z)\rangle _L=\langle Q^{(1)}_{ij}(z)\rangle \nonumber \\{} & {} \quad \text{ and }\quad \lim _{L\uparrow \infty }\langle Q^{(2)}_{ijm}(z)\rangle _L=\langle Q^{(2)}_{ijm}(z)\rangle \quad \text{ for } \text{ all }\;z, \end{aligned}$$

where also the r.h.s. integrands are defined by the formulas (36) and (37).

The important element of part i) of Corollary 1 is the stationarity of \(\phi _i^{(1)}\) itself, not just of \(\nabla \phi _i^{(1)}\). Note in particular that because of \(\frac{2d}{d+2}>1\), (78) implies \(\langle \phi _i^{(1)}\rangle =0\). Such a result was first established in [30, Proposition 2.1] in the case of a discrete medium, see [27, Proposition 1] for the first result for a continuum medium. For part ii), we note that we cannot expect \(\phi ^{(2)}_{ij}\) to be stationary unless \(d>4\). Part iii) is new and relies on a soft argument based on the uniform bounds of Proposition 3.

3.4 Estimate of Homogenization Error to Second Order, Application to the Green Function

A second main role of the corrector estimates of Proposition 3, in particular the estimate of the flux correctors, is to provide an estimate of the homogenization error. On our second-order level, this connection relies on identity (85) involving the two-scale expansion (84), which we recall now. Suppose that u and \({{\bar{u}}}\) are related via

$$\begin{aligned} \nabla \cdot a\nabla u=\nabla \cdot {{\bar{a}}}\nabla {{\bar{u}}} \end{aligned}$$

and that \({{\bar{u}}}^{(2)}\) is related to \({{\bar{u}}}\) via

$$\begin{aligned} \nabla \cdot ({{\bar{a}}}\nabla \bar{u}^{(2)}+{\overline{a}}^{(2)}_{i}\nabla \partial _{i}{{\bar{u}}})=0. \end{aligned}$$

Consider the error in the second-order two-scale expansion

$$\begin{aligned} w:=u-(1+\phi ^{(1)}_i\partial _i+\phi ^{(2)}_{ij}\partial _{ij})(\bar{u}+{{\bar{u}}}^{(2)}). \end{aligned}$$

Then, \(\sigma ^{(2)}_{ij}\) allows to write the residuum in divergence form:

$$\begin{aligned} -\nabla \cdot a\nabla w=\nabla \cdot \big ((\phi _{ij}^{(2)}a-\sigma _{ij}^{(2)}) \nabla \partial _{ij}({{\bar{u}}}+{{\bar{u}}}^{(2)})+\bar{a}_{i}^{(2)}\nabla \partial _i{{\bar{u}}}^{(2)}\big ). \end{aligned}$$

Now the advantage of A and thus a being symmetric becomes apparent: It implies that the symmetric part of the three-tensor with entries \({{\bar{a}}}^{(2)}_{imn}\) vanishes (see, e.g., [20, Lemma 2.4]). Since (83) may be rewritten as \(-\nabla \cdot {{\bar{a}}}\nabla \bar{u}^{(2)}={\bar{a}}^{(2)}_{imn}\partial _{imn}{{\bar{u}}}\), we may assume \({{\bar{u}}}^{(2)}=0\) under our symmetry assumption. Hence (84) simplifies to

$$\begin{aligned} w:=u-\big (1+\phi ^{(1)}_i\partial _i+(\phi ^{(2)}_{ij}-\phi ^{(2)}_{ij}(0))\partial _{ij}\big )\bar{u} \end{aligned}$$

and (85) may be rewritten as

$$\begin{aligned} -\nabla \cdot a\nabla w=\nabla \cdot \big ((\phi _{ij}^{(2)}-\phi _{ij}^{(2)}(0))a -(\sigma _{ij}^{(2)}-\sigma _{ij}^{(2)}(0))\big )\nabla \partial _{ij}\bar{u}. \end{aligned}$$

We are allowed to pass to the centered versions of the second-order (flux) corrector, by which we mean that \((\phi _{ij}^{(2)},\sigma _{ij}^{(2)})\) is replaced by \((\phi _{ij}^{(2)}-\phi _{ij}^{(2)}(0),\sigma _{ij}^{(2)}-\sigma _{ij}^{(2)}(0))\), which we do with (71) in mind, since a change by an additive constant does not affect anything stated so far, and in particular not formula (67), on which (85) solely relies. The upcoming lemma provides an estimate of the second-order stochastic homogenization error w; (89) is optimal since the rate is governed by the dimension-dependent expression \(\mu _{d}^{(2)}\) with its argument given by the scale of the r.h.s. h, which here is expressed by the diameter 2R of its support. Since the estimate is pointwise in the gradient, a logarithm is unavoidable,Footnote 12 and a r.h.s. norm marginally stronger than \(\sup |\nabla ^2h|\) has to be used.Footnote 13

Lemma 3

Let \(d>2\) and \(\langle \cdot \rangle \) satisfy Assumptions 1 with symmetric A; let \(\langle \cdot \rangle _L\) be defined as in Sect. 1.5. Given a deterministic and smooth function f supported in \(B_R(y)\) with \(y\in {\mathbb {R}}^d\) and some \(R<\infty \), let u and \({{\bar{u}}}\) be the decaying solutions of

$$\begin{aligned} -\nabla \cdot a\nabla u=f=-\nabla \cdot {{\bar{a}}}\nabla {{\bar{u}}}. \end{aligned}$$

Then, w defined in (86) satisfies for all \(p<\infty \)

$$\begin{aligned} \langle |\nabla w(0)|^p\rangle _L^\frac{1}{p} \lesssim _p \max \{\mu _{d}^{(2)}(R),\ln R\}R\sup |\nabla ^2 f|. \end{aligned}$$

This pointwise estimate (89) relies on a decomposition of the r.h.s. of (87) into pieces supported on dyadic annuli. For each piece, we first apply Lemma 5 combined with the energy estimate, into which we feed (71) and a pointwise bound on \(\nabla ^3{{\bar{u}}}\) relying on the bounds on the Green function of the constant coefficient operator \(\nabla \cdot {\overline{a}}\nabla \), see (88).

The main goal of this subsection is to estimate the homogenization error on the level of the Green function, see Proposition 4. This type of homogenization result with singular r.h.s. has been worked out on the level of the first-order approximation in [10, Corollary 3] and extended to second order in [9, Theorem 1], where these estimates are derived from estimates on \((\phi ^{(1)}_i,\sigma _i^{(1)})\) and \((\phi ^{(2)}_{ij},\sigma ^{(2)}_{ij})\) of the type of Proposition 3, however in a pathwise way, see [9, Proposition 1]. While equipped with Proposition 3, we could post-process [9, Theorem 1] to obtain Proposition 4, we take a different, and shorter, route in this paper. Note that [9, Theorem 1] is not formulated in terms of the Green function G, but in terms of decaying a-harmonic functions in exterior domains. Recovering a statement on the Green function would require [10, Lemma 4], which we restate as Lemma 4 below for the convenience of the reader.

Proposition 4

. Let \(d>2\) and \(\langle \cdot \rangle \) satisfy Assumptions 1 with symmetric A; let \(\langle \cdot \rangle _L\) be defined as in Sect. 1.5. Then we have for \({{\mathcal {E}}}\) defined in (58)

$$\begin{aligned} |y-x|^{d+2}\langle |{{\mathcal {E}}}(x,y)|^p\rangle _L^\frac{1}{p} \lesssim _p\max \{\mu _d^{(2)}(|y-x|),\ln \vert x-y\vert \} \end{aligned}$$

provided \(|y-x|\ge 2\) and for all \(p<\infty \).

Here comes the crucial Lemma that converts weak into strong control.

Lemma 4

Let \(\langle \cdot \rangle \) be an ensemble of \(\lambda \)-uniformly elliptic coefficient fields.Footnote 14 Let the random function u be a-harmonic in the ball \(B_R\) of radius R. Then, we have for all \(p<\infty \)


Here \(\lesssim _{p}\) has the same meaning as in Proposition 3.

Lemma 4 amounts to an inner regularity estimate for a-harmonic functions u, in terms of the norms \(L^p_{\langle \cdot \rangle }L^2_x\) and \(W^{-2,1}_xL^p_{\langle \cdot \rangle }\) on the level of the gradient \(\nabla u\). As [10, Lemma 4], estimate (91) is a consequence of an inner regularity estimate, uniform in a, with respect to norms \(L^2_x\) and \(H^{-n}_x\) (the case \(W^{-2,1}_x\) of (91) is obtained for \(n>\frac{d}{2}+2\)). However, it strengthens [10, Lemma 4] by restricting the r.h.s. functional to smooth functions g with compact support, i.e., functions that vanish to appropriate order at the boundary.

Nevertheless, it requires only a minor modification of the proof. It is obtained as a combination of two ingredients. First, by the Caccioppoli estimate and by an \(L^2_x\) interpolation estimate, we may estimate the l.h.s. of (91) by the \(L^2_x\) norm of w for \(\Delta ^{2n} w = u\). Second, appealing to the fact that the Dirichlet operator \(\Delta ^{2n}\) has finite trace for \(2n>d\), we may obtain (91). This second step differs from [10, Lemma 4], where the Fourier decomposition was explicitly used to solve \(\Delta ^{2n} w = u\) (thus, losing the property of compact support). This argument also shows that the second derivative on g, that we need here for our second-order homogenization, could be replaced by any order (properly non-dimensionalized).

We use Lemma 4 only in combination with a second inner regularity estimate, Lemma 5, which amounts to a Lipschitz estimate. Lipschitz estimates are central in the large-scale regularity theory in homogenization as initiated by Avellaneda and Lin in the periodic context, and as introduced by Armstrong and Smart [5] to the random context.

Lemma 5

Let \(d>2\) and \(\langle \cdot \rangle \) satisfy Assumptions 1; let \(\langle \cdot \rangle _L\) be defined as in Sect. 1.5. Let the random function u be a-harmonic in the ball \(B_R\) of radius R. Then, we have for all \(p',p<\infty \)


Here \(\lesssim _{p,p'}\) has the same meaning as in Proposition 3.

Lemma 5 is an easy consequence of the pathwise Lipschitz estimate [29, Theorem 1]. More precisely, we refer to [29, (16)], which takes the form of

with the random radius \(r_*\) defined in [29, (12)]. It easily follows from the estimates on \((\phi _i^{(1)},\sigma _i^{(1)})\) in Proposition 3 that \(\langle r_*^p\rangle _L^\frac{1}{p}\) \(\lesssim _{p}1\) for all \(p<\infty \). On the other hand, by standard Schauder theory in \(C^{\alpha '}\) we have

where C depends at most polynomially on the local Hölder norm \([a]_{\alpha ,B_1}\) (recalling that it satisfies (14)). Now (92) follows from combining both estimates; note that the loss in stochastic integrability is unavoidable, since it compensates the fact that both \(r_*\) and \([a]_{\alpha ,B_1}\) are not uniformly bounded.

As mentioned, we use Lemma 4 only in its form combined with Lemma 5

Corollary 2

Let \(d>2\) and \(\langle \cdot \rangle \) satisfy Assumptions 1; let \(\langle \cdot \rangle _L\) be defined as in Sect. 1.5. Let the random function u be a-harmonic in the ball \(B_R\) of radius R. Then, we have for all \(p',p<\infty \)


Corollary 2 amounts to an inner regularity estimate for a-harmonic functions u, in terms of the norms \(L^\infty _xL^p_{\langle \cdot \rangle }\) and \(W^{-2,1}_xL^p_{\langle \cdot \rangle }\) on the level of the gradient \(\nabla u\). We call this estimate an annealed estimate, since now on both sides of (93), the probabilistic norm is inside.

In this paper, we use Lemma 4, or rather Corollary 2, in a more substantial way than it is used in [10, Corollary 3]. Here comes an outline of the argument for Proposition 4: We apply Lemma 3 with the origin replaced by a general point \(x_0\). Writing \(u(x)=\int _{{\mathbb {R}}^d}\textrm{d}y\,h(y)\cdot \nabla _yG(x,y)\) and \(\bar{u}(x)=\int _{{\mathbb {R}}^d}\textrm{d}y\,h(y)\cdot \nabla _y{\overline{G}}(x,y)\), this provides control of

$$\begin{aligned} \int _{{\mathbb {R}}^d}\,\textrm{d}y (\nabla \cdot h)(y)\bigg (\nabla _xG(x_0,y)&-\partial _i{\overline{G}}(x_0-y)(e_i+\nabla \phi ^{(1)}_i(x_0))\\&-\partial _{ij}{\overline{G}}(x_0-y)(\phi _i^{(1)}(x_0)e_j+\nabla \phi ^{(2)}_{ij}(x_0))\bigg ), \end{aligned}$$

in terms of \(\mu ^{(2)}_d(R)\sup |\nabla ^2\,h|\) with 2R the diameter of \(\text {supp }g\); here we used the centering of \(\phi ^{(2)}_{ij}\) in \(x_0\). We now fix a point \(y_0\) with \(|y_0-x_0|\ge 4\) and replace both instances of \({\overline{G}}(x_0-y)\) by what we obtain from applying the two-scale expansion operator in the y-variable

$$\begin{aligned} 1+\phi _m^{*(1)}(y)\frac{\partial }{\partial y_m} +(\phi ^{*(2)}_{mn}(y)-\phi ^{*(2)}_{mn}(y_0))\frac{\partial ^2}{\partial y_m\partial y_n}. \end{aligned}$$

Provided h is supported in \(B_R(y_0)\) with \(R:=\frac{1}{2}|y_0-x_0|\ge 2\), this preserves the estimate: While for three out of the four extra terms, this follows directly from parts i) through iii) of Proposition 3, we need part iv) and an integration by parts in y for the contribution coming from \(\phi _i^{*(1)}(y)\partial _{im}{\overline{G}}(x_0-y)\) \((e_i+\nabla \phi _i^{(1)}(x_0))\). Keeping only first- and second-order terms and recalling the definition (58), this yields

$$\begin{aligned} \bigg \langle \bigg |\int _{{\mathbb {R}}^d}\textrm{d}y\,h(y) \cdot {\mathcal E}_m(x_0,y)\bigg |^p\bigg \rangle _L^\frac{1}{p}\lesssim \mu _d^{(2)}(R)\sup |\nabla ^2 h| \end{aligned}$$

for any h supported in \(B_R(y_0)\). By construction, up to third-order terms, \({\mathbb {R}}^d-\{x_0\}\ni y\mapsto {\mathcal E}_m(x_0,y)\) is a linear combination of a gradient of an \(a^*\)-harmonic function, namely \(\frac{\partial G}{\partial x_m}(x_0,y)\), and gradients of two-scale expansions of \(\bar{a}^*\)-harmonic functions, namely of \({{\bar{u}}}(y)\) \(=\partial _i{\overline{G}}(x_0-y)(\delta _{im}+\partial _m\phi ^{(1)}_i(x_0))\) and of \({{\bar{u}}}(y)\) \(=\partial _{ij}{\overline{G}}(x_0-y)\) \((\phi _i^{(1)}(x_0)\delta _{jm}\) \(+\partial _m\phi ^{(2)}_{ij}(x_0))\). Hence we may appeal once more to (87), this time in the y-variable and thus for the dual medium, and with the origin replaced by \(y_0\). We decompose the r.h.s. of (87) into a far field supported on \({\mathbb {R}}^d-B_R(y_0)\) and a near field supported on dyadic annuli centered at \(y_0\) of radii \(R,\frac{R}{2},\frac{R}{4},\cdots \). For the near-field contributions, we appeal to the energy estimate followed by Lemma 5. For the far-field contribution, we use (94) (in conjunction with the estimate of the near-field part) by appealing to Corollary 2, both with the origin replaced by \(y_0\). It is thus Corollary 2 that converts the weak control (94) into pointwise control (90).

4 Heuristic Result

In this section, we heuristically argue that the strategy of “periodizing the realizations” leads to a bias that is of order \(O(L^{-1})\), as announced in (4). We argue that this is the case even for an isotropicFootnote 15 range-one medium in the small contrast regime.Footnote 16 However, rather than extending the a restricted to the RVE periodically, we extend it by even reflection; this amounts to imposing flux boundary instead of periodic boundary conditions. More precisely, fixing a direction \(\xi =e_1\), we impose flux boundary conditions in just one of the directions orthogonal to \(e_1\), say \(e_d\), and resort to the strategy of “periodizing the ensemble” in the other \(d-1\) directions. Hence, we give the naive strategy a pole position: We implement it in the less intrusive form of reflection rather than periodization—less intrusive because it does not create discontinuities in the coefficient field. Nonetheless, treating just one of the directions in this naive way increases the bias scaling from \(O(L^{-d})\) to \(O(L^{-1})\). Admittedly, the heuristic analysis also becomes simpler by considering the reflective version, and by implementing it in just one direction. Incidentally, by a similar heuristic argument we also convinced ourselves that the Dirichlet boundary condition leads to a bias of the same order (but different sign).

We now make this more precise: Periodizing our stationary centered Gaussian ensemble in directions \(i=1,\ldots ,d-1\) on the level of the covariance function amounts to

$$\begin{aligned} c'_L(z):=\sum _{k'\in {\mathbb {Z}}^{d-1}\times \{0\}}c(z+Lk'), \end{aligned}$$

cf. (9). We pick a realization according to this \(\langle \cdot \rangle _L'\), restrict it to the stripe \({\mathbb {R}}^{d-1}\times [0,\frac{L}{2}]\), extend it by even reflection to \({\mathbb {R}}^{d-1}\times [-\frac{L}{2},\frac{L}{2}]\) and then extend it L-periodically in the \(e_d\)-direction to all of \({\mathbb {R}}^d\). This defines a (non-stationary) centered Gaussian ensemble \(\langle \cdot \rangle _{L}^{\textrm{sym}}\), as such determined by its covariance function \(c^{\textrm{sym}}_L(y,x)=c^{\textrm{sym}}_L(x,y):=\langle g(x) g(y)\rangle _L^{\textrm{sym}}\). The covariance function is characterized by its connection to \(c'_L\) via

$$\begin{aligned} c^{\textrm{sym}}_L(x,y)=c_L'(x-y)\quad \text{ provided }\;x,y\in {\mathbb {R}}^{d-1}\times [0,\tfrac{L}{2}] \end{aligned}$$

and its reflection and translation symmetriesFootnote 17

$$\begin{aligned} c^{\textrm{sym}}_L(x,y)&=c^{\textrm{sym}}_L(x,y-2y_de_d), \end{aligned}$$
$$\begin{aligned} c^{\textrm{sym}}_L(x,y)&=c^{\textrm{sym}}_L(x,y+Le_d), \end{aligned}$$

see Fig. 1.

Fig. 1
figure 1

Piecewise definition of \(c^{\textrm{sym}}_L(x,y)\) as a function of \(x_d, y_d\) (with the argument \(x'-y'\) of \(c'_L\) suppressed)

It obviously inherits stationarity and periodicity in directions \(i=1,\ldots ,d-1\) from \(c'_L\) so that

$$\begin{aligned} c^{\textrm{sym}}_L(x,y)&=c^{\textrm{sym}}_L(x+z',y+z')\quad \text{ for }\;z'\in {\mathbb {R}}^{d-1}\times \{0\},\nonumber \\ c^{\textrm{sym}}_L(x,y)&=c^{\textrm{sym}}_L(x,y+Le_i)\quad \text{ for }\;i=1,\cdots ,d. \end{aligned}$$

Hence comparing \(\langle \cdot \rangle _L\) to \(\langle \cdot \rangle _L^{\textrm{sym}}\), we keep (full) periodicity, lose stationarity in direction \(e_d\) but gain reflection symmetry in that direction. There are two derived symmetries that will play a role, namely

$$\begin{aligned} c^{\textrm{sym}}_L(x,y)&=c^{\textrm{sym}}_L(x,y+(L-2y_d)e_d), \end{aligned}$$
$$\begin{aligned} c^{\textrm{sym}}_L(x,y)&=c^{\textrm{sym}}_L(x+\tfrac{L}{2}e_d,y+\tfrac{L}{2}e_d). \end{aligned}$$

While (100) is an obvious combination of (97) and (98), (101) requires an argument, see Appendix B.1.

As for (7), we think of \(\langle \cdot \rangle _L^{\textrm{sym}}\) as denoting also the push-forward of the Gaussian ensemble under \(a=A(g)\). We now sample a from \(\langle \cdot \rangle _L^{\textrm{sym}}\). By construction, the scalar a is not only L-periodic in every direction, cf. (99), but in addition even under reflection along the hyper planes \(\{x_d=0\}\) and \(\{x_d=\frac{L}{2}\}\), cf. (97), (100) and Fig. 1. These invariances are transmitted to the solution \(\phi _i^{(1)}\) of (2), and to the flux components \(e_i\cdot a(\nabla \phi _i^{(1)}+e_i)\) for any \(i\ne d\). On the other hand, the flux component \(e_d\cdot a(\nabla \phi _1^{(1)}+e_1)\) is odd w.r.t.  these reflections and thus vanishes along these two hyper planes. Hence when it comes to \(\phi ^{(1)}_1\), the box \([-\frac{L}{2},\frac{L}{2}]^{d-1}\times [0,\frac{L}{2}]\) can be seen as an RVE with a flux boundary conditions in direction \(e_d\) and periodic boundary conditions in directions \(i=1,\cdots ,d-1\). By the above reflection symmetry, we have

where \({{\bar{a}}}\) is defined as in (3). We shall heuristically establish (4) in the form of

$$\begin{aligned} e_1\cdot (\langle {{\bar{a}}}\rangle _L^{\textrm{sym}}-a_{\textrm{hom}})e_1=O(L^{-1}). \end{aligned}$$

More precisely, we shall show that to leading order in \(1-\lambda \ll 1\) and \(L\gg 1\),

$$\begin{aligned} e_1\cdot (\langle {{\bar{a}}}\rangle _L^{\textrm{sym}}-a_{\textrm{hom}})e_1\approx -L^{-1}I \quad \text{ with }\;I>0. \end{aligned}$$

The sign of the leading-order correction I is consistent with the following heuristics: The no-flux boundary conditions means that the current is restricted to the stripe \({\mathbb {R}}^{d-1}\times [0,\frac{L}{2}]\); for \(d=2\) and as \(L\downarrow 0\), the medium thus is close to a one-dimensional medium, for which one has the effective conductivity \(\langle a^{-1}\rangle ^{-1}\le a_{\textrm{hom}}\) (note that by statistical isotropy, also \(a_{\textrm{hom}}\) is scalar).

As for \(\langle {{\bar{a}}}\rangle _L\), we shall establish (102) by monitoring its L-derivative. More precisely, appealing to (39), we shall establish (102) in form of

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}e_1\cdot (\langle {{\bar{a}}} \rangle _L^{\textrm{sym}} -\langle \bar{a}\rangle _L)e_1\approx L^{-2} I. \end{aligned}$$

The advantage of monitoring the difference between two ensembles is that we may use Price’s formula in a different, much less subtle, way than in Sect. 2.1. More precisely, in Appendix B.3, for a general \([0,L)^d\)-periodic centered Gaussian ensemble \(\langle \cdot \rangle _{{\mathfrak {c}}_L}\) we shall establish the formula

$$\begin{aligned}{} & {} {\frac{\textrm{d}}{\textrm{d}L}\langle e_1\cdot {{\bar{a}}} e_1\rangle _{{\mathfrak {c}}_L} = -L^{-d}\int _{[-\frac{L}{2},\frac{L}{2})^d}\textrm{d}x\int _{[-\frac{L}{2},\frac{L}{2})^d}\textrm{d}y} \nonumber \\{} & {} \times \big \langle \big (a'(\nabla \phi _1^{(1)}+e_1)\big )(x)\cdot \nabla \nabla G^{per}(x,y) \cdot \big ( a'(\nabla \phi _1^{(1)}+e_1)\big )(y)\big \rangle _{{\mathfrak {c}}_L}\frac{D{\mathfrak {c}}_L}{\partial L}(x,y), \nonumber \\ \end{aligned}$$

where the “material” derivative of the covariance function \({\mathfrak {c}}_L(x,y)\) is defined via:

$$\begin{aligned} \frac{D{\mathfrak {c}}_L}{\partial L}(L{{\hat{x}}},L{{\hat{y}}})=\frac{\textrm{d}}{\textrm{d}L}\big ({\mathfrak {c}}_L(L{{\hat{x}}},L{{\hat{y}}})\big ), \end{aligned}$$

and where \(G^{per}\) denotes the Green function associated with the operator \(-\nabla \cdot a\nabla \) on the torus \([0,L)^d\), which is unambiguously defined in terms of its first and mixed derivatives. The present version of (104) also relies on the assumption

$$\begin{aligned} \frac{D{\mathfrak {c}}_L}{\partial L}(x,y)=0\quad \text{ for }\;x=y. \end{aligned}$$

Note that definition (105) implies

$$\begin{aligned} \frac{D{\mathfrak {c}}_L}{\partial L}\quad \text{ is } [0,L)^d- \text{ periodic } \text{ in } \text{ both } \text{ arguments }, \end{aligned}$$
$$\begin{aligned} \frac{D{\mathfrak {c}}_L}{\partial L}=\frac{\partial {\mathfrak {c}}_L}{\partial L}+L^{-1}(x\cdot \nabla _x +y\cdot \nabla _y){\mathfrak {c}}_L. \end{aligned}$$

Let us compare formula (104), which in the presence of stationarity simplifies (in the sense that \(L^{-d}\int _{[-\frac{L}{2},\frac{L}{2})^d}\textrm{d}y\) is replaced by the evaluation at \(y=0\)), to (24). The main difference does not lie in the periodic setting (in view of (107), \(\int _{[-\frac{L}{2},\frac{L}{2})^d}\textrm{d}x\) and \(G^{per}\) can formally be replaced by \(\int _{{\mathbb {R}}^d}\textrm{d}x\) and G, respectively), but in the convective contribution to (108), which is the generator of rescaling the space variables of c, and thus describes a rescaling of a. Indeed, this contribution vanishes after applying \(\int _{{\mathbb {R}}^d}\textrm{d}x\) (only formally, since the integral does not converge absolutely) because the space average \({{\bar{a}}} e_1\) is invariant under rescaling of a.

As in Sect. 2.4, in the small contrast regime, we have

$$\begin{aligned}&\langle \big (a'(\nabla \phi _1^{(1)}+e_1)\big )(x)\cdot \nabla \nabla G^{per}(x,y) \cdot \big ( a'(\nabla \phi _1^{(1)}+e_1)\big )(y)\big \rangle _{{\mathfrak {c}}_L} \\&\approx -\partial _1^2G^{per}_\textrm{hom}(x-y) \langle a'(x) a'(y)\rangle _{{\mathfrak {c}}_L}, \end{aligned}$$

so that (104) simplifies to

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}\langle e_1\cdot {{\bar{a}}} e_1\rangle _{{\mathfrak {c}}_L}&\approx L^{-d}\int _{[-\frac{L}{2},\frac{L}{2})^d}\textrm{d}x\int _{[-\frac{L}{2},\frac{L}{2})^d}\textrm{d}y\partial _1^2 G^{per}_{\textrm{hom}}(x-y)\langle a'(x) a'(y)\rangle _{{\mathfrak {c}}_L} \frac{D{\mathfrak {c}}_L}{\partial L}(x,y). \end{aligned}$$

We apply (109) to both \([0,L)^d\)-periodic ensembles, \(\langle \cdot \rangle _L^{\textrm{sym}}\) and \(\langle \cdot \rangle _L\), which we may since (106) is satisfied (almost everywhere) for both: Indeed, by reflection symmetry (97) and periodicity (98), it is enough to consider \(x\in {\mathbb {R}}^{d-1}\times [0,\frac{L}{2})\). Then, for \(|y-x|\) sufficiently small we have \(c_L^{\textrm{sym}}(x,y)=c_L(x-y)\) by (96). By the finite-range assumption on c, this yields

$$\begin{aligned} c_L^{\textrm{sym}}(x,y)\!=\! c_L(x-y)\!=\!c(x-y)\,\text{ provided }\;|x-y|\;\text{ is } \text{ sufficiently } \text{ small }\; \text{ and }\;L\gg 1. \end{aligned}$$

Hence, we have for \(y=x\) that \(\frac{\partial }{\partial L}c_L^{\textrm{sym}}(x,y)\) \(=\frac{\partial }{\partial L}c_L(0)\) \(=0\) and \(-\nabla _yc^{\textrm{sym}}_L(x,y)\) \(=\nabla _xc^{\textrm{sym}}_L(x,y)\) \(=\nabla c_L(0)\) \(=0\). Introducing in addition the function \({{\mathcal {A}}}\) as in Sect. 2.4 for both \(\langle a'(x) a'(y)\rangle _L\) \(={{\mathcal {A}}}'(c_L(x,y))\) and \(\langle a'(x) a'(y)\rangle _L^{\textrm{sym}}\) \(={{\mathcal {A}}}'(c_L^{\textrm{sym}}(x,y))\), we obtain from (109)

$$\begin{aligned}&\frac{\textrm{d}}{\textrm{d}L}e_1\cdot (\langle {{\bar{a}}}\rangle _L^{\textrm{sym}}-\langle \bar{a}\rangle _L)e_1 \approx L^{-d}\int _{[-\frac{L}{2},\frac{L}{2})^d}\textrm{d}x \nonumber \\&\int _{[-\frac{L}{2},\frac{L}{2})^d}\textrm{d}y\partial _1^2 G^{per}_{\textrm{hom}}(x-y)\frac{D}{\partial L}\big ({\mathcal A}(c_L^{\textrm{sym}}) -{{\mathcal {A}}}(c_L)\big )(x,y). \end{aligned}$$

There is a cancellation when considering the difference in (110): Indeed, by (96) and (97) (see also Fig. 1) we have

$$\begin{aligned} c_L^{\textrm{sym}}(x,y)=c_L'(x-y)\quad \text{ provided }\;(x_d,y_d)\in [-\frac{L}{2},0]^2\cup [0,\frac{L}{2}]^2. \end{aligned}$$

Likewise, we obtain from (9), (95) and the finite range of dependence assumption

$$\begin{aligned} c_L(x-y)=c_L'(x-y)\quad \text{ provided }\;(x_d,y_d)\in [-\frac{L}{2},0]^2\cup [0,\frac{L}{2}]^2. \end{aligned}$$

Hence the integral in (110) reduces to \((x_d,y_d)\) \(\in ([-\tfrac{L}{2},0]\times [0,\tfrac{L}{2}])\) \(\cup ([0,\tfrac{L}{2}]\times [-\tfrac{L}{2},0])\). Moreover, since the integrand in (109) is invariant under permuting x and y, we obtain

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}&e_1\cdot (\langle {{\bar{a}}}\rangle _L^{\textrm{sym}} -\langle \bar{a}\rangle _L)e_1\approx 2L^{-d}\int _{[-\frac{L}{2},\frac{L}{2})^{d-1}\times [0,\frac{L}{2}]}\textrm{d}x \int _{[-\frac{L}{2},\frac{L}{2})^{d-1}\times [-\frac{L}{2},0]}\textrm{d}y\nonumber \\&\times \partial _1^2 G^{per}_{\textrm{hom}}(x-y) \frac{D}{\partial L}\big ({{\mathcal {A}}}(c_L^{\textrm{sym}})-{{\mathcal {A}}}(c_L)\big )(x,y). \end{aligned}$$

It follows from a combination of symmetries (97) and (98) and (101) that \(c_L^{\textrm{sym}}\) is invariant under the inversion at \((\frac{L}{4},-\frac{L}{4})\) in the \((x_d,y_d)\) plane, that is,

$$\begin{aligned} (x,y)\mapsto ((x',\tfrac{L}{2}-x_d),(y',-\tfrac{L}{2}-y_d)). \end{aligned}$$

By stationarity, periodicity, and (236), \(c_L\) has the same symmetry (112). By periodicity and radial symmetry, \((x,y)\mapsto \partial _1^2 G^{per}_{\textrm{hom}}(x-y)\) also has symmetry (112). Since the triangle in the \((x_d,y_d)\) plane

$$\begin{aligned} \Delta :=\big \{x_d\ge 0,\quad y_d\le 0,\quad x_d-y_d\le \tfrac{L}{2}\big \} \end{aligned}$$

is such that its (disjoint) union with its image under (112) renders the rectangle \([0,\frac{L}{2}]\times [-\frac{L}{2},0]\), (111) may be rewritten as

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}&e_1\cdot (\langle {{\bar{a}}}\rangle _L^{\textrm{sym}} -\langle \bar{a}\rangle _L)e_1\approx 4L^{-d}\int _{[-\frac{L}{2},\frac{L}{2})^{d-1}}\textrm{d}x' \int _{[-\frac{L}{2},\frac{L}{2})^{d-1}}\textrm{d}y'\int _{\Delta }\textrm{d}x_d\textrm{d}y_d\nonumber \\&\times \partial _1^2 G^{per}_{\textrm{hom}}(x-y) \frac{D}{\partial L}\big ({{\mathcal {A}}}(c_L^{\textrm{sym}})-{{\mathcal {A}}}(c_L)\big )(x,y). \end{aligned}$$

By the finite range assumption and for \(L\gg 1\), we have (where for a stationary ensemble we identify \(c(x,y)=c(x-y)\))

$$\begin{aligned} c_L(x,y)=c_L'(x,y)\quad \text{ provided }\;(x_d,y_d)\in \Delta , \end{aligned}$$

so that in (114), we may replace \(c_L\) by \(c_L'\). Likewise, by definition (96) and (97) (see also Fig. 1) we have

$$\begin{aligned} c_L^{\textrm{sym}}(x,y)=c_L'(x,y-2y_de_d)\quad \text{ provided }\;(x_d,y_d)\in \Delta , \end{aligned}$$

so that in (114), we may express \(c_L^{\textrm{sym}}\) in terms of \(c_L'\). Since both ensembles \(\langle \cdot \rangle _L'\) and \(\langle \cdot \rangle _L^{\textrm{sym}}\) are stationary in directions \(i=1,\cdots ,d-1\), (114) may be rewritten as

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}&e_1\cdot (\langle {{\bar{a}}}\rangle _L^{\textrm{sym}} -\langle \bar{a}\rangle _L)e_1\approx 4L^{-1}\int _{[-\frac{L}{2},\frac{L}{2}]^{d-1}}\textrm{d}x'\int _{\Delta }\textrm{d}x_d\textrm{d}y_d\, \partial _1^2 G^{per}_{\textrm{hom}}(x-(0,y_d))\nonumber \\&\times \big (\frac{D}{\partial L}{{\mathcal {A}}}(c_L')(x,(0,-y_d)) -\frac{D}{\partial L}{{\mathcal {A}}}(c_L')(x,(0,y_d))\big ). \end{aligned}$$

By the finite range assumption and for \(L\gg 1\), we have

$$\begin{aligned} c_L'(x,(0,y_d))=c(x,(0,y_d))\quad \text{ provided }\;x\in [-\tfrac{L}{2},\tfrac{L}{2}]^{d-1}, \end{aligned}$$

so that the material derivative in (115) reduces to the convective derivative; on stationary c’s, the convective derivative \(L^{-1}(x\cdot \nabla _x+y\cdot \nabla _y)\) acts as \(L^{-1}z\cdot \nabla \) with \(z=x-y\), and thus as the radial derivative \(L^{-1}r\partial _r\) with \(R=|z|\). Hence, (115) takes the form

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}L}&e_1\cdot (\langle {{\bar{a}}}\rangle _L^{\textrm{sym}} -\langle \bar{a}\rangle _L)e_1\approx 4L^{-2}\int _{[-\frac{L}{2},\frac{L}{2}]^{d-1}}\textrm{d}x'\int _{\Delta }\textrm{d}x_d\textrm{d}y_d\, \partial _1^2 G^{per}_{\textrm{hom}}(x',x_d-y_d)\nonumber \\&\times \big ((r\partial _r){{\mathcal {A}}}(c)(x',x_d+y_d) -(r\partial _r){{\mathcal {A}}}(c)(x',x_d-y_d)\big ). \end{aligned}$$

We now proceed to a second (and last) approximation. Recall our assumption that c is supported on the unit ball, hence the effective domain of integration in (116) is \(|x'|\le 1\), next to \(0\le x_d-y_d\le \frac{L}{2}\), cf. (113). In this range, we may approximate \(G_{\textrm{hom}}^{per}(x',x_d-y_d)\) by \(G_\textrm{hom}(x',x_d-y_d)\). After this substitution, we may neglect the restriction \(x_d-y_d\le \frac{L}{2}\). Hence (116) implies (103) where the L-independent quantity I is defined via

$$\begin{aligned} I&:=4\int _{{\mathbb {R}}^{d-1}}\textrm{d}x'\int _{0}^{\infty }\textrm{d}x_d\int _{-\infty }^{0}\textrm{d}y_d\, \partial _1^2 G_{\textrm{hom}}(x',x_d-y_d)\nonumber \\&\quad \times \big ((r\partial _r){{\mathcal {A}}}(c)(x',x_d+y_d) -(r\partial _r){{\mathcal {A}}}(c)(x',x_d-y_d)\big ). \end{aligned}$$

In Appendix B.2, we compute this integral:

$$\begin{aligned} I=\frac{32}{(d+1)(d-1)}\frac{|B_1'|}{|\partial B_1|} \int _0^\infty dr{{\mathcal {A}}}(c), \end{aligned}$$

where \(B_1\) is the unit ball in \({\mathbb {R}}^{d-1}\) and \(B'_1\) in \({\mathbb {R}}^{d-1}\).

In particular, we have \(I>0\), see Sect. 2.4 for the explanation why \({\mathcal {A}}\) is a nonnegative function (different from 0).

5 Proofs

5.1 Proof of Theorem 2: Asymptotic of the Bias

The goal is to pass from the representation in Proposition 2 to the asymptotics in Theorem 2. To do so, we have to show that the first r.h.s. term of (62), up to the factor \(L^{d+1}\), converges to the r.h.s. term of (39), and that the remaining terms are \(o(L^{-(d+1)})\). Note that by integration, (10) implies

$$\begin{aligned} \sup _{x}(1+\vert x\vert ^2)^{\frac{d+1}{2}+\alpha }\vert \nabla c(x)\vert <\infty , \end{aligned}$$

so that from (25) and Proposition 3i), the fifth term is directly of order \(L^{-(d+1+2\alpha )}\) which as desired is \(o(L^{-(d+1)})\). We now discuss the first four terms. Without loss of generality, we henceforth assume that the exponent \(\alpha >0\) is (sufficiently) small.

We start with the second term and estimate \(\epsilon ^{(1)}\), see (60): In the range \(|k|\ge \frac{|z|}{L}\), we obtain from Taylor applied to \((1-\eta _L)\partial _{ij}{{\bar{G}}}\) that the summand is estimated by \(\vert k\vert |z|^2(L|k|)^{-(d+2)}\) \(\le |z|(L|k|)^{-(d+1)}\). Hence, the contribution to the sum from this range is dominated by \(\min \{|z|^2\,L^{-(d+2)},\) \(|z|L^{-(d+1)}\}\). In the other range \(|k|\le \frac{|z|}{L}\), the contribution from the middle term vanishes by parity, the contribution from the last term is estimated by \(|z|L^{-(d+1)}\) (by a similar argument to the one that shows that the limit (59) exists), and the first term in the summand is estimated by \(|k|(\vert k\vert L)^{-d}\) so that its contribution to the sum is also dominated by \(|z|L^{-(d+1)}\). Since this second range is only present for \(|z|\ge L\), we obtain in conclusion

$$\begin{aligned} |\epsilon ^{(1)}_{Lijn}(z)|&\lesssim \min \{|z|^2L^{-(d+2)},|z|L^{-(d+1)}\}. \end{aligned}$$

For the estimate of \(\epsilon ^{(2)}\), see (61), we proceed in a similar way and obtain the stronger estimate

$$\begin{aligned} |\epsilon ^{(2)}_{Lijmn}(z)|&\lesssim |z|L^{-(d+2)}. \end{aligned}$$

We combine the estimates (120) and (121) with the corrector estimates of Proposition 3i), which by definitions (36) and (37) yield for all \(p<\infty \)

$$\begin{aligned} \langle |Q^{(1)}_{ij}(z)|^p\rangle ^{\frac{1}{p}}_L+\langle |Q^{(2)}_{ijm}(z)|^p\rangle ^{\frac{1}{p}}_L\lesssim 1. \end{aligned}$$

We now see that Assumption 1 is just what we need: By (119), we obtain for the second term in (62)

$$\begin{aligned} \Big |\int _{{\mathbb {R}}^d}\text {d}z \Big \langle \epsilon ^{(2)}_{Lijmn}(z)Q^{(2)}_{ijm}(z) +\epsilon ^{(1)}_{Lijn}(z)Q^{(1)}_{ij}(z)\Big \rangle _L \partial _nc(z)\Big |\lesssim L^{-(d+2)}+L^{-(d+1+2\alpha )}, \end{aligned}$$

which as desired is \(o(L^{-(d+1)})\). In this subsection, \(\lesssim \) means \(\le \) up to a multiplicative constant that only depends on d, \(\lambda \), and the constants implicit in (10) and (11) of Assumption 1.

We now turn to the third term on the r.h.s. of (62). It follows from Proposition 3i) and Proposition 4, together with (11) in Assumption 1, that

$$\begin{aligned}&\Big |\Big \langle (\nabla \phi ^{*(1)}+\xi ^*)(0)\cdot a'(0){\mathcal E}(0,z) a'(z)(\nabla \phi ^{(1)}+\xi )(z)\Big \rangle _L\Big | \nonumber \\ {}&\lesssim \max \{\mu ^{(2)}_d(\vert z\vert ),\ln \vert z\vert \}\vert z\vert ^{-(d+2)}\lesssim |z|^{-(d+\frac{3}{2})}. \end{aligned}$$

Inserting (25), we obtain the following estimate

$$\begin{aligned}&\Big |\int _{{\mathbb {R}}^d}\text {d}z(1-\eta _L(z))\Big \langle (\nabla \phi ^{*(1)}+\xi ^*)(0)\cdot a'(0){{\mathcal {E}}}(0,z) a'(z)(\nabla \phi ^{(1)}+\xi )(z)\Big \rangle _L\frac{\partial c_L}{\partial L}(z)\Big |\nonumber \\ {}&\lesssim \sum _{k}|k|\int _{{\mathbb {R}}^d}\text {d}z(1-\eta _L)(z)|z|^{-(d+\frac{3}{2})}|\nabla c(z+Lk)|. \end{aligned}$$

Using (119) and splitting the integral into \(\{\vert z\vert \le \frac{1}{2}L\vert k\vert \}\) and its complement, we obtain that the z-integral is estimated by \((\vert k\vert L)^{-(d+1+2\alpha )}\), which implies that the sum converges and is estimated by \(L^{-(d+1+2\alpha )}\), which as desired is \(o(L^{-(d+1)})\).

We now address the first term in (62). The argument is based on the qualitative result of Corollary 1 in the following subsection. By the first item in (82) we obtain, by the explicit dependence of \({{\bar{G}}}_T\) and thus \({{\bar{\Gamma }}}\) on \({{\bar{a}}}\),

$$\begin{aligned} \lim _{L\uparrow \infty }\langle |{{\bar{\Gamma }}}-\Gamma _\textrm{hom}|\rangle _L=0. \end{aligned}$$

Since, on the other hand, \({{\bar{\Gamma }}}\) is uniformly bounded (recall that \({{\bar{a}}}\) is confined to the set (1)), and by (122), the convergence of the first term in (62) to the r. h. s of (39) follows from the two last items in (82), the definition (38) and Lebesgue’s convergence theorem.

We finally turn to the fourth r. h. s term of (62). We first reinterpret and bound this term using the solution of a PDE: considering u the decaying solution of

$$\begin{aligned} -\nabla \cdot a\nabla u=\nabla \cdot \Big (\eta _L a'(\nabla \phi ^{(1)}+\xi )\frac{\partial c_L}{\partial L}\Big ), \end{aligned}$$

we have from Proposition 3i) and (11)

$$\begin{aligned}&\Big \vert \int _{{\mathbb {R}}^d}\text {d}z\eta _L(z) \Big \langle (\nabla \phi ^{*(1)}+\xi ^*)(0)\cdot a'(0)\nabla \nabla G(0,z) a'(z)(\nabla \phi ^{(1)}+\xi )(z)\Big \rangle _L \frac{\partial c_L}{\partial L}(z)\Big \vert \nonumber \\&\quad =\vert \langle (\nabla \phi ^{*(1)}+\xi ^*)(0)\cdot a'(0)\nabla u(0)\rangle _L\vert \lesssim \langle \vert \nabla u(0)\vert ^2\rangle _L^{\frac{1}{2}}. \end{aligned}$$

We split u into the near-origin and the far-origin contribution \(u=u_N+\sum _{1\le 2^k\le L}u_{kF}\) with

$$\begin{aligned}{} & {} -\nabla \cdot a\nabla u_N=\nabla \cdot \Big (\eta _1 a'(\nabla \phi ^{(1)}+\xi )\frac{\partial c_L}{\partial L}\Big ), \\{} & {} \quad -\nabla \cdot a\nabla u_{kF}=\nabla \cdot \Big ((\eta _{2^k}-\eta _{2^{k-1}}) a'(\nabla \phi ^{(1)}+\xi )\frac{\partial c_L}{\partial L}\Big ). \end{aligned}$$

The near-origin contribution is directly estimated using Schauder’s theory. Indeed, making use of the \(\alpha \)-Hölder regularity (14) and \(\nabla \phi ^{(1)}\) (itself a consequence of Schauder’s theory applied to the equation (2)), the moment bounds Proposition 3i) as well as (25), (10) and (119) imply:

$$\begin{aligned} \langle \Vert \eta _1 a'(\nabla \phi ^{(1)}+\xi )\frac{\partial c_L}{\partial L}\Vert ^p_{C^{0,\alpha }(B_1)}\rangle _L^{\frac{1}{p}}\lesssim \sup _{B_1}\Bigg \vert \frac{\partial c_L}{\partial L}\Bigg \vert + \sup _{B_1}\Bigg \vert \nabla \frac{\partial c_L}{\partial L}\Bigg \vert \lesssim L^{-(d+1+2\alpha )}. \end{aligned}$$

Therefore, from Schauder’s theory and the energy estimate we deduce

$$\begin{aligned} \langle \vert \nabla u_N(0)\vert ^2\rangle _L^{\frac{1}{2}}&\lesssim \Big \langle \Big (\int _{B_1}\vert \nabla u_N\vert ^2\Big )^2\Big \rangle _L^{\frac{1}{4}}+\langle \Vert \eta _1 a'(\nabla \phi ^{(1)}+\xi )\frac{\partial c_L}{\partial L}\Vert ^4_{C^{0,\alpha }(B_1)}\rangle _L^{\frac{1}{4}} \\ {}&\lesssim \Big (\int _{{\mathbb {R}}^d} \eta ^2_1 \Big \vert \frac{\partial c_L}{\partial L}\Big \vert ^2\Big )^{\frac{1}{2}}+L^{-(d+1+2\alpha )} \\ {}&\lesssim L^{-(d+1+2\alpha )}. \end{aligned}$$

We now turn to the far-field contribution \(\nabla u_F(0):=\sum _{1\le 2^k\le L}\nabla u_{kF}(0)\). Using the Lipschitz estimate of Lemma 5 together with an energy estimate and Proposition 3i) as well as (25), and (119), we derive

This shows that the fourth r. h. s term is \(o(L^{-(d+1)})\).

5.2 Proof of Proposition 2: Limit \(T\uparrow \infty \)

The strategy of proof is as follows. First, we reorder the terms of the derivative of \(\frac{d}{dL}\langle \xi ^*\cdot {\bar{a}}_T\xi \rangle _L\) in order to make appear the “massive” analogue (that is, involving the massive operator \(\frac{1}{T}-\nabla \cdot a\nabla \)) of the r.h.s. of (62). For this first step, we essentially make rigorous the computations done in Sect. 2.2. Second, we systematically make use of the dominated convergence theorem to obtain the convergence of each term to its massless counterpart, yielding the formula (62).

Step 1. Formula for \(\frac{\textrm{d}}{\textrm{d}L}\langle \xi ^*\cdot {\bar{a}}_T\xi \rangle _L\). We establish the “massive analogue” of (62), namely

$$\begin{aligned}{} & {} {\frac{\textrm{d}}{\textrm{d}L}\langle \xi ^*\cdot {\bar{a}}_T\xi \rangle _L} \nonumber \\= & {} L^{-(d+1)}\int _{{\mathbb {R}}^d} \textrm{d}z\xi ^*\cdot \big \langle {\overline{\Gamma }}_{T/L^2ijmn}\,\big (-z_m Q^{(1)}_{Tij}(z) +Q^{(2)}_{Tijm}(z)\big )\big \rangle _L\xi \partial _nc(z) \nonumber \\{} & {} \quad +\int _{{\mathbb {R}}^d} \textrm{d}z \xi ^*\cdot \big \langle \epsilon ^{(1)}_{TLijn}(z) Q^{(1)}_{Tij}(z) +\epsilon ^{(2)}_{TLijmn}(z) Q^{(2)}_{Tijm}(z)\big \rangle _L\xi \partial _nc(z) \nonumber \\{} & {} \quad -\int _{{\mathbb {R}}^d} \textrm{d}z(1-\eta _L)(z) \big \langle (\nabla \phi ^{*(1)}_T+\xi ^*)(0)\cdot a'(0){\mathcal E}_T(0,z) a'(z)(\nabla \phi ^{(1)}_T+\xi )(z)\big \rangle _L \frac{\partial c_L}{\partial L}(z) \nonumber \\{} & {} \quad -\int _{{\mathbb {R}}^d} \textrm{d}z\eta _L(z) \big \langle (\nabla \phi ^{*(1)}_T+\xi ^*)(0)\cdot a'(0)\nabla \nabla G_T(0,z) a'(z)(\nabla \phi ^{(1)}_T+\xi )(z)\big \rangle _L \frac{\partial c_L}{\partial L}(z) \nonumber \\{} & {} \quad +\frac{1}{2}\big \langle (\nabla \phi ^{*(1)}_T+\xi ^*)\cdot a'' (\nabla \phi ^{(1)}_T+\xi )\big \rangle _L\frac{\partial c_L}{\partial L}(0), \end{aligned}$$

where \(\epsilon ^{(1)}_{TLijn}\) & \(\epsilon ^{(2)}_{TLijlm}\) are defined as in (60) & (61) with \({\bar{G}}\) replaced by \({\bar{G}}_T\), where \({{\mathcal {E}}}_T\) is defined like in (58) with G & \({\bar{G}}\) replaced by \(G_T\) & \({\bar{G}}_T\) (but with non-massive first- and second-order correctors), and where \(Q^{(1)}_T\) & \(Q^{(2)}_T\), which are quartic expressions in the correctors, are defined like in (36) & (37) with the first-order correctors \(\phi ^{(1)}\) & \(\phi ^{*(1)}\) (those linear in \(\xi \) & \(\xi ^*\)) replaced by their massive counterparts \(\phi _T^{(1)}\) & \(\phi _T^{*(1)}\) (but keeping the non-massive first and second order correctors \(\phi _i^{(1)}\), \(\phi _j^{*(1)}\), \(\phi _{im}^{(2)}\), \(\phi _{jm}^{*(2)}\)). Recall that \({\overline{\Gamma }}_{T/L^2ijmn}\) is defined in (59).

The starting point is (57); the second r.h.s. term remains untouched and reappears as the last term in (123). Writing \(\int \textrm{d}z\) \(=\int \textrm{d}z(1-\eta _L)(z)\) \(+\int \textrm{d}z\eta _L(z)\), we split the first r.h.s. term in (57) into a far- and near-field part. The near-field part remains untouched and reappears as the previous to last term in (123). In the far-field part, we replace \(\nabla \nabla G_T(0,z)\) according to the massive version of (58) by \({{\mathcal {E}}}_T(0,z)\) and terms involving \({\bar{G}}_T\). The contribution with \({{\mathcal {E}}}_T(0,z)\) reappears as the third r.h.s. term in (123). By the massive version of the definition (36) & (37) specified above, the terms involving \({\bar{G}}_T\) give rise to

$$\begin{aligned} -\int _{{\mathbb {R}}^d} \textrm{d}z (1-\eta _L)(z)\xi ^*\cdot \big ( -\big \langle \partial _{ij}{\bar{G}}_T(z) Q^{(1)}_{Tij}(z)\big \rangle _L +\big \langle \partial _{ijm}{\bar{G}}_T(z) Q^{(2)}_{Tijm}(z)\big \rangle _L\big )\xi \frac{\partial c_L}{\partial L}(z). \end{aligned}$$

We now insert (25) and perform the resummation at the end of Sect. 2.1, which is based on the periodicity of \(Q^{(1)}\) and \(Q^{(2)}\) under \(\langle \cdot \rangle _L\), and now is legitimate in view of the good decay properties of \(\nabla \nabla G_T\) (see (55)):

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^d} \textrm{d}z\sum _{k}k_n(1-\eta _L)(z)\xi ^*\cdot \big ( -\big \langle \partial _{ij }{\bar{G}}_T(z+Lk) Q^{(1)}_{Tij }(z)\big \rangle _L \\{} & {} +\big \langle \partial _{ijm}{\bar{G}}_T(z+Lk) Q^{(2)}_{Tijm}(z)\big \rangle _L\big )\xi \partial _nc(z). \end{aligned}$$

Using that by parity, \(\sum _{k}k_n\partial _{ij}{\bar{G}}_T(Lk)\) \(=0\), we now appeal to the massive version of the definitions (60) & (61). This gives rise to the second r.h.s. of (123) and the leading-order term

$$\begin{aligned} \int _{{\mathbb {R}}^d} \textrm{d}z\xi ^*\cdot \Big \langle \sum _{k}k_n\partial _{ijm}{\bar{G}}_T(Lk) \,\big (-z_m Q^{(1)}_{Tij}(z)+Q^{(2)}_{Tijm}(z)\big )\Big \rangle _L\xi \partial _n c(z). \end{aligned}$$

It remains to insert the definition (59) and appeal to the scaling \({\bar{G}}_T(Lx)=L^{2-d}{\bar{G}}_{T/L^2}(x)\).

Step 2. Limit \(T\uparrow \infty \). We now show that each term in (123) passes to the limit as \(T\uparrow \infty \) and converges to its massless counterpart. To do so, we need to establish that this limit makes sense for each of the five r.h.s. terms of (123); in this task, the dominated convergence theorem is our main tool. Note that (120) and (121) hold uniformly in T, at the level of the massive quantities. Therefore, combined with the bounds and convergences of the massive quantities (52), (53), (54), (55) and Proposition 5, the second, third, fourth and fifth r.h.s. terms of (123) converge to their massless counterparts as \(T\uparrow \infty \). Consequently, the subtle part is in the first r.h.s. term of (123), that we treat in detail.

In the sequel, \(L \ge 1\) is fixed. We prove that

$$\begin{aligned} \begin{aligned}&\lim _{T \uparrow \infty } \int _{{\mathbb {R}}^d} \textrm{d}z \big \langle {\overline{\Gamma }}_{T/L^2 ijmn}\big (\xi ^*\cdot Q^{(2)}_{Tijm}(z)\xi -z_m \xi ^*\cdot Q^{(1)}_{Tij}(z)\xi \big )\big \rangle _L \partial _nc(z)\\&= \int _{{\mathbb {R}}^d} \textrm{d}z \big \langle {\overline{\Gamma }}_{ijmn}\big (\xi ^*\cdot Q^{(2)}_{ijm}(z)\xi -z_m \xi ^*\cdot Q^{(1)}_{ij}(z)\xi \big )\big \rangle _L \partial _nc(z). \end{aligned} \end{aligned}$$

We claim that the only additional ingredient is the well-posedness of \({\overline{\Gamma }}_{ijmn}:=\lim _{T\uparrow \infty }{\overline{\Gamma }}_{Tijmn}\) along with the bound

$$\begin{aligned} \left|{\overline{\Gamma }}_{ijmn}\right| \le \sup _{T \ge 1} |{\overline{\Gamma }}_{Tijmn}| \lesssim 1 \langle \cdot \rangle _L-\text { almost-surely}, \end{aligned}$$

where we recall (59)

$$\begin{aligned} {\overline{\Gamma }}_{Tijmn}:=\sum _{k\in {\mathbb {Z}}^d} k_n\partial _{ijm}{\bar{G}}_T(k). \end{aligned}$$

Indeed, thanks to the assumption (10) on c, the bounds on the correctors (52), and (125), the integrand of the l. h. s integral in (124) are bounded (uniformly in T) by \((1+\vert z\vert )^{-d-2\alpha }\). We then conclude using the convergences (53) together with the Lebesgue convergence theorem.

Here comes the argument for (125). We fix a smooth compactly supported \(\eta \) with \(\eta =1\) on the unit cube \((-\frac{1}{2},\frac{1}{2})^d\) that we use it to split the lattice, which we interpret as a Riemann sum:

$$\begin{aligned}&{\sum _{k\not =0}k_n\partial _{ijm}{\bar{G}}_T(k)} \nonumber \\&=\int _{{\mathbb {R}}^d\backslash (-\frac{1}{2},\frac{1}{2})^d}\textrm{d}x\eta (x)x_n\partial _{ijm}{\bar{G}}_T(x) +\int _{{\mathbb {R}}^d} \textrm{d}x (1-\eta )(x)x_n\partial _{ijm}{\bar{G}}_T(x)\nonumber \\&\quad +\sum _{k\not =0}\big (k_n\partial _{ijm}{\bar{G}}_T(k) -\int _{k+(-\frac{1}{2},\frac{1}{2})^d}\textrm{d}x x_n\partial _{ijm}{\bar{G}}_T(x)\big ). \end{aligned}$$

The first r.h.s. integral effectively extends over a compact subset of \({\mathbb {R}}^d\backslash \{0\}\) and thus obviously converges for \(T\uparrow \infty \), thanks to (53). On the second r.h.s. integral in (126), we perform two integrations by parts:

$$\begin{aligned} \int _{{\mathbb {R}}^d} \textrm{d}x (1-\eta )(x)x_n\partial _{ijm}{\bar{G}}_T(x)=\int _{{\mathbb {R}}^d} \textrm{d}x\big (-\partial _j\eta (x)\delta _{mn}\partial _i{\bar{G}}_T(x) +\partial _m\eta (x)x_n\partial _{ij}{\bar{G}}_T(x)\big ). \end{aligned}$$

Again, the r.h.s. integral effectively extends over a compact subset of \({\mathbb {R}}^d\backslash \{0\}\) and converges for \(T\uparrow \infty \), thanks to (53). We finally turn to the last contribution in (126) where each summand has a limit \(T\uparrow \infty \). This extends to the sum because of dominated convergence: Each summand is dominated, in absolute value, by the Lipschitz norm of \(x\mapsto x_n\partial _{ijm}{\bar{G}}_T(x)\) on the translated cube \(k+(-\frac{1}{2},\frac{1}{2})^d\), which by the uniform-in-T decay of the derivatives of \({\bar{G}}_T\) (see (56)), gives an expression that is summable in \(k\in {\mathbb {Z}}^d\backslash \{0\}\).

5.3 Proof of Lemma 1: Fluctuation Estimates

As announced above, we show only (75) by closely following [39]. The only difference is that we appeal not only to the annealed Calderón–Zygmund estimates as in [39], but also to the annealed weighted estimates contained in Lemma 2.

For a deterministic and periodic vector field h, we consider the random variable of zero average

$$\begin{aligned} F:=\int _{[0,L)^d} h\cdot \nabla \phi _{ij}^{(2)}. \end{aligned}$$

We employ on it the spectral gap inequality (cf. [39, Lem. 3.1]), which, combined with Minkowski’s integral inequality (assuming that \(p \ge 2\)), reads

$$\begin{aligned} \langle |F|^{p}\rangle _L^{\frac{1}{p}} \lesssim _p \Big ( \int _{[0,L)^d} \big \langle \big | \frac{\partial F}{\partial g}\big |^p\big \rangle _L^{\frac{2}{p}} \Big )^{\frac{1}{2}}, \end{aligned}$$

where \(\frac{\partial F}{\partial g}=\frac{\partial F(g)}{\partial g(x)}\) is the Fréchet (or functional or vertical or Malliavin) derivative on \(L^2([0,L)^d)\) of F w.r.t. g defined by, for all periodic perturbation \(\delta g\in L^2([0,L)^d)\)

$$\begin{aligned} \lim _{\varepsilon \downarrow 0}\frac{F(g+\varepsilon \delta g)-F(g)}{\varepsilon }:=\int _{[0,L)^d} \textrm{d}x\,\delta g(x)\frac{\partial F(g)}{\partial g(x)}. \end{aligned}$$

(Since \(L^2([0,L)^d)\) is a Hilbert space, this Fréchet derivative is actually a gradient.) We split the proof into three steps. First, we establish that the Fréchet derivative of F is given by

$$\begin{aligned} \frac{\partial F}{\partial g} =\nabla v\cdot a'(\nabla \phi ^{(2)}_{ij}+\phi ^{(1)}_ie_j) -(\nabla w_j+ve_j)\cdot a'(\nabla \phi _i^{(1)}+e_i), \end{aligned}$$

where v and \(w_j\) are defined through (133) and (135). Next, we show that the annealed estimates of Lemma 2 imply

$$\begin{aligned}&\bigg (\int _{[0,L)^d}\langle |\nabla v|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2} \lesssim _p\bigg (\int _{[0,L)^d}|h|^2\bigg )^\frac{1}{2}, \end{aligned}$$
$$\begin{aligned}&\bigg (\int _{[0,L)^d}\langle |\nabla w_j+ve_j|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2} \lesssim _p\bigg (\int _{[0,L)^d}|x|^2_L|h|^2\bigg )^\frac{1}{2}, \end{aligned}$$

where we recall that \(\vert x\vert _L=\inf _{k\in {\mathbb {Z}}^d}\vert x+kL\vert \). Last, we insert (128) into (127), and we appeal to the Cauchy–Schwarz inequality [we also employ (11)], to the effect of

$$\begin{aligned} \langle |F|^{p}\rangle ^{\frac{1}{p}}&\lesssim _p \Big ( \int _{[0,L)^d} \big (\langle |\nabla v ^{2p}|\rangle ^{\frac{1}{p}} \langle |\nabla \phi ^{(2)}_{ij}+\phi ^{(1)}_ie_j|^{2p}\rangle ^{\frac{1}{p}} \\&\quad + \langle |\nabla w_j+ve_j|^{2p}\rangle ^{\frac{1}{p}} \langle |\nabla \phi _i^{(1)}+e_i|^{2p}\rangle ^{\frac{1}{p}}\big ) \Big )^{\frac{1}{2}}. \end{aligned}$$

Invoking (69) and recalling (129) and (130) finally yields the desired estimate (75).

Step 1. Argument for (128). We give ourselves infinitesimal (periodic) perturbation \(\delta g \in L^2([0,L)^d)\) of g. In view of (2) and (29), it generates a perturbation \(\delta \phi _i^{(1)}\) characterized by


In view of (31), this in turn generates the perturbation \(\nabla \delta \phi _{ij}^{(2)}\) characterized by

$$\begin{aligned}{} & {} -\nabla \cdot \big (a(\nabla \delta \phi _{ij}^{(2)}+\delta \phi _i^{(1)}e_j) +\delta g a'(\nabla \phi ^{(2)}_{ij}+\phi ^{(1)}_ie_j)\big ) \nonumber \\= & {} Pe_j\cdot \big (a\nabla \delta \phi _i^{(1)}+\delta g a'(\nabla \phi _i^{(1)}+e_i)\big ), \end{aligned}$$

where P denotes the (\(L^2\)-orthogonal) projection onto functions of vanishing spatial mean, i.e., . The form of (132) motivates the introduction of the periodic function v defined through


so that, by testing (133) with \(\delta \phi ^{(2)}_{ij}\) and (132) with v, we obtain the representation for \(\delta F:=\int _{[0,L)^d}h\cdot \nabla \delta \phi ^{(2)}_{ij}\):

$$\begin{aligned} \delta F= & {} \int _{[0,L)^d}\Big (\nabla v\cdot \big (a \delta \phi _i^{(1)}e_j +\delta g a'(\nabla \phi ^{(2)}_{ij}+\phi ^{(1)}_ie_j)\big ) \nonumber \\{} & {} -ve_j\cdot \big (a\nabla \delta \phi _i^{(1)}+\delta g a'(\nabla \phi _i^{(1)}+e_i)\big )\Big ). \end{aligned}$$

This in turn prompts the introduction of a second auxiliary periodic function \(w_j\) of zero mean

$$\begin{aligned} -\nabla \cdot a^*(\nabla w_j+ve_j)=Pe_j\cdot a^*\nabla v, \end{aligned}$$

so that by testing (131) with \(w_j\) and (135) with \(\delta \phi _i^{(1)}\), we may eliminate \(\delta \phi _i^{(1)}\) in (134) and recover (128) in form of

$$\begin{aligned} \delta F =\int _{[0,L)^d}\delta g\Big ( \nabla v\cdot a'(\nabla \phi ^{(2)}_{ij}+\phi ^{(1)}_ie_j) -(ve_j + \nabla w_j)\cdot a'(\nabla \phi _i^{(1)}+e_i) \Big ). \end{aligned}$$

Step 2. Argument for (129) and (130). Notice first that (129) is a direct consequence of Lemma 2 with weight \(w=1\) applied to v satisfying (133). Therefore, it remains to establish (130). In this perspective, we introduce the (gradient) field \(h_j\) such that the r.h.s. of (135) reads \(Pe_j\cdot a^*\nabla v\) \(=\nabla \cdot h_j\). As a consequence of annealed unweighted estimates on \(\nabla (-\nabla \cdot a^*\nabla )^{-1}\nabla \cdot \) (namely, Lemma 2 with weight \(w=1\)), we get

$$\begin{aligned} \bigg (\int _{[0,L)^d}\langle |\nabla w_j|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2} \lesssim _p \bigg (\int _{[0,L)^d}\langle |h_j|^{2p} + |v|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2}. \end{aligned}$$

We now claim the following annealed Hardy inequality:

$$\begin{aligned} \bigg (\int _{[0,L)^d}\langle |v|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2} \lesssim \bigg (\int _{[0,L)^d}|x|^2_L\langle |\nabla v|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2}. \end{aligned}$$

As a consequence of annealed weighted estimates on \(\nabla (-\nabla \cdot a^*\nabla )^{-1}\nabla \cdot \) (namely, Lemma 2 with weight \(w=\vert \cdot \vert ^2_L\)) applied to (133), we have

$$\begin{aligned} \bigg (\int _{[0,L)^d}|x|^2_L\langle |\nabla v|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2} \lesssim _p\bigg (\int _{[0,L)^d}|x|^2_L|h|^{2}\bigg )^\frac{1}{2}, \end{aligned}$$

and therefore, by (137), there holds

$$\begin{aligned} \bigg (\int _{[0,L)^d}\langle |v|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2} \lesssim _p\bigg (\int _{[0,L)^d}|x|^2_L|h|^{2}\bigg )^\frac{1}{2}. \end{aligned}$$

Moreover, by the annealed weighted estimates on \(\nabla ^2(-\triangle )^{-1}\) [46, Theorem 7.1], we obtain

$$\begin{aligned} \bigg (\int _{[0,L)^d}|x|^2_L\langle |\nabla h_j|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2} \lesssim _p\bigg (\int _{[0,L)^d}|x|^2_L\langle |\nabla v|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2}. \end{aligned}$$

Combining it with the Hardy inequality (137) for \(h_j\) and with (138) yields

$$\begin{aligned} \bigg (\int _{[0,L)^d}\langle |h_j|^{2p}\rangle _L^\frac{1}{p}\bigg )^\frac{1}{2} \lesssim _p\bigg (\int _{[0,L)^d}|x|^2_L|h|^{2}\bigg )^\frac{1}{2}. \end{aligned}$$

Inserting this and (139) into (136), and employing once more (139) in the triangle inequality gives (130).

Step 3. Argument for (137). W. l. o. g. we may assume that \(L=1\), and we consider random periodic functions of vanishing average v. The annealed Hardy inequality (137) relies on three ingredients. First, if \(\langle |u|^{2p}\rangle ^{\frac{1}{p}}\) is compactly supported, we have

$$\begin{aligned} \int _{{\mathbb {R}}^d} \langle |u|^{2p}\rangle ^{\frac{1}{p}} \lesssim \int _{{\mathbb {R}}^d} |x|^2 \langle |\nabla u|^{2p}\rangle ^{\frac{1}{p}}. \end{aligned}$$

Next, for \(\Omega := [-\frac{1}{2},\frac{1}{2})^d \backslash [-\frac{1}{4},\frac{1}{4})^d\), the following annealed Poincaré estimate holds:

$$\begin{aligned} \Bigg (\int _{\Omega } \langle |v|^{2p}\rangle ^{\frac{1}{p}}\Bigg )^{\frac{1}{2}} \lesssim \Bigg (\int _{\Omega } \langle |\nabla v|^{2p}\rangle ^{\frac{1}{p}}\Bigg )^{\frac{1}{2}} + \int _{\Omega } \langle |v|^{2p}\rangle ^{\frac{1}{2p}}. \end{aligned}$$

Last, we make use of an annealed Poincaré–Wirtinger estimate

$$\begin{aligned} \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \langle |v|^{2p}\rangle ^{\frac{1}{2p}} \lesssim \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \langle |\nabla v|^{2p}\rangle ^{\frac{1}{2p}}. \end{aligned}$$

Using (140) for \(u:=\eta v\) where \(\eta \) is a cutoff function of \([-\small \frac{1}{2},\small \frac{1}{2})^d\) into \([-\small \frac{3}{4},\small \frac{3}{4})^d\), we have by periodicity of v

$$\begin{aligned} \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d}\langle |v|^{2p}\rangle ^{\frac{1}{p}} \lesssim \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} |x|^2_1 \langle |\nabla v|^{2p}\rangle ^{\frac{1}{p}} + \int _{\Omega } \langle |v|^{2p}\rangle ^{\frac{1}{p}}, \end{aligned}$$

Inserting (141) and then (142) into the above estimate yields

$$\begin{aligned} \begin{aligned} \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d}\langle |v|^{2p}\rangle ^{\frac{1}{p}}&\lesssim \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} |x|^2_1 \langle |\nabla v|^{2p}\rangle ^{\frac{1}{p}} + \Big (\int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \langle |\nabla v|^{2p}\rangle ^{\frac{1}{2p}}\Big )^{2}. \end{aligned} \end{aligned}$$

Since \(d>2\), we may employ the Cauchy–Schwarz inequality to get

$$\begin{aligned} \Bigg (\int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \langle |\nabla v|^{2p}\rangle ^{\frac{1}{2p}}\Bigg )^{2}&\le \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} |x|^2_1 \langle |\nabla v|^{2p}\rangle ^{\frac{1}{p}} \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} |x|^{-2}_1 \\&\lesssim \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} |x|^2_1 \langle |\nabla v|^{2p}\rangle ^{\frac{1}{p}}. \end{aligned}$$

Inserting this into (143) yields the desired (137) (noting that by periodicity, we can replace \([-\tfrac{1}{2},\tfrac{1}{2})\) by \([0,1)^d\)).

We now establish successively (140), (141), and (142). First, (140) comes by applying the following Hardy inequality for compactly supported functions \(\phi \) (see [6, Theorem 1.2.8] with \(p=2\) and \(V=|x|^2\)):

$$\begin{aligned} \int _{{\mathbb {R}}^d} |\phi |^2 \lesssim \int _{{\mathbb {R}}^d} |x|^2 |\nabla \phi |^2, \end{aligned}$$

to \(\phi \rightsquigarrow \langle |u|^{2p}\rangle ^{\frac{1}{2p}}\), and noticing that by the Hölder inequality with exponents \((\frac{2p}{2p-1},2p)\)

$$\begin{aligned} \vert \nabla \langle |u|^{2p}\rangle ^{\frac{1}{2p}}\vert = \vert \langle |u|^{2p}\rangle ^{\frac{1}{2p}-1} \langle |u|^{2p-1} \nabla |u|\rangle \vert \le \langle |\nabla u|^{2p}\rangle ^{\frac{1}{2p}}. \end{aligned}$$

Similarly, we get (141) from the usual Poincaré inequality applied to the function \(\langle |v|^{2p}\rangle ^{\frac{1}{2p}}\). Last, we get (142) by recalling that v is periodic of vanishing average in \([-\small \frac{1}{2},\small \frac{1}{2})^d\), so that

$$\begin{aligned}&\int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \langle |v|^{2p}\rangle ^{\frac{1}{2p}} = \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \big \langle \big |v -\int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} v \big |^{2p}\big \rangle ^{\frac{1}{2p}} \\&\qquad \le \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \textrm{d}x \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \textrm{d}y \langle |v(x) - v(x+y) |^{2p}\rangle ^{\frac{1}{2p}} \\&\qquad \le \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \textrm{d}x \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \textrm{d}y \big \langle \big (|y|\int _0^1 d s |\nabla v(x+sy)|\big )^{2p}\big \rangle ^{\frac{1}{2p}} \\&\qquad \lesssim \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d}\textrm{d}x \int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \textrm{d}z \langle | \nabla v(z)|^{2p}\rangle ^{\frac{1}{2p}} =\int _{[-\small \frac{1}{2},\small \frac{1}{2})^d} \langle | \nabla v|^{2p}\rangle ^{\frac{1}{2p}}. \end{aligned}$$

5.4 Proof of Corollary 1: Limit \(L\uparrow \infty \)

In view of (2) and (29), we may consider the field \(\phi ^{(1)}\) as a function of g, provided the latter is periodic. The same applies to \(\phi ^{(2)}\), cf. (31), provided we make it unique through

$$\begin{aligned} \phi ^{(2)}_{ij}(0)=0. \end{aligned}$$

We will monitor the joint distribution of the triplet of fields \((g,\phi ^{(1)}_i(g),\phi ^{(2)}_{ij}(g))\) under \(\langle \cdot \rangle _L\). This amounts to the push-forward \(\langle \cdot \rangle _{L,ext}\) of \(\langle \cdot \rangle _L\) under the map \(g\mapsto (g,\phi ^{(1)}_i(g),\phi ^{(2)}_{ij}(g))\), which we denote by \((\text {Id},\phi ^{(1)}_i,\phi ^{(2)}_{ij})\):

$$\begin{aligned} \langle \cdot \rangle _{L,ext}:=(\text {Id},\phi ^{(1)}_i,\phi ^{(2)}_{ij})\#\langle \cdot \rangle _L; \end{aligned}$$

the index “ext” hints to the fact that \(\langle \cdot \rangle _{L,ext}\) is an extension of \(\langle \cdot \rangle _L\) in the sense that the latter is the marginal of the former w.r.t. the first component. As will become apparent in Step 1, \(\langle \cdot \rangle _{L,ext}\) is a probability measure on the product space \({\mathcal {C}}^{0,\alpha ,\beta }\times {\mathcal {C}}^{1,\alpha ,\beta }\times {\mathcal {C}}^{1,\alpha ,\beta }\), provided \(\alpha \in (0,1)\) and \(\beta \in (\frac{1}{2},\infty )\), where \({\mathcal {C}}^{n,\alpha ,\beta }\) denotes the space of functions that are locally in \({\mathcal {C}}^{n,\alpha }\) but are allowed to grow at rate \(\beta \):

$$\begin{aligned} {\mathcal {C}}^{n,\alpha ,\beta }:=\{\,\zeta :{\mathbb {R}}^d\rightarrow {\mathbb {R}} \,\vert \,\Vert \zeta \Vert _{n,\alpha ,\beta }:=\sup _{x\in {\mathbb {R}}^d}(1+\vert x\vert )^{-\beta } \Vert \zeta \Vert _{C^{n,\alpha }(B_1(x))}<\infty \}. \end{aligned}$$

On the one hand, this norm is weak enough so that Proposition 3 implies that the family \(\{\langle \cdot \rangle _{L,ext}\}_{L\uparrow \infty }\) is tight, see Step 1. On the other hand, it is (obviously) strong enough so that q, \(Q^{(1)}_{ij}(z)\) and \(Q^{(2)}_{ijm}(z)\) are continuous functions on the product space \({\mathcal {C}}^{0,\alpha ,\beta }\times {\mathcal {C}}^{1,\alpha ,\beta }\times {\mathcal {C}}^{1,\alpha ,\beta }\), cf. (16), (36) and (37), which will imply part iii) of the corollary. In Step 2, we show that any (weak) limitFootnote 18\(\langle \cdot \rangle _{\textrm{ext}}\) is stationary, satisfies the moment bounds of Corollary 1, and is supported on fields that satisfy the relations of Corollary 1. In Step 3, we identify the first marginal of \(\langle \cdot \rangle _{\textrm{ext}}\) with our whole-space ensemble \(\langle \cdot \rangle \). In Step 4, we construct \(\phi ^{(1)}_i\) and \(\phi ^{(2)}_{ij}\), now only defined almost surely, satisfying the requirements of Corollary 1. Finally, in Step 5, we argue that \(\langle \cdot \rangle _{\textrm{ext}}\) is the push-forward of the whole-space ensemble \(\langle \cdot \rangle \) under the map \((\text {Id},\phi ^{(1)}_i,\phi ^{(2)}_{ij})\). In the following, we drop the indices i and j.

Step 1. Compactness result. We show that Proposition 3 implies for \(\alpha \in (0,1)\), \(\beta \in (\frac{1}{2},\infty )\), and \(p<\infty \)

$$\begin{aligned} \sup _{L\ge 1}\langle (\Vert \text {Id}\Vert _{0,\alpha ,\beta }+\Vert (\phi ^{(1)},\phi ^{(2)})\Vert _{1,\alpha ,\beta })^p\rangle _L<\infty . \end{aligned}$$

We combine this with the compact embedding \({\mathcal {C}}^{n,\alpha ',\beta '}\subset {\mathcal {C}}^{n,\alpha ,\beta }\) for \(\alpha <\alpha '\) and \(\beta '<\beta \), which is a consequence of Arzelà-Ascoli’s theorem. This implies by Prohorov’s theorem [13, Theorem 3.8.4] that there exists a probability measure \(\langle \cdot \rangle _{\textrm{ext}}\) on \({\mathcal {C}}^{0,\alpha ',\beta '}\times {\mathcal {C}}^{1,\alpha ',\beta '} \times {\mathcal {C}}^{1,\alpha ',\beta '}\) such that, up to a subsequence that we do not relabel,

$$\begin{aligned} \langle \cdot \rangle _{L,ext}\underset{L\uparrow \infty }{\rightharpoonup }\langle \cdot \rangle _{\textrm{ext}}. \end{aligned}$$

We argue for (146). Since the balls \(\{B_1(x)\}_{x\in \frac{1}{\sqrt{d}}{\mathbb {Z}}^d}\) cover \({\mathbb {R}}^d\), we have by a union bound argument

$$\begin{aligned}{} & {} \big \langle \big (\Vert g\Vert _{0,\alpha ,\beta }+\Vert (\phi ^{(1)},\phi ^{(2)})\Vert _{1,\alpha ,\beta } \big )^p\big \rangle _L \nonumber \\{} & {} \qquad \lesssim \big \langle \big (\sup _{x\in \frac{1}{\sqrt{d}}{\mathbb {Z}}^d}(1+\vert x\vert )^{-\beta } (\Vert g\Vert _{C^{0,\alpha }(B_1(x))} +\Vert (\phi ^{(1)},\phi ^{(2)})\Vert _{C^{1,\alpha }(B_1(x))})\big )^p\big \rangle _L \nonumber \\{} & {} \qquad \le \sum _{x\in \frac{1}{\sqrt{d}}{\mathbb {Z}}^d} (1+\vert x\vert )^{-p\beta } \langle (\Vert g\Vert _{C^{0,\alpha }(B_1(x))} +\Vert (\phi ^{(1)},\phi ^{(2)})\Vert _{C^{1,\alpha }(B_1(x))})^p\rangle _L.\qquad \quad \end{aligned}$$

By local Schauder theory, we obtain from (2) and (31)

$$\begin{aligned} \Vert \phi ^{(1)}\Vert _{C^{1,\alpha }(B_1(x))}&\lesssim C(\Vert a\Vert _{C^{0,\alpha }(B_2(x))}) \Big (1+\Big (\int _{B_2(x)}|\phi ^{(1)}|^2\Big )^{\frac{1}{2}}\Big ),\\ \Vert \phi ^{(2)}\Vert _{C^{1,\alpha }(B_1(x))}&\lesssim C(\Vert a\Vert _{C^{0,\alpha }(B_2(x))}) \Big (\Vert \phi ^{(1)}\Vert _{C^{1,\alpha }(B_2(x))}+\Big (\int _{B_2(x)}|\phi ^{(2)}|^2\Big )^{\frac{1}{2}}\Big ) \end{aligned}$$

with an at most polynomial dependence of the constant on \(\Vert a\Vert _{C^{0,\alpha }(B_2(x))}\). By (13), this yields

$$\begin{aligned} \big \langle \big (\Vert g\Vert _{C^{0,\alpha }(B_1(x))} +\Vert (\phi ^{(1)},\phi ^{(2)})\Vert _{C^{1,\alpha }(B_1(x))}\big )^p\big \rangle _L^\frac{1}{p} \lesssim _{p,p'}1+ \int _{B_2(x)}\langle |(\phi ^{(1)},\phi ^{(2)})|^{p'}\rangle _L^\frac{1}{p'}, \end{aligned}$$

provided \(p<p'\). By Proposition 3i) for \(\phi ^{(1)}\) and by iv) for \(\phi ^{(2)}\) together with (144), we have that the r.h.s. of (149) is estimated by \((1+|x|)^\frac{1}{2}\). Hence, the summand on the r.h.s. of (148) is estimated by \((1+|x|)^{-p(\beta -\frac{1}{2})}\), which is summable provided \(p>\frac{d}{\beta -\frac{1}{2}}\). The remaining range is then obtained by Jensen’s inequality.

Step 2. Stationarity, moment bounds, and PDE-constrained support of \(\langle \cdot \rangle _{\textrm{ext}}\). Here and in the sequel, we denote by \((g,\phi ,\psi )\in {{\mathcal {C}}}^{0,\alpha ,\beta }\times {{\mathcal {C}}}^{1,\alpha ,\beta }\times {{\mathcal {C}}}^{1,\alpha ,\beta }\) the integration variables of \(\langle \cdot \rangle _{L,ext}\) and its limit \(\langle \cdot \rangle _{\textrm{ext}}\). First, because \({\mathcal C}^{1,\alpha ,\beta }\ni \psi \mapsto \psi ^2(0)\) is continuous, (144) is preserved under (147):

$$\begin{aligned} \psi (0)=0\quad \text {for }\langle \cdot \rangle _{\textrm{ext}}-\text {a. e.} \psi . \end{aligned}$$

Next, note that \(\phi ^{(1)}\) and \(\nabla \phi ^{(2)}\) are stationaryFootnote 19 (the anchoring (144) of \(\phi ^{(2)}\) does not admit stationarity, but does not affect the stationarity property of \(\nabla \phi ^{(2)}\)) and \(\langle \cdot \rangle _L\) is stationary.Footnote 20 Hence, we may introduce the push-forward of \(\langle \cdot \rangle _{L,ext}\) under the map \((g,\phi ,\psi )\mapsto (g,\phi ,\nabla \psi )\), which we call \((\text {Id},\text {Id},\nabla )\) and which is stationary, a linear constraint that is preserved under the weak convergence (147):

$$\begin{aligned} (\text {Id},\text {Id},\nabla )\#\langle \cdot \rangle _{\textrm{ext}} \text { is stationary.} \end{aligned}$$

We now turn to the estimates of Proposition 3. Clearly, the bounds (69), the second bound in (70) for any compactly supported function \(\eta \) and (71) is preserved under (147):

$$\begin{aligned}{} & {} \langle \vert \nabla \phi \vert ^p+\vert \phi \vert ^p+\vert \nabla \psi \vert ^p\rangle _{\textrm{ext}}\lesssim _p 1, \nonumber \\{} & {} \quad \big \langle \big |\int _{{\mathbb {R}}^d}\eta \phi \big |^p\big \rangle _{\textrm{ext}}^\frac{1}{p} \lesssim _p \big (\int _{{\mathbb {R}}^d}|\eta |^\frac{2d}{d+2}\big )^\frac{d+2}{2d}, \nonumber \\{} & {} \quad \text{ and }\quad \langle \vert \psi (z)\vert ^p\rangle _{\textrm{ext}}^\frac{1}{p} \lesssim _p \mu ^{(2)}_d(\vert z\vert ). \end{aligned}$$

Finally, from the weak convergence (147), the definition of \({\overline{a}}\) in (3) together with the stationarity of \(\langle \cdot \rangle _L\) in form of \(\langle {\overline{a}}\rangle _L=\langle a(\nabla \phi ^{(1)}+e_i)(0)\rangle _L\) and the decay of averages (70) for the flux (applied with \(\eta =L^{-d}\mathbb {1}_{[0,L)^d}\)), one has

$$\begin{aligned} \langle {\overline{a}}\rangle _L\underset{L\uparrow \infty }{\rightarrow }\langle a(\nabla \phi +e_i)(0)\rangle _{\textrm{ext}} \quad \text{ and }\quad \langle \vert {\overline{a}}-\langle {\overline{a}}\rangle _L\vert \rangle _L\lesssim L^{-\frac{d}{2}}, \end{aligned}$$

so that \(\langle \vert {\overline{a}}-\langle a(\nabla \phi +e_i)\rangle _{\textrm{ext}}\vert \rangle _L\underset{L\uparrow \infty }{\rightarrow } 0\). Therefore, introducing the notation \(a_{\text {hom},ext}:=\langle a(\nabla \phi +e_i)(0)\rangle _{\textrm{ext}}\), we obtain under the weak convergence (147) that the Eqs. (2) and (31), when tested against smooth compactly supported functions, are preserved in the sense that

$$\begin{aligned} \nabla \cdot a(\nabla \phi +e_i)= & {} 0\quad \text {and}\quad -\nabla \cdot a(\nabla \psi +\phi e_j)\nonumber \\= & {} e_j\cdot (a(\nabla \phi +e_i)-a_{\text {hom},ext}e_i) \quad \text{ for }\;\langle \cdot \rangle _{\textrm{ext}}\text{-a. } \text{ e. }\;(g,\phi ,\psi ).\nonumber \\ \end{aligned}$$

Step 3. Identification of the first marginal of \(\langle \cdot \rangle _{\textrm{ext}}\). We show that the first marginal of \(\langle \cdot \rangle _{\textrm{ext}}\) is given by \(\langle \cdot \rangle \). By the definition (145), the first marginal of \(\langle \cdot \rangle _{L,ext}\) is the Gaussian measure \(\langle \cdot \rangle _L\). By (147), the sequence \(\{\langle \cdot \rangle _L\}_{L\uparrow \infty }\) of Gaussian measures is tight on \({\mathcal {C}}^{0,\alpha ,\beta }\). By [13, Corollary 3.8.5], it is thus enough to prove the weak convergence of \(\langle \cdot \rangle _L\) to \(\langle \cdot \rangle \) on squares of bounded linear forms:

$$\begin{aligned} \lim _{L\uparrow \infty }\langle \ell ^2\rangle _L=\langle \ell ^2\rangle \quad \text {for all }\ell \in ({\mathcal {C}}^{0,\alpha ,\beta })^{*}. \end{aligned}$$

By density and tightness, it is enough to check (154) for linear forms \(\ell \) of the form \(g\mapsto \int \zeta g\) for an arbitrary Schwartz function \(\zeta \). The definition (9) of \(c_L\) can also be stated in terms of the (distributional) Fourier transform of the (periodic) \(c_L\) as

$$\begin{aligned} {{\mathcal {F}}}c_L=\big (\frac{2\pi }{L}\big )^d \sum _{q\in \frac{2\pi }{L}{\mathbb {Z}}^d}{\mathcal F}c(q)\delta (\cdot -q). \end{aligned}$$

Hence, the l.h.s. of (154) assumes the form of a Riemann sum:

$$\begin{aligned} \langle \ell ^2\rangle _L=\big (\frac{2\pi }{L}\big )^{d} \sum _{q\in \frac{2\pi }{L}{\mathbb {Z}}^d}{\mathcal {F}}c(q)\vert {\mathcal {F}}\zeta (q)\vert ^2. \end{aligned}$$

Since by Assumption 1 in form of the integrability of c, see (12), \({\mathcal {F}}c\) is continuous, we obtain (154):

$$\begin{aligned} \lim _{L\uparrow \infty }\langle \ell ^2\rangle _L =\int _{{\mathbb {R}}^d} {\mathcal {F}}c\vert {\mathcal {F}}\zeta \vert ^2=\langle \ell ^2\rangle . \end{aligned}$$

Step 4. Construction of \(\phi ^{(1)}\) and \(\phi ^{(2)}\). We show that there exist \(\phi ^{(1)}\) and \(\phi ^{(2)}\) satisfying Corollary 1i) and ii), respectively. We construct these random variables via disintegration of the measure \(\langle \cdot \rangle _{\textrm{ext}}\) with respect to its first marginal \(\langle \cdot \rangle \), which amounts to conditional expectation w.r.t. g. By [44, Theorem 6.4] there exists a family of measures \(\{\langle \cdot |g\rangle _{\textrm{ext}}\}_{g\in {\mathcal {C}}^{0}_{\alpha ,\beta }}\) on \({\mathcal {C}}^{1,\alpha ,\beta }\times {\mathcal {C}}^{1,\alpha ,\beta }\) such that for all \(\langle \cdot \rangle _{\textrm{ext}}\)-integrable functions F,

$$\begin{aligned} \langle F\rangle _{\textrm{ext}}=\langle \langle F|g\rangle _{\textrm{ext}}\rangle . \end{aligned}$$

We now define the \(\langle \cdot \rangle \)-integrable functions \(\phi ^{(1)}\) and \(\phi ^{(2)}\) through conditional expectation

$$\begin{aligned} \phi ^{(1)}(g):=\langle \phi |g\rangle _{\textrm{ext}}\quad \text{ and }\quad \phi ^{(2)}(g):=\langle \psi |g\rangle _{\textrm{ext}}\quad \text{ for } \text{ all }\;g\in {\mathcal {C}}^0_{\alpha ,\beta } \end{aligned}$$

and verify that they satisfy all the requirements of Corollary 1i) and ii). Since the conditioning w.r.t. g commutes with the multiplication by \(a=A(g)\), the linear equations (153) are preserved and give rise to (79) and (81). It is easy to check that the stationarity (151) translates into stationarity of \(\phi ^{(1)}\) and \(\nabla \phi ^{(2)}\). It follows from (150), via Jensen’s inequality applied to the conditional expectation, that \(\phi ^{(1)}\) and \(\phi ^{(2)}\) satisfy the moment bounds of Corollary 1i) and ii), and also the decay bound (78) and the growth bound (80).

Step 5. Identification of \(\langle \cdot \rangle _{\textrm{ext}}\). We now establish

$$\begin{aligned} \langle \cdot \rangle _{\textrm{ext}}=(\text {Id},\phi ^{(1)},\phi ^{(2)})\#\langle \cdot \rangle . \end{aligned}$$

by showing

$$\begin{aligned} u^{(1)}:=\phi -\phi ^{(1)}(g)=0\quad \text{ and }\quad u^{(2)}:=\psi -\phi ^{(2)}(g)=0\quad \text{ for }\;\langle \cdot \rangle _{\textrm{ext}}- \text{ a. } \text{ e. }\;(g, \phi ,\psi ). \nonumber \\ \end{aligned}$$

By (79) and (153), we have \(-\nabla \cdot a\nabla u^{(1)}=0\) \(\langle \cdot \rangle _{\textrm{ext}}\)-a. s. By Caccioppoli’s inequality, we thus obtain

Taking the expectation \(\langle \cdot \rangle _{\textrm{ext}}\) and using the stationarity of \(\phi ^{(1)}\) and (151), this implies

$$\begin{aligned} \langle \vert \nabla u^{(1)}\vert ^2\rangle _{\textrm{ext}} \lesssim \frac{1}{R^2}\langle \vert u^{(1)}\vert ^2\rangle _{\textrm{ext}}\quad \text{ for } \text{ all }\;R<\infty . \end{aligned}$$

Letting \(R\uparrow \infty \) while appealing to (69) and (152) yields \(\nabla u^{(1)}=0\). We now use (78) and the associated property in (152), for the averaging function \(\eta =R^{-d}\mathbb {1}_{B_R}\). Because of \(\frac{2d}{d+2}>1\), this yields . Hence, we obtain the first claim of (156). Note that this implies in particular \(a_{\text {hom},ext}=a_{\text {hom}}\), namely the first claim of Corollary 1iii).

It now follows from (81) and (153) that \(-\nabla \cdot a\nabla u^{(2)}=0\) \(\langle \cdot \rangle _{\textrm{ext}}\)-a. s.. Starting again with Caccioppoli’s inequality, followed by the combination of the stationarity of \(\nabla \phi ^{(2)}\) and (151), finally followed by the combination of (80) and (152), we obtain

Letting \(R\uparrow \infty \) we conclude \(\nabla u^{(2)}=0\). Using once more the combination of (80) and (152), this time for \(z=0\), we find \(u^{(2)}(0)=0\), which gives the second claim of (156). The same argument shows that there is at most one pair \((\phi ^{(1)},\phi ^{(2)})\) of random variables satisfying the properties of Corollary 1i) and ii). Therefore, \(\langle \cdot \rangle _{\textrm{ext}}\) is unique and thus the limit of \(\langle \cdot \rangle _{L,\textrm{ext}}\) for the entire sequence \(L\uparrow \infty \).

5.5 Proof of Lemma 4: Improved Caccioppoli Inequality

By scaling, it is enough to consider \(R=2\); we fix a smooth cutoff function \(\eta \) for \(B_1\) in \(B_2\). Our starting point is the following localized version of a standard \(L^2\)-based interpolation estimate, with \(n\in {\mathbb {N}}\) to be fixed later, which we take from the proof of [10, Lemma 4]:

$$\begin{aligned} \int _{{\mathbb {R}}^d}(\eta ^{4n}\triangle ^{2n}w)^2 \lesssim \bigg (\int _{{\mathbb {R}}^d}|\eta ^{4n+1}\nabla \triangle ^{2n}w|^2\bigg )^\frac{4n}{4n+1} \bigg (\int _{{\mathbb {R}}^d} w^2\bigg )^\frac{1}{4n+1}+\int _{{\mathbb {R}}^d} w^2. \end{aligned}$$

We apply it to \(w\in H^{2n}_0(B_2)\) (as usual, \(H^{2n}_0(B_2)\) denotes the closure of \(C^\infty _0(B_2)\) w.r.t. the \(H^{2n}(B_2)\)-norm) that (weakly) solves

$$\begin{aligned} \triangle ^{2n}w=u\quad \text{ in }\quad B_2. \end{aligned}$$

We construct w with help of the Riesz representation theorem, so that we automatically have

$$\begin{aligned} \int _{{\mathbb {R}}^d} u w=\int _{{\mathbb {R}}^d}(\triangle ^nw)^2\sim \int _{{\mathbb {R}}^d}|\nabla ^{2n}w|^2\gtrsim \int _{{\mathbb {R}}^d}w^2, \end{aligned}$$

where we used higher-order \(L^2\)-regularity and a higher-order Poincaré estimate. We obtain from inserting (158) into (157)

$$\begin{aligned} \int _{{\mathbb {R}}^d}(\eta ^{4n}u)^2 \lesssim \bigg (\int _{{\mathbb {R}}^d}|\eta ^{4n+1}\nabla u|^2\bigg )^\frac{4n}{4n+1} \bigg (\int _{{\mathbb {R}}^d} w^2\bigg )^\frac{1}{4n+1}+\int _{{\mathbb {R}}^d} w^2. \end{aligned}$$

Combining this with Caccioppoli’s estimate \(\int _{{\mathbb {R}}^d}|\eta ^{4n+1}\nabla u|^2\lesssim \int _{{\mathbb {R}}^d}(\eta ^{4n}u)^2\) and Young’s inequality, we obtain

$$\begin{aligned} \int _{{\mathbb {R}}^d} (\eta ^{4n}u)^2\lesssim \int _{{\mathbb {R}}^d} w^2, \end{aligned}$$

so that, by the choice of \(\eta \), we deduce

$$\begin{aligned} \int _{B_1}|\nabla u|^2\le \int _{{\mathbb {R}}^d}(\eta ^{4n}u)^2\lesssim \int _{{\mathbb {R}}^d} w^2. \end{aligned}$$

It remains to post-process this inner regularity estimate for an a-harmonic function u.

In route to an annealed estimate, we express the r.h.s. of (160) in terms of u, which is conveniently done in terms of the complete orthonormal system of eigenfunctions \(\{w_k\}_{k\in {\mathbb {N}}}\subset H_0^{2n}(B_2)\) and eigenvalues \(\{\lambda _k\}_k\subset (0,\infty )\) of the Dirichlet-\(\triangle ^{2n}\), which is a positive operator with compact inverse:

$$\begin{aligned} \int _{{\mathbb {R}}^d} w^2=\sum _{k}\frac{1}{\lambda _k^2}\bigg (\int _{{\mathbb {R}}^d} u w_k\bigg )^{2} =\sum _{k}\frac{1}{\lambda _k}\frac{\bigg (\int _{{\mathbb {R}}^d} u w_k\bigg )^{2}}{\int _{{\mathbb {R}}^d}(\triangle ^{n}w_k)^2} {\mathop {\lesssim }\limits ^{(159)}} \sum _{k}\frac{1}{\lambda _k}\frac{\bigg (\int _{{\mathbb {R}}^d} u w_k\bigg )^{2}}{\int _{{\mathbb {R}}^d}|\nabla ^{2n}w_k|^2}. \end{aligned}$$

We insert this into (160)

$$\begin{aligned} \int _{B_1}|\nabla u|^2 \lesssim \sum _{k}\frac{1}{\lambda _k} \frac{\bigg (\int _{{\mathbb {R}}^d} uw_k\bigg )^2}{\int _{{\mathbb {R}}^d}|\nabla ^{2n}w_k|^2} \end{aligned}$$

and apply \(\langle (\cdot )^\frac{p}{2}\rangle \). By Hölder’s inequality in k, we obtain

$$\begin{aligned} \big \langle \bigg (\int _{B_1}|\nabla u|^2\bigg )^\frac{p}{2}\bigg \rangle \lesssim \bigg (\sum _{k'}\frac{1}{\lambda _{k'}}\bigg )^{\frac{p}{2}-1} \sum _{k}\frac{1}{\lambda _k} \frac{\bigg \langle \bigg |\int _{{\mathbb {R}}^d} uw_k\bigg |^{p}\bigg \rangle }{\bigg (\int _{{\mathbb {R}}^d}|\nabla ^{2n}w_k|^2\bigg )^\frac{p}{2}}. \end{aligned}$$

In order to proceed, we need \(\sum _{k}\frac{1}{\lambda _k}<\infty \), which means that the inverse of the Dirichlet-\(\triangle ^{2n}\) has finite trace, which in turn follows from the finiteness of the corresponding Green function along the diagonal, which requires that Dirac distributions are in \(H^{-2n}(B_2)\), which amounts to the Sobolev embedding \(H^{2n}_0(B_2)\subset C^0_0(B_2)\) and thus holds provided \(2n>d\), which we henceforth assume. Hence by the density of \(C^\infty _0(B_2)\) in \(H^{2\,m}_0(B_2)\ni w_k\), we obtain the annealed inner regularity estimate

$$\begin{aligned} \big \langle \bigg (\int _{B_1}|\nabla u|^2\big )^\frac{p}{2}\bigg \rangle \lesssim \sup _{w\in C^\infty _0(B_2)} \frac{\bigg \langle \bigg |\int _{{\mathbb {R}}^d} uw\bigg |^{p}\bigg \rangle }{\bigg (\int _{{\mathbb {R}}^d}|\nabla ^{2n}w|^2\bigg )^\frac{p}{2}}. \end{aligned}$$

It remains to post-process (161). Provided \(2n>\frac{d}{2}+3\), we may appeal to Sobolev’s embedding applied to \(\nabla ^3 w\) in order to upgrade (161) to

$$\begin{aligned} \bigg \langle \bigg (\int _{B_1}|\nabla u|^2\bigg )^\frac{p}{2}\bigg \rangle ^\frac{1}{p} \lesssim \sup _{w\in C^\infty _0(B_2)} \frac{\bigg \langle \bigg |\int _{{\mathbb {R}}^d} uw\bigg |^{p}\bigg \rangle ^\frac{1}{p}}{\sup |\nabla ^{3}w|}. \end{aligned}$$

Since we may w. l. o. g. assume \(\int _{B_2}u=0\), we may restrict to w with \(\int w=0\). A standard argumentFootnote 21 in the theory of distributions yields the existence of a vector field \(g\in C^\infty _0((-2,2)^d)\subset C^\infty _0(B_{2\sqrt{d}})\) such that

$$\begin{aligned} \nabla \cdot g=w\quad \text{ and }\quad \sup |\nabla ^3 g|\lesssim \sup |\nabla ^3 w|. \end{aligned}$$

Hence (162) may be upgraded to the desired

$$\begin{aligned} \bigg \langle \bigg (\int _{B_1}|\nabla u|^2\bigg )^\frac{p}{2}\bigg \rangle ^\frac{1}{p} \lesssim \sup _{g\in C^\infty _0(B_{2\sqrt{d}})} \frac{\bigg \langle \bigg |\int _{{\mathbb {R}}^d} g\cdot \nabla u\bigg |^p\bigg \rangle ^\frac{1}{p}}{\sup |\nabla ^3g|}. \end{aligned}$$

\(\square \)

5.6 Proof of Lemma 3: Annealed Estimate on the Two-Scale Expansion Error

In Step 1, we establish a pointwise bound on \(\nabla ^3{\overline{u}}\), which in Step 2 we combine with a dyadic decomposition argument.

Step 1. Pointwise bound on \(\nabla ^3{{\bar{u}}}\). We claim that

$$\begin{aligned} \langle \vert \nabla ^3{{\bar{u}}} (x)\vert ^p\rangle _L^{\frac{1}{p}} \lesssim R\sup \vert \nabla ^2 f\vert \bigg (\frac{R}{R+\vert x-y\vert }\bigg )^d. \end{aligned}$$

Indeed, by (88) we have \({{\bar{u}}}(x)=\int _{B_R(y)}\textrm{d}z\,f(z) {{\bar{G}}}(x-z)\). From the bounds on the constant-coefficient Green function \({{\bar{G}}}\), which are uniform in the random coefficient \(\lambda \textrm{id}\le {{\bar{a}}}\le \textrm{id}\), we obtain

$$\begin{aligned}{} & {} \langle \vert \nabla ^3{{\bar{u}}}(x)\vert ^p\rangle _L^{\frac{1}{p}} \\ {}{} & {} \lesssim \left\{ \begin{array}{lll}\text{ for } |x-y|\le 2R:&{} \int _{B_R(y)}\textrm{d}z\,\vert \nabla ^2 h(z)\vert \langle \vert \nabla \bar{G}(x-z)\vert ^p\rangle ^{\frac{1}{p}}&{} \lesssim \sup |\nabla ^2 f|R\\ \text{ for } |x-y|\ge 2R:&{} \int _{B_R(y)}\textrm{d}z\,\vert \nabla f(z)\vert \langle \vert \nabla ^2{{\bar{G}}}(x-z)\vert ^p\rangle ^{\frac{1}{p}}&{} \lesssim \sup |\nabla f|(\frac{R}{|x-y|})^d \end{array}\right\} . \end{aligned}$$

It remains to appeal to \(\sup |\nabla f|\le R\sup |\nabla ^2 f|\).

Step 2. Dyadic decomposition. We restrict the r.h.s. of (87) to dyadic annuli:

$$\begin{aligned} \begin{aligned} h_k:= \mathbb {1}_{B_{2^k}\setminus B_{2^{k-1}}} \bigg ((\phi _{ij}^{(2)}-\phi _{ij}^{(2)}(0))a -(\sigma _{ij}^{(2)}-\sigma _{ij}^{(2)}(0))\bigg ) \nabla \partial _{ij} {\bar{u}}; \end{aligned} \end{aligned}$$

this induces the decomposition \(\nabla w=\sum _{k\in {\mathbb {Z}}} \nabla w_k\), where \(\nabla w_k\) is the square-integrable solution of

$$\begin{aligned} -\nabla \cdot a\nabla w_k = \nabla \cdot h_k. \end{aligned}$$

We observe from (165) that \(w_k\) is a-harmonic in \(B_{2^{k-1}}\).

Due to the above decomposition, the desired estimate (89) is reduced to estimating \(\nabla w_k(0)\) for each k. Using first Lemma 5, then the energy estimate, and finally Minkowski’s inequality, we obtain provided \(p'>p\ge 2\)


In view of the definition (165) of \(h_k\), (71) in Proposition 3, and (164), we have for \(p''>p'\)


We now distinguish the two cases of \(2^k\le R\), where we use , and of \(2^k>R\), where we use

The combination of (166), (167) and the two last estimates yields:

$$\begin{aligned} \big \langle |\nabla w(0)|^{p} \big \rangle _L^{\frac{1}{p}} \lesssim \bigg (\sum _{2^k\le R}\mu _d^{(2)}(2^k)+\sum _{2^k>R}(\frac{R}{2^k})^{\frac{d}{2}} \mu _d^{(2)}(2^k)\bigg ) R\sup |\nabla ^2 h|. \end{aligned}$$

Since for any \(d>2\), \(\mu _d^{(2)}(r)\) is non-decreasing in r, linear for \(r \le 1\), and not increasing faster than \(r^\frac{1}{2}\) for \(r\ge 1\), we recover (89).

5.7 Proof of Proposition 4: Annealed Error Estimate on the Expansion of the Green Function

Throughout the proof, we fix two “base points” \(x_0,y_0\in {\mathbb {R}}^d\) with \(|x_0-y_0|\ge 2\).

Step 1. Passage to the full error in the two-scale expansion. Recall that \({{\mathcal {E}}}\), cf. (58) for its definition, is the truncated version, on the level of the mixed derivatives, of the full error of the second-order two-scale expansion of the constant-coefficient fundamental solution \({{\bar{G}}}\), which is given byFootnote 22

$$\begin{aligned} \begin{aligned} w_{x_0,y_0}(x,y)&:= G(x,y)-\bigg (1+\phi ^{(1)}_{i}(x)\partial _i +(\phi ^{(2)}_{im}-\phi ^{(2)}_{im}(x_0))(x)\partial _{im}\bigg ) \\&\quad \times \bigg (1-\phi ^{*(1)}_j(y)\partial _j+(\phi _{jn}^{*(2)} \\&\quad -\phi _{jn}^{*(2)}(y_0))(y) \partial _{jn}\bigg ){\bar{G}}(x-y). \end{aligned} \end{aligned}$$

We now find the mixed derivative:

$$\begin{aligned} \begin{aligned} \nabla \nabla w_{x_0,y_0}(x,y)&=\nabla \nabla G(x,y) \\&\quad -\big [(e_i+\nabla \phi _i^{(1)})(x)\partial _i +(\nabla \phi _{im}^{(2)}+\phi _{i}^{(1)}e_m)(x)\partial _{im} +(\phi _{im}^{(2)}\\&\quad -\phi _{im}^{(2)}(x_0))(x)e_k\partial _{imk}\big ]\\&\quad \otimes \big [-(e_j+\nabla \phi _{j}^{*(1)})(y)\partial _j + (\nabla \phi _{jn}^{*(2)} +\phi _{j}^{*(1)}e_n)(y)\partial _{jn} \\&\quad -(\phi _{jn}^{*(2)}-\phi _{jn}^{*(2)}(y_0))(y)e_l\partial _{jnl} \big ]{\bar{G}}(x-y). \end{aligned} \end{aligned}$$

We further consider the difference between the mixed derivative of full error and its truncated version \({\mathcal {E}}\) by setting \(x=x_0\) and \(y=y_0\), which gives, together with Proposition 3,

$$\begin{aligned}{} & {} \big \langle \left|\nabla \nabla w_{x_0,y_0}(x_0,y_0) - {\mathcal {E}}(x_0,y_0)\right|^p \big \rangle _L^{\frac{1}{p}} \nonumber \\{} & {} \lesssim \mu ^{(2)}_d(\vert x_0-y_0\vert )|x_0-y_0|^{-d-2}\quad \text { for any } p<\infty . \end{aligned}$$

Therefore, to obtain the desired estimate (90) it suffices to show

$$\begin{aligned} \big \langle |\nabla \nabla w_{x_0,y_0}(x_0,y_0)|^{p} \big \rangle _L^{\frac{1}{p}} \lesssim \max \{\mu ^{(2)}_d(|x_0-y_0|),\ln \vert x_0-y_0\vert \} |x_0-y_0|^{-d-2}. \nonumber \\ \end{aligned}$$

Step 2. A decomposition of the full error \(\nabla \nabla w_{x_0,y_0}(x_0,y_0)\). In this step, we shall derive a characterizing PDE (175) of the full error (169) in order to split it into a far-field part \(w_{x_0,y_0,\infty }\) and dyadic near-field parts \(w_{x_0,y_0,k}(x,\cdot )\), which will be explicitly given later on. The distinction between far and near fields refers to the scale \(R:=|x_0-y_0|/2\). Recall that \(w_{x_0,y_0}(x,y)\) involves the two-scale expansion in both the x and y variables; we now freeze \(x=x_0\) and consider y as the “active” variable. For the ease of the statement, we make use of the notation

$$\begin{aligned} {{\bar{u}}}_{x_0}(x,\cdot ):= \bigg (1+\phi _i^{(1)}(x)\partial _i+(\phi _{im}^{(2)}(x)-\phi _{im}^{(2)}(x_0))\partial _{im}\bigg ) {{\bar{G}}}(x-\cdot ). \end{aligned}$$

This amounts to rewriting \( w_{x_0,y_0}(x_0,\cdot )\) as follows:

$$\begin{aligned} w_{x_0,y_0}(x_0,\cdot )= G(x_0,\cdot )- \bigg (1-\phi ^{*(1)}_{j}\partial _j+ (\phi _{jn}^{*(2)}-\phi _{jn}^{*(2)}(y_0)) \partial _{jn}\bigg ) {\bar{u}}_{x_0}(x_0,\cdot ).\nonumber \\ \end{aligned}$$

We note that \(G(x_0,\cdot )\) is \(a^{*}\)-harmonic, whereas \({\overline{u}}_{x_0}(x_0,\cdot )\) is \({\overline{a}}^{*}\)-harmonic in \({\mathbb {R}}^d\backslash \{x_0\}\). Hence, the representation of the error in the second-order two-scale expansion introduced in Sect. 3.4, we thus have

$$\begin{aligned} -\nabla \cdot a^* \nabla w_{x_0,y_0}(x_0,\cdot )= \nabla \cdot h_{x_0,y_0}(x_0,\cdot ) \quad \text {in}\quad {\mathbb {R}}^d \setminus \{x_0\}, \end{aligned}$$

where the vector field \(h_{x_0,y_0}\) is given by

$$\begin{aligned} h_{x_0,y_0}(x,\cdot ):= \bigg ((\phi ^{*(2)}_{jn}-\phi ^{*(2)}_{jn}(y_0))a^* -(\sigma ^{*(2)}_{jn}-\sigma _{jn}^{*(2)}(y_0))\bigg ) \nabla \partial _{jn}{\bar{u}}_{x_0}(x,\cdot ).\qquad \end{aligned}$$

Next we define the dyadic near-field (scalar) functions:

$$\begin{aligned} \begin{aligned} -\nabla \cdot a^*\nabla w_{x_0,y_0,k}(x,\cdot ) =\nabla \cdot \mathbb {1}_{B_{2^k}(y_0) \backslash B_{2^{k-1}}(y_0)}h_{x_0,y_0}(x,\cdot ), \end{aligned} \nonumber \\ \end{aligned}$$

and the far-field function:

$$\begin{aligned} \begin{aligned} w_{x_0,y_0,\infty }:=w_{x_0,y_0}-\sum _{2^k\le R}w_{x_0,y_0,k}, \end{aligned} \end{aligned}$$

In fact, we are interested in the quantities \(\nabla _{x} w_{x_0,y_0,k}(x_0,\cdot )\) and \(\nabla _{x} w_{x_0,y_0,\infty }(x_0,\cdot )\), which we address in two steps.

Step 3. Estimate of the near-field parts \(\nabla \nabla _{x} w_{x_0,y_0,k}(x_0,\cdot )\). Note that applying \(\nabla _x\) and evaluating at \(x=x_0\) commutes with the differential operator \(\nabla \cdot a^*\nabla \). For the ease of notation, we fix an arbitrary coordinate direction \(i=1,\cdots ,d\) and introduce the abbreviationFootnote 23\(w_{k,i}(y):=\partial _{x_i} w_{x_0,y_0,k}(x_0,y)\). We start by estimating its constitutive element \({{\bar{u}}}_{x_0}\) (see (173)) and we obtain from Proposition 3 and the \(-d\)-homogeneity of \({{\bar{G}}}\)

$$\begin{aligned}{} & {} \sup _{y\in B_{R}(y_0)}\big \langle \left|\nabla ^j \partial _{x_i}{\bar{u}}_{x_0}(x_0,y)\right|^{p}\big \rangle _L^{\frac{1}{p}} \nonumber \\{} & {} \lesssim R^{-j-d+1} \bigg (1+ \bigg \langle \bigg \vert \bigg (\nabla \phi ^{(1)}(x_0),\frac{\phi ^{(1)}(x_0)}{R}, \frac{\nabla \phi ^{(2)}(x_0)}{R}\bigg ) \bigg \vert ^p \bigg \rangle _L^{\frac{1}{p}}\bigg ) \lesssim R^{-j-d+1} \nonumber \\ \end{aligned}$$

for any \(j\ge 0\) and \(p<\infty \) (we also used \(R\ge 1\) in the last estimate). As in the proof of Lemma 3, \(2<p<p'\) denote generic exponents for stochastic integrability. Thus, using (92) in Lemma 5 and the energy estimate, we have


By Minkowski’s inequality and Proposition 3, we also have


The combination of (180) and (181) leads to

$$\begin{aligned} \sum _{2^k\le R} \big \langle \vert \nabla w_{k,i}(y_0) \vert ^{p} \big \rangle _L^{\frac{1}{p}} \lesssim \max \{\mu _d^{(2)}(R),\ln R\} R^{-2-d}. \end{aligned}$$

Step 4. Estimate of the near-field parts \(\nabla \nabla _{x} w_{x_0,y_0,k}(x_0,\cdot )\) in a weak norm. Let \(p<p'<p''\) be three stochastic exponents. In the sequel, \(h=h(y)\) always denotes an arbitrary smooth vector field compactly supported in \(B_R(y_0)\). We now justify a weak control on \(\nabla \nabla _{x} w_{x_0,y_0,\infty }(x_0,\cdot )\) that will appear useful in Step 5 when appealing to Corollary 2:

$$\begin{aligned} \sum _{2^k\le R}\big \langle \big | \int _{{\mathbb {R}}^d} h\cdot \nabla w_{k,i}\big |^{p}\big \rangle ^\frac{1}{p}_L \lesssim \mu _{d}^{(2)}(R)R\sup |\nabla ^3 h|. \end{aligned}$$

We start with a strong estimate of \(w_{k,i}\) on this set. As opposed to (180), we use Jensen’s inequality to pass to the spatial \(L^2\)-norm and then replace the energy estimate by the annealed Calderón–Zygmund estimate [39, Proposition 7.1],

Then, to bound the r.h.s., we appeal to the definition of \(h_{x_0,y_0}\) in (176), Proposition 3 and (179) to get the following estimate similar to (181),

This shows (183) in form of

$$\begin{aligned} \sum _{2^k\le R}\big \langle \big | \int _{{\mathbb {R}}^d} h\cdot \nabla w_{k,i}\big |^{p'}\big \rangle ^\frac{1}{p'}_L\le & {} \sup |h| \sum _{2^k\le R} \bigg \langle \bigg (\int _{B_R(y_0)}|\nabla w_{k,i}|\bigg )^{p}\bigg \rangle ^{\frac{1}{p}}_L \\\lesssim & {} \mu _{d}^{(2)}(R)R\sup |\nabla ^3 h|. \end{aligned}$$

Step 5. Estimate of the far-field part \(\nabla \nabla _{x} w_{x_0,y_0,\infty }(x_0,\cdot )\) by a duality argument. Again, for the ease of notation we introduce the abbreviation \(w_{\infty ,i}(y):=\partial _{x_i}w_{x_0,y_0,\infty }(x_0,y)\) with an arbitrary coordinate direction \(i=1,\cdots ,d\), which is \(a^*\)-harmonic on \(B_{2^{k_0}}(y_0)\). While in the previous two steps (mostly) relied on homogenization in the y-variable in form of control of \((\phi ^{*(2)},\sigma ^{*(2)})\), we now (primarily) need homogenization in the x-variable, in form of Lemma 3, next to control of \(\phi ^{*(2)}\). of Corollary 2: there hoWe start with an application lds


While the second contribution has been estimated in (183), we now need a similar estimate on the first contribution, namely,

$$\begin{aligned}{} & {} \Bigg \langle \Bigg | \int _{{\mathbb {R}}^d} h\cdot \nabla \partial _{x_i} w_{x_0,y_0}(x_0,\cdot ) \Bigg |^p \Bigg \rangle _L^{\frac{1}{p}}\nonumber \\{} & {} \qquad \lesssim _{p} \max \{\mu ^{(2)}_3(R),\ln R\}R\sup |\nabla ^3 h|\quad \text {for any }p<\infty . \end{aligned}$$

Equipped with (185), (172) follows from the combination of (184), (183), (182) and (178).

Now, we focus on the argument for (185). Let \(h\in C_0^\infty (B_R(y_0))\) be arbitrary, with u and \({\bar{u}}\) satisfying (88). We recall the definition of the error in the two-scale expansion that we express in terms of the Green functions \(G,{{\bar{G}}}\) using (88):

$$\begin{aligned} \begin{aligned} w_{x_0}(x)&:= u(x) - \big (1+\phi _{i}^{(1)}(x)\partial _i+ (\phi _{im}^{(2)}-\phi _{im}^{(2)}(x_0))(x) \partial _{im} \big ) {\bar{u}}(x) \\&\overset{(88)}{=}\ \int _{{\mathbb {R}}^d} (\nabla \cdot h)(G(x,\cdot ) - {\overline{u}}_{x_0}(x,\cdot )), \end{aligned} \end{aligned}$$

where we recall that \({{\bar{u}}}_{x_0}\) is defined in (173). Then, by taking derivatives on the both sides of the above equation with respect to the x-variable and by integrating by parts with respect to the y-variable lead to

$$\begin{aligned} \partial _{x_i} w_{x_0}(x)= & {} \int _{{\mathbb {R}}^d} (\nabla \cdot h) \big (\partial _{x_i} G(x,\cdot ) - \partial _{x_i}{\bar{u}}_{x_0}(x,\cdot )\big ) \nonumber \\= & {} - \int _{{\mathbb {R}}^d} h\cdot \nabla \big ( \partial _{x_i} G(x,\cdot ) - \partial _{x_i}{\bar{u}}_{x_0}(x,\cdot )\big ). \end{aligned}$$

We now express the integral in the l.h.s. of (185) with help of (186). First, by applying \(\nabla \) to (174), we obtain

$$\begin{aligned} \nabla \partial _{x_i} w_{x_0,y_0}(x_0,\cdot )= & {} \nabla \partial _{x_i} G(x_0,\cdot )-\nabla \partial _{x_i} {\bar{u}}_{x_0}(x_0,\cdot ) \nonumber \\{} & {} +\nabla \big [\big (\phi ^{*(1)}_{j}\partial _j -(\phi _{jn}^{*(2)}-\phi _{jn}^{*(2)}(y_0))\partial _{jn} \big )\partial _{x_i} {\bar{u}}_{x_0}(x_0,\cdot )\big ]. \nonumber \\ \end{aligned}$$

Second, we split the integral in the l.h.s. of (185) as follows:

$$\begin{aligned}{} & {} \int _{{\mathbb {R}}^d} h\cdot \nabla \partial _{x_i} w_{x_0,y_0}(x_0,\cdot ) \overset{(187)}{=}\int _{{\mathbb {R}}^d} h\cdot \nabla \big (\partial _{x_i} G(x_0,\cdot )- \partial _{x_i}{\bar{u}}_{x_0}(x_0,\cdot ) \big )\nonumber \\{} & {} \qquad \qquad + \int _{{\mathbb {R}}^d} h\cdot \nabla \big [\big (\phi ^{*(1)}_{j}\partial _j-(\phi _{jn}^{*(2)} -\phi _{jn}^{*(2)}(y_0)) \partial _{jn}\big ) \partial _{x_i}{\bar{u}}_{x_0}(x_0,\cdot )\big ] \nonumber \\{} & {} \quad \overset{(186)}{=}\ -\partial _{x_i} w_{x_0}(x_0)-\int _{{\mathbb {R}}^d} (\nabla \cdot h) \phi ^{*(1)}_j \partial _j \partial _{x_i}{\bar{u}}_{x_0}(x_0,\cdot ) \nonumber \\{} & {} \qquad \qquad + \int _{{\mathbb {R}}^d} (\nabla \cdot h) (\phi _{jn}^{*(2)} -\phi _{jn}^{*(2)}(y_0))\partial _{jn} \partial _{x_i}{\bar{u}}_{x_0}(x_0,\cdot ). \end{aligned}$$

For the first term r.h.s. term of (188), it follows from Lemma 3 applied with \(f=\nabla \cdot h\) that

$$\begin{aligned} \big \langle |\nabla _x w_{x_0}(x_0) |^p\big \rangle ^{\frac{1}{p}}_L \lesssim \max \{\mu _{d}^{(2)}(R),\ln R\}R\sup |\nabla ^3 h|. \end{aligned}$$

For the second term r.h.s. term of (188), we exploit the structure (173) of \(\partial _{x_i}{\bar{u}}_{x_0}(x_0,\cdot )\), the random part of which is independent of the integration variable y. Hence, we may use the Cauchy–Schwarz inequality, and then appealing to the definition (68) of \(\omega ^*_j\) (with \(\phi ^{(1)}_j\) replaced by \(\phi ^{*(1)}\)) together with (73) and (69), and finally recall that h is supported in \(B_R\), to the effect of

$$\begin{aligned} \begin{aligned}&\Big \langle \Big |\int _{{\mathbb {R}}^d} (\nabla \cdot h) \phi ^{*(1)}_j \partial _j \partial _{x_i}{\bar{u}}_{x_0}(x_0,\cdot ) \Big |^p\Big \rangle ^{\frac{1}{p}}_L \\ {}&\quad \le \Big \langle \Big |\int _{{\mathbb {R}}^d} (\nabla \cdot h) \phi ^{*(1)}_j \partial _{jk} {\bar{G}}(x_0-\cdot ) \Big |^{2p} \Big \rangle ^{\frac{1}{2p}}_L \Big \langle |\delta _{ik}+\partial _i \phi ^{(1)}_k(x_0)|^{2p} \Big \rangle ^{\frac{1}{2p}}_L \\ {}&\qquad + \Big \langle \bigg |\int _{{\mathbb {R}}^d} (\nabla \cdot h) \phi ^{*(1)}_j \partial _{jkm} {\bar{G}}(x_0-\cdot ) \Big |^{2p} \Big \rangle ^{\frac{1}{2p}}_L \Big \langle \Big |(\phi _k^{(1)} \delta _{im} + \partial _i \phi ^{(2)}_{km})(x_0) \bigg |^{2p} \Big \rangle ^{\frac{1}{2p}}_L \\ {}&\quad \lesssim \bigg (\Big \langle \Big |\int _{{\mathbb {R}}^d} \nabla (\nabla \cdot h)\cdot (\nabla \omega ^*_j-\nabla \omega ^*_j(x_0)) \partial _{jk} {\bar{G}}(x_0-\cdot ) \Big |^{2p} \Big \rangle ^{\frac{1}{2p}}_L\\ {}&\qquad +\Big \langle \Big |\int _{{\mathbb {R}}^d} (\nabla \cdot h)(\nabla \omega ^*_j-\nabla \omega ^*_j(x_0))\cdot \nabla \partial _{jk} {\bar{G}}(x_0-\cdot ) \Big |^{2p} \Big \rangle ^{\frac{1}{2p}}_L\bigg )\\ {}&\qquad \Big \langle |\delta _{ik}\!+\!\partial _i \phi ^{(1)}_k(x_0)|^{2p} \Big \rangle ^{\frac{1}{2p}}_L \\ {}&\qquad + \Big \langle \Big |\int _{{\mathbb {R}}^d} (\nabla \cdot h) \phi ^{*(1)}_j \partial _{jkm} {\bar{G}}(x_0-\cdot ) \Big |^{2p} \Big \rangle ^{\frac{1}{2p}}_L \Big \langle \Big |(\phi _k^{(1)} \delta _{im} + \partial _i \phi ^{(2)}_{km})(x_0) \Big |^{2p} \Big \rangle ^{\frac{1}{2p}}_L\\ {}&\quad \lesssim \sup |\nabla ^2 h| \mu ^{(2)}_{d}(R)+\sup \vert \nabla h\vert (\mu ^{(2)}_d(R)+1)R^{-1} \lesssim \mu _d^{(2)}(R)R\sup |\nabla ^3h|. \end{aligned} \nonumber \\ \end{aligned}$$

The third r.h.s. term in (188) is easily dealt with by recalling that h is of compact support, using Jensen’s inequality and the Cauchy–Schwarz inequality, and then a combination of (71) and (179)

$$\begin{aligned} \begin{aligned}&\bigg \langle \bigg | \int _{{\mathbb {R}}^d} (\nabla \cdot h) (\phi _{jn}^{*(2)} -\phi _{jn}^{*(2)}(y_0))\partial _{jn} \partial _{x_i}{\bar{u}}_{x_0}(x_0,\cdot ) \bigg |^{p} \bigg \rangle ^{\frac{1}{p}}_L \\&\lesssim \sup |\nabla h| \int _{B_R}\langle |\phi _{jn}^{*(2)} -\phi _{jn}^{*(2)}(y_0)|^{2p} \rangle ^{\frac{1}{2p}} \langle | \partial _{jn} \partial _{x_i}{\bar{u}}_{x_0}(x_0,\cdot )|^{2p}\rangle ^{\frac{1}{2p}}_L \\&\lesssim _p R\sup |\nabla ^2h| R^d \mu _d^{(2)}(R)R^{-d-1} \\&\lesssim \mu _d^{(2)}(R)R\sup |\nabla ^3 h|. \end{aligned}\nonumber \\ \end{aligned}$$

Inserting the estimates (189), (190), and (191), into (188) entails (185). \(\square \)