Abstract
We consider solutions of an elliptic partial differential equation in \(\mathbb{R }^d\) with a stationary, random conductivity coefficient that is also periodic with period \(L\). Boundary conditions on a square domain of width \(L\) are arranged so that the solution has a macroscopic unit gradient. We then consider the average flux that results from this imposed boundary condition. It is known that in the limit \(L \rightarrow \infty \), this quantity converges to a deterministic constant, almost surely. Our main result is that the law of this random variable is very close to that of a normal random variable, if the domain size \(L\) is large. We quantify this approximation by an error estimate in total variation. The error estimate relies on a second order Poincaré inequality developed recently by Chatterjee.
1 Introduction
Elliptic partial differential equations of the form
arise in many physical applications where the coefficient \(a(x)\) may be modeled best as a random field, due to inherent uncertainty and complexity of the physical medium [23]. In this situation, the solutions \(u\) are also random objects. Homogenization theory for these equations [14, 16, 20] shows that, although the coefficient \(a(x)\) may be highly irregular, a solution \(u\) may be approximated well by the solution of an “effective” elliptic equation having a more regular coefficient, perhaps a deterministic coefficient. Quantifying the error in such an approximation and understanding the statistical structure of the random solution is very important.
Here we consider solutions of the elliptic equation
where the scalar function \(a(x)\in L^\infty (\mathbb{R }^d)\) is a stationary random field satisfying the uniform ellipticity condition \(0 < a_* \le a(x) \le a^*\), with \(a_*\) and \(a^*\) being deterministic constants. The parameter \(\beta \ge 0\) is deterministic. The set \(D_L = [0,L)^d\) is the domain, and we require that \(\phi \) satisfies periodic boundary conditions on the boundary of \(D_L\). If we interpret (1.1) in terms of electrical conductivity, then \(\phi \) is a potential, \(a(x)\) is the conductivity, and the vector field \(-a(x) (\nabla \phi + e_1)\) is a current density. The unit vector \(e_1\) is deterministic, and it is the gradient of the linear potential \(x \cdot e_1\).
The Eq. (1.1) plays an important role in the homogenization theory for the random elliptic operator \(u \mapsto - \nabla \cdot (a(x/\epsilon ) \nabla u)\) in the limit \(\epsilon \rightarrow 0\) [14, 20]. It is well-known that the homogenized conductivity tensor \(\bar{a}\) for that operator can be expressed in terms of functions \(\phi \), called “correctors”, which solve (1.1) with \(e_1\) being one of the \(d\) standard basis vectors and which have stationary gradient. On the other hand, in a numerical computation of \(\bar{a}\) one must approximate the true correctors by solving (1.1) in a bounded domain \(D_L\) with suitable boundary condition. The periodic boundary condition that we impose here is one choice that allows accurate approximation of the effective coefficient \(\bar{a}\) in the limit \(L \rightarrow \infty \) [6, 19].
The focus of this paper is on the statistical behavior of the quantity
for large \(L\). Using (1.1) and the periodicity of \(\phi \) we see that \(\Gamma _{L,\beta }\) may also be written as
This is a random variable since \(a(x)\) and the solution \(\phi \) are random. In terms of conductivity, \(\Gamma _{L,\beta }\) may be interpreted as an average flux in the direction \(e_1\) that results from a macroscopic potential gradient imposed in the direction of \(e_1\). The results of [6, 19] imply that for \(\beta \ge 0\) fixed, \(\Gamma _{L,\beta }\) converges almost surely, as \(L \rightarrow \infty \), to a deterministic constant \(\bar{\Gamma }_\beta > 0\). For \(\beta = 0\), the limit \(\bar{\Gamma }_0\) is one of the diagonal entries of the homogenized tensor \(\bar{a}\). For finite \(L\), it is interesting to understand how \(\Gamma _{L,\beta }\) and \(\phi \) fluctuate around their means. However, not much is known about the distribution of \(\Gamma _{L,\beta }\) or the distribution of \(\phi \). Our main result is an estimate showing that for \(L > \!\!> 1\), the distribution of \(\Gamma _{L,\beta }\) is very close to that of a normal random variable.
Before we present the main result and explain its relation to other works, let us define the problem precisely and establish notation. For \(L \in \mathbb{Z }^+\), let \(D_L = [0,L)^d \subset \mathbb{R }^d\) and let \(L^\infty _{per}(D_L)\) denote the set of functions in \(L^\infty (\mathbb{R }^d)\) which are periodic with period \(L\) in each direction. That is, for all \(a \in L^\infty _{per}(D_L),\,a(x + Lk) = a(x)\) holds for all \(k \in \mathbb{Z }^d\) and almost every \(x \in \mathbb{R }^d\). The coefficient \(a(x)\) in (1.1) will be a random function in \(L^\infty _{per}(D_L)\). We also require that \(a(x)\) is stationary with respect to integer shifts: for every \(k \in \mathbb{Z }^d\) and \(a(\cdot + k)\) is equal in law to \(a(\cdot )\).
We suppose that the random nature of \(a(x)\) comes from its dependence on a random vector \(\zeta = (\zeta _k)_{k \in \mathbb{Z }^d \cap D_L}\) whose \(L^d\) components are independent and identically distributed real-valued random variables, defined over a probability space \((\Omega ,\mathcal F ,\mathbb{P })\), where \(\Omega = \mathbb{R }^{L^d}\) and \(\mathbb{P }\) is a product measure on \(\Omega \). We often will write \(a(x)\) for the random function \(a(x,\zeta )\), the dependence on \(\zeta \) being understood. Let \(\mathbb E [Y]\) denote expectation with respect to the probability measure \(\mathbb{P }\) defining the law of \(\zeta \) (there will be another assumption about the law of \(\zeta \) below). We will make the following structural assumptions about the random function \(a(x,\zeta )\). First, we suppose that the map \(\zeta \rightarrow a(\cdot ,\zeta )\) from \(\mathbb{R }^{L^d} \rightarrow L^{\infty }_{per}(D_L)\) is a twice Fréchet differentiable map. For each \(k \in \mathbb{Z }^d\), let \(Q_k = k + [0,1)^d \subset \mathbb{R }^d\) denote the cube of size 1 with a corner at \(k\). We suppose there are positive constants \(\tau , C_1, C_2, C_3, a^*, a_* > 0\) such that the following hold \(\mathbb{P }\)-almost surely:
The function \(\mathbb{I }_{S}(x)\) is the indicator of the set \(S\), and \(B_{\tau }(y)\) is the ball of radius \(\tau \) centered at \(y\), in the metric topology of the torus on \(D_L\). That is, \(x \in B_{\tau }(y)\) if and only if \(x = y + Lk + z\) for some \(k \in \mathbb{Z }^d\) and \(|z| < \tau \). We suppose the constants \(\tau ,\,C_1,\,C_2,\,C_3,\,a^*\), and \(a_*\) are independent of \(L\). Notice that the bound (1.3) implies that \(a(x,\zeta )\) depends only on those \(\zeta _k\) for \(k\) in a \(\tau \)-neighborhood of \(x\) (fixed with respect to \(L\)). Thus, \(a(x,\zeta )\) and \(a(y,\zeta )\) are independent if \(|x - y| > 2\tau \). Also, (1.2) and (1.3) imply that there are constants \(\zeta _{min} < \zeta _{max}\) such that \(\zeta _k \in [\zeta _{min},\zeta _{max}]\) holds with probability one.
For clarity, let us highlight a simple example for which these assumptions hold. Suppose that \(0 < \zeta _{min} \le \zeta _k \le \zeta _{max}\) for all \(k \in \mathbb{Z }^d \cap D_L\) holds with probability one. For \(x \in \mathbb{R }^d\), define the piecewise constant function
where the sets \(M_k = k + M_0\) are translates of a given bounded and measureable set \(M_0\) satisfying \(Q_0 \subset M_0 \subset B_\tau (0)\). The notation \((k \; \text {mod} \;L)\) refers to the point \((k_1\; \text {mod} \;L, \dots , k_d\; \text {mod} \;L) \in D_L \cap \mathbb{Z }^d\). It is easy to see that \(a(x) \in L^\infty _{per}(D_L)\) with probability one. Conditions (1.2), (1.3), and (1.4) hold with \(a_* = \zeta _{min},\,a^* = \zeta _{max}O(\tau ^d)\), and \(C_1 = C_2 = C_3 = 1\). Moreover, for each \(k \in \mathbb{Z }^d,\,a(\cdot + k)\) has the same law as \(a(\cdot )\), since \(a(x + k, \zeta ) = a(x,\hat{\zeta })\) where \(\hat{\zeta }_j = \zeta _{(j + k) \;\text {mod}\; L }\).
Let \(H^1_{per}(D_L)\) denote the set of \(L\)-periodic functions in \(H^1_{loc}(\mathbb{R }^d)\). That is, \(\phi \in H^1_{per}(D_L)\) if \(\phi \in H^1_{loc}(\mathbb{R }^d)\) and \(\phi (x + Lk) = \phi (x)\) a.e. \(\mathbb{R }^d\) for every \(k \in \mathbb{Z }^d\). If \(a(x) \in L^\infty (D_L)\) and satisfies \(0 < a_* \le a(x) \le a^*\) almost everywhere, then there exists a weak solution \(\phi \in H^1_{per}(D_L)\) to (1.1):
For \(\beta > 0\), the solution is unique. For \(\beta = 0\), the solution is not unique, but any two solutions in \(H^1_{per}(D_L)\) must differ by a constant. So, under the normalization condition
and for fixed \(L\), the solution is unique in \(H^1_{per}(D_L)\) for all \(\beta \ge 0\). With \(a(x) = a(x,\zeta )\) satisfying the conditions above, this unique solution \(\phi (x) = \phi (x,a,L,\beta )\) depends on the parameters \(L\) and \(\beta \), on \(x \in D_L\), and on the random variables \((\zeta _j)_{j \in D_L \cap \mathbb{Z }^d}\) which determine \(a\). The uniqueness of the solution implies that \(\phi (x)\) is statistically stationary with respect to integer shifts: the law of \(\phi (x)\) is the same as that of \(\phi (x + k)\) for any \(k \in \mathbb{Z }^d\), since the variables \(\zeta _j\) are identically distributed.
Having defined both \(a(x)\) and \(\phi (x)\), we now define the random variable
which also is a function of the \(L^d\) random variables \(\{ \zeta _j \;|\; j \in \mathbb{Z }^d \cap D_L \}\). It is known that \(\Gamma _{L,\beta }\) has a variational representation:
The Euler–Lagrange equation for this variational problem is (1.1), and \(\phi \) is the unique minimizer (unique up to addition of a constant if \(\beta = 0\)).
We make one more technical assumption about the variables \(\zeta _k\). We suppose that the law of \(\zeta _k\) is that of \(h(Z_k)\) where \(Z_k\) is a standard normal random variable and \(h:\mathbb{R }\rightarrow \mathbb{R }\) is twice differentiable and satisfies \(|h^{\prime }(z)| \le c_1\) and \(|h^{\prime \prime }(z)| \le c_2\). While this assumption excludes some interesting choices for the law of \(\zeta _k\), it does not imply that the law of \(\zeta _k\) has a density with respect to Lebesgue measure on \(\mathbb{R }\). We suppose that \(h\) is not a constant, so that \(\text {Var}(\zeta _k) > 0\).
Our main result is the following theorem. Recall that the total variation distance \(d_{TV}(X,Y)\) between the laws of two real-valued random variables \(X\) and \(Y\) is defined as
where the supremum is over all Borel sets \(A \subset \mathbb{R }\). This quantity is invariant under centering and scaling: \(d_{TV}(X,Y) = d_{TV}((X- \mu )/\sigma ,(Y - \mu )/\sigma )\) for all \(\mu \in \mathbb{R },\,\sigma \!>\! 0\).
Theorem 1.1
Let \(d \ge 1\). There is a constant \(C > 0\) and \(q > 4\) such that
holds for all \(L > 1\) and \(\beta \ge 0\), where \(W_{L,\beta }\) is a normal random variable having the same mean and variance as \(\Gamma _{L,\beta }\), and
Observe that the random variable \(\Phi _0\) which appears in Theorem 1.1 depends on both \(L\) and \(\beta \). It is easy to see that \(\mathbb E [ \Phi _0] \ge 1\) holds for all \(L > 1\) and \(\beta \ge 0\) and all \(d \ge 1\) [see (2.8)]. So, when it is also true that \(\mathbb E [\Phi _0^q]\) is bounded by a constant, independent of \(L > 1\), then the right side of (1.9) is bounded by \(O(L^{-d/2})\); in particular, the distribution of \(\Gamma _{L,\beta }\) approaches that of a normal random variable. As explained below, \(O(L^{-d/2})\) is the optimal bound on \(d_{TV}(\Gamma _{L,\beta },W_{L,\beta })\), in the sense that this is the expected bound if \(\Gamma _{L,\beta }\) behaves like the average of \(O(L^d)\) independent random variables. For all dimensions \(d \ge 1\), if \(\beta \ge \beta _0 > 0\) is bounded away from zero independently of \(L\), then all moments \(\mathbb E [\Phi _0^q]\) are bounded independently of \(L > 1\) (for example, see Corollary 5.5). In this case, Theorem 1.1 implies that \(d_{TV}(\Gamma _{L,\beta },W_{L,\beta }) = O(L^{-d/2})\) as \(L \rightarrow \infty \), which is the optimal bound.
If \(\beta = 0\) or if \(\beta > 0\) is allowed to vanish as \(L \rightarrow \infty \), estimating the moments \(\mathbb E [\Phi _0^q]\) is a delicate issue. To estimate \(\mathbb E [\Phi _0^q]\) in this situation one can use the arguments developed recently by Gloria and Otto [13], which is the work most directly related to this article. In [13], the authors derive variance bounds for a discrete functional similar to \(\Gamma _{L,\beta }\), involving an infinite network of random resistors on the bonds of the integer lattice \(\mathbb{Z }^d\). The PDE (1.1) is replaced by a discrete difference equation on all of \(\mathbb{Z }^d\), without the periodicity assumption. The stationary potential field \(\phi (x)\) is defined at points \(x \in \mathbb{Z }^d\); the gradient and divergence have interpretations as difference operators. For each edge \(e\) in the integer lattice, the conductivity \(A = A(e)\) is a random variable which is stationary with respect to lattice translation, but it is not periodic. Consequently, \(\phi \) depends nontrivially on the infinite set of conductances \(A(e)\). Gloria and Otto then consider the random variable
where \(\eta _L(x) \ge 0\) is a deterministic weight function that is supported on a cube of size \(L\), and having total mass 1. In present setting, the periodization of the random field \(a(x)\) over \(D_L\) serves a similar purpose to the weight function \(\eta _L\). One of the main results of [13] is that there is a constant \(C > 0\) such that
holds for all \(\beta > 0\) and \(L > 1\). Another important result from [13], and a key step in the analysis of \(\text {Var}(\tilde{\Gamma }_{L,\beta })\), is the following bound on moments of the discrete corrector \(\phi \):
The constants \(C_q, \gamma _q > 0\) are independent of \(L > 1\) and \(\beta > 0\). Observe that in dimension \(d = 2\), there is an extra factor that diverges as \(\beta \rightarrow 0\). The extension of the analysis of [13] to the present setting (spatial continuum, with periodicity on \(D_L\)) can be carried out to estimate moments of both \(\int _{Q_0} \phi (x) \,dx\) and \(\Phi _0\). In particular, the argument shows that for \(d \ge 3\) all moments \(\mathbb E [\Phi ^q]\) are bounded independently of \(L > 1\) and \(\beta \ge 0\). Therefore, for \(d \ge 3\), Theorem 1.1 implies that \(d_{TV}(\Gamma _{L,\beta },W_{L,\beta }) = O(L^{-d/2})\). In the case \(d = 2\), however, the argument shows that \(\mathbb E [\Phi ^q]\) is bounded by \(C |\log \beta |^\gamma \) for certain constants \(C_q,\gamma _q > 0\) independent of \(L > 1\) and \(\beta > 0\). So, in the case \(d = 2\), if \(\beta = 0\) or if \(|\log \beta |^{\gamma _q} \rightarrow \infty \) faster than \(L^{d/2}\) as \(L \rightarrow \infty \), we cannot conclude from this bound that \(d_{TV}(\Gamma _{L,\beta },W_{L,\beta }) \rightarrow 0\) as \(L \rightarrow \infty \). In Sect. 5 we explain a few points about this method of bounding \(\mathbb E [\Phi _0^q]\) and its relation to the present setting. However, the extension of the results of [13] to the periodic setting is being worked out in [12], so we do not pursue it further.
Other works related to Theorem 1.1 include those of Naddaf and Spencer [18], Conlon and Naddaf [8], and Boivin [4] in the discrete case and Yurinskii [26] in the continuum setting; they also derive upper bounds on the variance of quantities similar to \(\tilde{\Gamma }_{L,\beta }\) and \(\Gamma _{L,\beta }\). Komorowski and Ryzhik [15] have proved some related moment bounds on \(\phi \) in the discrete case when \(d = 1\). In the discrete setting the work of Wehr [24] contains a lower bound on the variance of a quantity analogous to \(\Gamma _{L,0}\). However, none of the works we have mentioned address the issue of a central limit theorem: whether the distribution of \(\Gamma _{L,\beta }\) is approximately normal for \(L > \!\!> 1\). If \(\beta = 0\) and the dimension is \(d=1\), then Eq. (1.1) can be integrated, with the solution \(\phi \) written in terms of integrals of \(1/a(x)\). In that case it is known that the solution itself may satisfy a central limit theorem after suitable renormalization; see Borgeat and Piatnitski [5], Bal et al. [1] for precise statement of these results. In the multidimensional setting, however, those techniques do not apply.
The basis for our proof of Theorem 1.1 is the following general inequality developed recently by Chatterjee [7], based on Stein’s method of normal approximation. From now on, we often use \(\Gamma \) for \(\Gamma _{L,\beta }\), the dependence on \(L\) and \(\beta \) being understood (\(\phi \) also depends on both \(L\) and \(\beta \)).
Theorem 1.2
([7], Theorem 2.2) Let \(h \in C^2(\mathbb{R };\mathbb{R })\). Let \(\{Z_k\}_{k \in \mathcal{I }}\) be a collection of independent, standard normal random variables, where \(\mathcal{I }\) is a finite index set. For \(k \in \mathcal{I }\), let \(\zeta _k = h(Z_k)\), and let \(\Gamma = \Gamma (\zeta ):\mathbb{R }^{|\mathcal{I }|} \rightarrow \mathbb{R }\) be a function of the random vector \(\zeta = (\zeta _k)_{k \in \mathcal{I }}\). Define constants
and
where \(\tilde{\zeta }(t) = (\tilde{\zeta }_k(t))_{k \in \mathcal{I }}\) is the random vector defined by
and \(Z^{\prime } = (Z_k^{\prime })_{k \in \mathcal{I }}\) is an independent copy of the random vector \(Z = (Z_k)_{k \in \mathcal{I }}\). If \(W\) is a normal random variable having the same mean and variance as \(\Gamma \), then
where \(\sigma ^2 = \text {Var}(\Gamma ),\,c_1 = ||h^{\prime } ||_\infty \), and \(c_2 = ||h^{\prime \prime } ||_\infty \).
We have stated this theorem differently from its statement in [7], yet the bound (1.13) follows directly from the analysis proving Theorem 2.2 of [7] (see pp. 33–34 therein). One way to bound (1.12) uses the operator norm for the Hessian of \(\Gamma \), as in [7]. With this approach, one obtains from (1.12) the estimate
which implies that
where
Here \(\nabla _\zeta \Gamma \) refers to the gradient with respect to the variables \(\zeta _k\) for \(k \in D_L\), and \(\nabla ^2_\zeta \Gamma \) is the Hessian, an \(L^d \times L^d\) matrix. The norm \(||\nabla _\zeta ^2 \Gamma ||\) is the \(L^2\) operator norm. The bound (1.15) is precisely the bound stated in Theorem 2.2 of [7]. Instead of using (1.14) and (1.15), however, we will use a different approach to bounding \(\kappa _3\) that allows us to make better use of the structure of \(\Gamma \).
For the moment, let us consider (1.15) instead of (1.13). What should we expect of the scaling of each of the terms in (1.15)? Consider a sum of random variables
where \(Z_j\) are independent standard normal random variables. Then \(\partial _j S = L^{-d} g^{\prime }(Z_j) \), so that \(\kappa _0 = O(L^{-3d/2})\) if \(g^{\prime }\) is bounded. Also, \(||\nabla S ||^4 = L^{-4d} (\sum _{j} g^{\prime }(Z_j)^2 )^2 \), so that \(\kappa _1 = O(L^{-d/2})\). For \(\kappa _2\) notice that \(\partial _j \partial _k S = L^{-d} \delta _{jk}g^{\prime \prime }(Z_j)\), so that \(\kappa _2 = O(L^{-d})\) if \(g^{\prime \prime }\) is bounded. Thus \(\kappa _3 = O(L^{-3d/2})\). Finally, the variance is \(\sigma ^2 = O(L^{-d})\), so that the bound (1.15) is \(O(L^{-d/2})\) for this simple sum of independent random variables. In general, if the dependence relations are sufficiently local, in the sense that \(L^{d} \partial _j \partial _k S\) is typically small for \(|j - k | >\!> 1\), we could still have \(\kappa _3 = O(L^{-3d/2})\) and \(d_{TV}(S, W) = O(L^{-d/2})\). Obviously \(\Gamma _{L,\beta }\) can be written as the normalized sum
where the random variables \(\eta _j\) are
Although the variables \(\eta _j\) in (1.17) are identically distributed, each \(\eta _j\) depends on the \(O(L^d)\) variables \(\zeta _k\) in a nonlinear way through solution of the PDE (1.1). Consequently, the terms in the sum (1.17) are mutually dependent, which makes the analysis of \(\Gamma _{L,\beta }\) challenging.
Starting from (1.13), a proof of Theorem 1.1 will follow from a suitable upper bound on \(\kappa _0\) and \(\kappa _3\) as well as a lower bound on \(\sigma ^2 = \text {Var}(\Gamma _{L,\beta })\), which appears in the denominator of (1.13). We will show that \(\kappa _0 = O(L^{-3d/2})\) and \(\kappa _3 = O(L^{-3d/2})\). We will also prove the following lower bound on \(\sigma ^2\), which is similar to a result in [24]:
Theorem 1.3
Let \(d \ge 1\). There is a constant \(C > 0\) such that
holds for all \(L \ge 1\) and \(\beta \ge 0\).
Although the proof of Theorem 1.1 does not require an upper bound on the variance of \(\Gamma _{L,\beta }\), the variance of \(\Gamma _{L,\beta }\) can also be estimated from above in terms of \(\Phi _0\):
Proposition 1.4
Let \(d \ge 1\). There is a constant \(C \ge 0\) such that
holds for all \(L \ge 1\) and \(\beta \ge 0\).
As we have mentioned, if \(d \ge 3\), or if \(d = 2\) and \(\beta > 0\) is fixed, then \(\mathbb E [\Phi _0^2]\) is bounded as \(L \rightarrow \infty \), and Proposition 1.4 implies that \(\text {Var}(\Gamma _{L,\beta }) = O(L^{-d})\).
Let us point out that the proof of Theorem 1.3 makes use of the structural assumptions on the coefficient \(a(x)\). Specifically, the lower bound in (1.3) enables us to estimate \(\frac{\partial \Gamma }{\partial \zeta _k}\) from below, a key step in proof of Theorem 1.3. This structural condition is not used anywhere else in the analysis. If (1.3) were replaced with
then (1.9) may be replaced with
The rest of the paper is organized as follows: in Sect. 2 we prove Theorem 1.3, the lower bound on \(\sigma ^2\). Upper bounds on the constants \(\kappa _0\) and \(\kappa _3\) and the rest of the proof of Theorem 1.1 are developed in Sects. 3 and 4. Section 3 contains some deterministic PDE estimates (Caccioppoli’s inequality and a version of Meyers’ estimate) which are useful in bounding \(\kappa _3\) and do not rely on the statistical structure of the coefficients. Section 4 contains the main argument bounding \(\kappa _0\) and \(\kappa _3\). Finally, in Sect. 5, we prove Proposition 1.4 and Corollary 5.5, which is an estimate of \(\mathbb E [\Phi _0^q]\) in the case that \(\beta > 0\) is fixed. We also make some remarks about estimating \(\mathbb E [\Phi _0^q]\) using the method of [13] to deal with the case that \(\beta \) vanishes as \(L \rightarrow \infty \).
A few more comments about notation: throughout the article we will use the convention that summation over indices \(j \in D_L\) means a summation over \(j \in \mathbb{Z }^d \cap D_L\), with \(j \in \mathbb{Z }^d\) being understood. For a given measureable set \(A \subset \mathbb{R }^d\), we define the normalized integral

We also use \(C\) to denote deterministic constants that may change from line to line, but do not depend on \(L\) or \(\beta \). We will use \(\Phi _j\) and \(\Phi _j^{\prime }\) to refer to the integrals
which appear frequently in the analysis. Recall that \(B_\tau (j) \supset Q_j\), so \(\Phi _j^{\prime } \ge \Phi _j\).
After this paper was submitted for publication, we learned of two other related works on discrete resistor network models. By making use of the martingale central limit theorem, Biskup et al. [2] have proved a central limit theorem for a discrete quantity similar to \(\tilde{\Gamma }_{L,\beta }\) when \(\phi \) satisfies linear Dirichet boundary conditions on a square box, in the regime of small ellipticity contrast (i.e. \(|\frac{a^*}{a_*} - 1|\) is sufficiently small). Using different techniques, including generalized Walsh decomposition and concentration bounds, Rossignol [21] has proved a variance bound and a central limit theorem for effective resistance of a resistor network on the discrete torus. We refer to the recent review paper [3] for many other references on the random conductance model.
2 A lower bound on the variance \(\sigma ^2\)
In this section we prove Theorem 1.3, the lower bound on \(\sigma ^2 = \text {Var}(\Gamma _{L,\beta })\) which appears in the denominator of (1.13). One approach to proving the lower bound is to use the argument of Wehr [24] who considered a functional similar to \(\Gamma \) for a discrete resistor network with random conductance (without a uniform ellipticity constraint). If we assume (1.5) with \(M_k = Q_k\) and that the law of \(\zeta _k\) is absolutely continuous with respect to Lebesgue measure on \([\zeta _{min},\zeta _{max}]\), that argument can be adapted to the present setting, under the constraint
where \(\nu \) is the density for the law of \(\zeta _k\). Here we give a proof that allows for the more general structural condition (1.3) and allows for the law of \(\zeta _k\) to be singular with respect to Lebesgue measure.
First, since the variables \(\{\zeta _k\}_{k \in \mathbb{Z }^d}\) are independent, we have the lower bound
where \(\mathbb E [\,\Gamma \,|\,\zeta _k\,]\) is the conditional expectation of \(\Gamma \), conditioned on the value of \(\zeta _k\). This inequality is proved in [25] (see Proposition 3.1, therein). Since the \(\zeta _k\) are identically distributed, we have \(\text {Var}\left( \mathbb E [\,\Gamma \,|\,\zeta _k\,] \right) = \text {Var}\left( \mathbb E [\,\Gamma \,|\,\zeta _j\,] \right) \) for all \(j,k \in D_L\), so that
Next, observe that
where
and \(\nu (ds)\) is the probability measure supported on \([\zeta _{min},\zeta _{max}]\) which is the law of the random variable \(\zeta _0\).
Since \(\zeta \mapsto a(x,\zeta )\) is nondecreasing with respect to each coordinate \(\zeta _k\), it follows from (1.8) that \(\Gamma \) is a nondecreasing function of each \(\zeta _k\), so we have \(g^{\prime }(s) \ge 0\). We will establish the following lower bound on the difference \(g(s) - g(s^{\prime })\):
Lemma 2.1
Define \(\rho (s) = \mathbb E \left[ \Phi _0 \;|\; \zeta _0 = s \right] \ge 0\). There is a constant \(\theta > 0\) such that
holds for all \(s^{\prime },s \in [\zeta _{min},\zeta _{max}]\), for all \(L > 1,\,\beta \ge 0\).
Therefore, from (2.3) we have
Since \(\int _{\mathbb{R }} (s - s^{\prime })^2 \nu (ds^{\prime }) = \mathbb E [(s - \zeta _0)^2] \ge \text {Var}(\zeta _0)\), this implies
Hence, (2.2) implies that
Let us observe that \(\mathbb E [\Phi _0] \ge 1\). Indeed, by stationarity of \(\phi \) we have

Since \(\phi \) is periodic, \(\int _{D_L} \nabla \phi \cdot e_1 \,dx = 0\). Hence,

Except for a proof of Lemma 2.1, this establishes Theorem 1.3. To prove Lemma 2.1, we will make use of the following lemma:
Lemma 2.2
Let \(\beta \ge 0\). Suppose that \(a(x)\) and \(a^{\prime }(x)\) are two measurable functions satisfying (1.2). Let \(\phi , \phi ^{\prime } \in H^1_{per}(D_L)\) satisfy
respectively. If \((a - a^{\prime })\) is supported on a measureable set \(M \subset D_L\), then
and
In particular, if the vectors \(\zeta \) and \(\zeta ^{\prime }\) differ only in the \(jth\) coordinate (i.e. \(\zeta _k = \zeta _k^{\prime }\) if \(k \ne j\)). Then \(a(\cdot ) = a(\cdot ,\zeta )\) and \(a^{\prime }(\cdot ) = a(\cdot ,\zeta ^{\prime })\) differ only on the set \(B_\tau (j)\) (by 1.3), so (2.9) and (2.10) hold with \(M = B_\tau (j)\).
Proof of Lemma 2.2
The function \(v(x) = \phi - \phi ^{\prime } \in H^1_{per}(D_L)\) is a weak solution to
Multiply by \(v\) and integrate by parts. The uniform ellipticity implies:
Since \(a^{\prime } - a = 0\) outside the set \(M\), the Cauchy–Schwarz inequality leads to
which is (2.9). The bound (2.10) now follows by the triangle inequality in \((L^2(M))^d\). \(\square \)
Proof of Lemma 2.1
Recall that \(g(s^{\prime }) - g(s) = \mathbb E [ \Gamma \;|\; \zeta _0 = s^{\prime }] - \mathbb E [\Gamma \;|\; \zeta _0 = s]\). Suppose the vectors \(\zeta \) and \(\zeta ^{\prime }\) differ only in the \(j^{th}\) coordinate (i.e. \(\zeta _k = \zeta _k^{\prime }\) if \(k \ne j\)) and that \(s^{\prime } = \zeta _j^{\prime } > \zeta _j = s\). Define \(a(\cdot ) = a(\cdot ,\zeta )\) and \(a^{\prime }(\cdot ) = a(\cdot ,\zeta ^{\prime })\). The difference \(a^{\prime } - a\) is supported in \(B_\tau (j)\), but its support may not be confined to \(\bar{Q}_j\). For this reason, we also define a function \(a^{\prime \prime }\) according to
Since \(a^{\prime }(x) \ge a^{\prime \prime }(x) \ge a(x)\) almost everywhere, we must have \(\Gamma (a^{\prime }) \ge \Gamma (a^{\prime \prime }) \ge \Gamma (a)\).
Let \(\phi , \phi ^{\prime \prime } \in H^1_{per}(D_L)\) satisfy
respectively. From the variational representation (1.8), we know that
Therefore,
Since \((a^{\prime \prime }- a) \ge 0\) is supported on \(\overline{Q_j}\), then by (2.12) and Lemma 2.2,
Hence by using (1.3) and (2.13) we obtain
By the lower bound in (1.3), this implies
On the other hand, arguing as at (2.12) and using (1.3) we also have
Lemma 2.1 now follows from (2.15) and (2.16) and the definition of \(g(s)\). \(\square \)
3 Deterministic estimates for solutions of the elliptic equation
Our next goal is to prove Theorem 1.1 by using Theorem 1.2. In this section, however, we first establish some regularity estimates that apply to solutions of elliptic equations. These estimates will be used in the process of bounding the constants \(\kappa _0\) and \(\kappa _3\) which appear in Theorem 1.2. These estimates rely only on the uniform ellipticity assumption, not on the statistical structure of the coefficient \(a(x)\).
3.1 Caccioppoli’s inequality
Recall that the Poincaré inequality tells us that for sufficiently regular sets \(D\) there is a constant \(C_D > 0\) such that
holds for all \(u \in H^1(D)\), where

For solutions of elliptic equations Caccioppoli’s inequality gives the reverse inequality, enabling us to control moments of \(\nabla \phi \) by moments of \(\phi \) itself. Here and at other points in the paper it will be convenient to use the notation \(3Q_j\) and \(5Q_j\) to refer to the cubes
which are concentric cubes of width 3 and 5, respectively, and containing \(Q_j = j + [0,1)^d\) in their center. We also define the random variables

Here is Caccioppoli’s inequality, presented in two different forms for convenient reference later:
Lemma 3.1
Let \(d \ge 1\) and let \(u \in H^1(3Q_j)\) be a weak solution to \(- \nabla \cdot (a \nabla u) + \beta u = \nabla \cdot \xi \) for \(x \in 3Q_j\), with \(\xi \in (L^2(3Q_j))^d\). There is a constant \(K\), depending only on \(a^*\) and \(a_*\) such that
holds for any constant \(b \in \mathbb{R }\). Similarly, there is a constant \(K\) such that if \(R > 0\) and \(u \in H^1(B_R(x_0))\) is a weak solution to \(- \nabla \cdot (a \nabla u) + \beta u= \nabla \cdot \xi \) for \(x \in B_R(x_0)\), with \(\xi \in (L^2(B_R))^d\), then
holds for any constant \(b \in \mathbb{R }\).
Lemma 3.1 and variants are a consequence of the following:
Lemma 3.2
Let \(K_1 = 2/a_*,\,K_2 = (2/a_*) + 8(a^*/a_*)^2\), and \(K_3 = (2/a_*) + 2/(a_*)^2\). Let \(Q\) be a bounded open subset of \(\mathbb{R }^{d}\) with smooth boundary. If \(\beta \ge 0\) and \(u \in H^1(Q)\) is a weak solution to \(- \nabla \cdot (a \nabla u) + \beta u = f + \nabla \cdot \xi \) for \(x \in Q\), with \(f \in L^2(Q)\) and \(\xi \in (L^2(Q))^d\), then
holds for any smooth function \(\varphi \ge 0\) which vanishes on the boundary of \(Q\), and any constant \(b \in \mathbb{R }\).
Proof of Lemma 3.2
A proof of this sort can be found in various texts, for example Chapter III of [9]. Suppose that \(u \in H^1(Q)\) solves \(-\nabla \cdot (a \nabla u) + \beta u = \nabla \cdot \xi \) in the weak sense:
By choosing a test function \(v = (u-b)\varphi ^2 \in H^1_0(Q)\), using the uniform ellipticity, and the fact that \(\varphi \ge 0\), we obtain from (3.4) the bound:
By the Cauchy–Schwarz inequality, for any \(\epsilon > 0\),
and
and
Now by choosing \(\epsilon = a_*/4\), we infer from (3.5) the bound
This completes the proof. \(\square \)
Remark 3.3
The conclusion of Lemma 3.2 also holds if we assume that \(u \in H^1_{per}(D_L)\) and \(\varphi \ge 0\) is periodic. In that case, \(\varphi (x)\) need not vanish at any point \(D_L\); the integration-by-parts is made possible by the periodicity.
Proof of Lemma 3.1
Let \(u(x)\) and \(b\) be as in Lemma 3.1. We apply Lemma 3.2 with \(Q = 3Q_j\). We may choose \(0\le \varphi (x) \le 1\) to be a smooth function with support in \(Q = 3Q_j \subset D_L\) and satisfying \(\varphi \equiv 1\) in \(Q_j\) and \(|\nabla \varphi | \le C\). In this case, observe that since \(\beta \ge 0\) and \(0 \le \varphi \le 1\),
This proves the first part of Lemma 3.1. The second part also follows by a similar argument with \(Q = B_R(x_0)\). In that case, we choose the test function \(\varphi (x)\) to be a smooth function with support in \(Q = B_R(x_0) \subset D_L\) and satisfying \(\varphi \equiv 1\) in \(B_R/2\) with \(|\nabla \varphi | \le C/R\). \(\square \)
3.2 Higher regularity—Meyers’ estimate
If \(u \in H^1_{per}(D_L)\) satisfies \(- \nabla \cdot (a \nabla u) + \beta u = \nabla \cdot v\) with \(v \in (L^2(D_L))^d\) and \(\beta \ge 0\), then it is easy to see that
must hold. If \(v \in (L^p(D_L))^d\) for some \(p > 2\), a well-known result of Meyers [17] implies higher integrability of \(\nabla u\), as described by the following lemma.
Lemma 3.4
For all \(s > 2\), there is a constant \(p^* > 2\) and \(C > 0\) such that the following holds: If \(L > 1,\,\beta \ge 0,\,v \in (L^{s}(D_L))^d\), and if \(u \in H^1_{per}(D_L)\) satisfies \(- \nabla \cdot (a \nabla u) + \beta u = \nabla \cdot v\), then

for all \(p \in [2,p^*]\).
Proof of Lemma 3.4
For \(\beta = 0\), (3.9) can be derived directly from Theorem 2 of Meyers [17] by using the periodicity of \(u\). However, for \(\beta > 0\) it is more convenient to give a proof based on a result of Giaquinta and Modica [10]. By the Caccioppoli inequality (Lemma 3.1) applied to \(u(x)\) we have
where
and \(K_R(x_0)\) is a cube of width \(2R\) centered at a point \(x_0\). By the Poincaré-Sobolev inequality, this implies

Now we apply Proposition 5.1 of [10] (see also Theorem V.1.2 of [9]) with \(g = |\nabla u|^{2d/(d+2)},\,q = (d+2)/d\), and \(f = (C \beta u^2 + C|v|^2)^{1/q}\). Since

holds for all \(x_0 \in D_L\) and \(R > 0\), that proposition implies that for some \(\epsilon > 0\) and \(p = q(1 + \epsilon )\), we must have

for any \(R > 0\) and \(x_0 \in D_L\). This means that

where \(r = 2(1 + \epsilon ) > 2\). In particular, we may choose \(R = L\) so that \(K_{R/2}(x_0) = D_L\). Since \(u\) and \(v\) are periodic over \(D_L\), we conclude that

If \(\beta > 0\), the last term in (3.13) may be absorbed into the others, as follows. Since \(r > 2\), we may use the test function \(\eta = |u|^{r-1} \text {sign}(u) \in H^1_{per}(D_L)\) in the equality
to obtain:
for any \(\epsilon > 0\). Therefore, with \(\epsilon = 2a_*\), we obtain
By Hölder’s inequality, this implies
Now we substitute this bound for the last term in (3.13), and we conclude that

holds for all \(L > 1\).
Since \(u\) satisfies \(- \nabla \cdot (a \nabla u) + \beta u = \nabla \cdot v\), we have
Therefore, since \(r > 2\), Jensen’s inequality implies

Combining this with (3.14) we conclude the proof. \(\square \)
4 The proof of Theorem 1.1
In this section we prove that the constants \(\kappa _0\) and \(\kappa _3\) appearing in Theorem 1.2 are bounded according to
for all \(L > 1\) and \(\beta \ge 0\), if \(q > 4\) is sufficiently large. By combining this with Theorem 1.3 and Theorem 1.2, we obtain Theorem 1.1.
To obtain these bounds, we will need to compute derivatives of \(\Gamma \) and \(\phi \) with respect to the variables \(\zeta _k\). First, we establish the differentiability of \(\phi (x,\zeta )\) with respect to \(\zeta _k\).
Lemma 4.1
For \(\beta \ge 0\), the function
is in \(H^1_{per}(D_L)\) and it is a weak solution of the equation
where \(\xi _k \in (L^2_{per}(D_L))^d\) is the vector field
Proof of Lemma 4.1
For \(\epsilon > 0\) small, let \(\zeta _j^{\prime } = \zeta _j\) for all \(j \ne k\) and let \(\zeta _k^{\prime } = \zeta _k - \epsilon \). Let
where \(\phi ^{\prime } = \phi (x,\zeta ^{\prime })\). The function \(\phi - \phi ^{\prime } \in H^1_{per}(D_L)\) satisfies
Using \(\phi - \phi ^{\prime }\) as a test function against (4.3), we integrate by parts and apply the Cauchy–Schwarz inequality to obtain
As \(\epsilon \rightarrow 0,\,||a(x,\zeta ) - a(x,\zeta ^{\prime }) ||^2_\infty \rightarrow 0\). So, combining (4.4) and the fact that \(\int _{D_L} \phi \,dx = \int _{D_L} \phi ^{\prime } \,dx = 0\), we conclude that \(\phi ^{\prime } \rightarrow \phi \) strongly in \(H^1_{per}(D_L)\) as \(\epsilon \rightarrow 0\).
Let \(w_k\) be the unique weak solution of (4.2) satisfying \(\int _{D_L} w_k \,dx = 0\). The function \(v^\epsilon \in H^1_{per}(D_L)\) satisfies
Therefore, \(v^\epsilon - w_k\) satisfies
Using \(v^\epsilon - w_k\) as a test function against (4.5), we obtain
Since \(\zeta \mapsto a(\cdot ,\zeta )\) is Fréchet differentiable, we know that \(||\frac{a(x,\zeta ) - a(x,\zeta ^{\prime })}{\epsilon } - \frac{\partial a}{\partial \zeta _k}(x,\zeta ) ||_{\infty } \rightarrow 0\) as \(\epsilon \rightarrow 0\). Furthermore, \(||\frac{a(x,\zeta ) - a(x,\zeta ^{\prime })}{\epsilon } ||\) is bounded as \(\epsilon \rightarrow 0\). Since \((\phi ^{\prime } - \phi ) \rightarrow 0\) in \(H^1_{per}\) as \(\epsilon \rightarrow 0\), the right side of (4.7) vanishes as \(\epsilon \rightarrow 0\). This and the Poincaré inequality implies that \(v^\epsilon \rightarrow w_k\) strongly in \(H^1_{per}\). Elliptic regularity implies that the limit holds pointwise in \(x\) and that \(w_k(x)\) is a continuous function. \(\square \)
Using the dominated convergence theorem and the fact that \(\zeta \mapsto a(\cdot ,\zeta ) \in L^\infty _{per}(D_L)\) is Fréchet differentiable, we find that
where \(w_j(x) = \frac{\partial }{\partial \zeta _j}\phi (x,\zeta ) \in H^1_{per}(D_L)\) was established in Lemma 4.1. Then, using integration by parts and (1.1), we see that the last term vanishes so that
In particular, the structural assumption (1.3) implies
Recall \(\Phi _j^{\prime }\) defined at (1.19). From this and the stationarity of \(\phi \) we see immediately that
Moments of \(\Phi _0^{\prime }\) are related to moments of \(\Phi _0\), as follows:
Lemma 4.2
For any power \(p \ge 1\), there is a constant \(C\) such that \(\mathbb E [(\Phi _0^{\prime })^p] \le C \mathbb E [(\Phi _0)^p]\) for all \(\beta \ge 0,\,L \ge 1\).
Proof
By Minkowski’s inequality and the stationarity of \(\phi \):
\(\square \)
By combining (4.11) and Lemma 4.2, we now have
which is the first estimate in (4.1).
Now we bound \(\kappa _3\). The term \(\kappa _3\) involves the Hessian \(\nabla ^2_\zeta \Gamma \), and from (4.9) we compute
where \(\partial _i \phi \) denotes the function \(\partial _i \phi = \frac{\partial }{\partial \zeta _i} \phi \). Recall that the function \(x \mapsto \frac{\partial a}{\partial \zeta _j}\) is supported in \(B_\tau (j)\). In particular, the bounds (1.3) and (1.4) imply
We will make use of the following observations:
Lemma 4.3
There is a constant \(p^* > 2\) and \(C > 0\) such that
holds for all \(L > 1,\,\beta \ge 0\) and \(p \in [2,p^*]\).
Proof of Lemma 4.3
This is a consequence of Lemma 3.4 and the stationarity of \(\phi \). Applying Lemma 3.4 to \(\phi \), with \(v(x) = a(x) e_1 \in (L^\infty (D_L))^d\) we conclude that for some \(p^* > 2\), there is \(C > 0\) such that, almost surely,

holds for all \(p \in [2,p^*],\,L > 1\), and \(\beta \ge 0\). Now, by the stationarity of \(\phi \),

\(\square \)
Lemma 4.4
Let \(\tilde{\zeta }(t) = h(\tilde{Z}(t))\) be the random vector defined in Theorem 1.2, and let \(\tilde{a}_t = a(x,\tilde{\zeta }(t))\) denote the associated conductivity. There are constants \(q > 4,\,C > 0\) such that
holds for all \(L > 1,\,\beta \ge 0,\,t \in [0,1]\).
Proof of Lemma 4.4
For each index \(i \in D_L\), let
We claim that there is a constant \(C\), independent of \(t,\,\beta \), and \(L\), such that
where \(u(x) \in H^1_{per}(D_L)\) satisfies
and the random variables \(\{s_{j}\}_{j \in D_L}\) are defined by
These variables are identically distributed and satisfy \(|s_j| \le C \Phi _j^{\prime }(\tilde{a}_t)\), by (4.9) and the fact that \(|h^{\prime }| \le c_1\). If \(\beta = 0\), we may assume \(\int _{D_L} u \,dx = 0\), so that \(u\) is uniquely defined. To see why (4.19) must be true, observe from (4.14) that
The second sum in (4.21) is bounded by
The first sum in (4.21) is exactly
where the vector field \(v \in (L^2(D_L))^d\) is
which depends on the random vectors \(Z\) and \(\tilde{Z}(t)\). Therefore,
Using Eq. (4.20) for \(u(x)\) and Eq. (4.2) for \(w_i(x) = \partial _i \phi (x)\), we have
Hence
This combined with (4.24) establishes (4.19). Observe that the vector field \(v\) is stationary with respect to integer shifts in \(x\), and it is independent of the index \(i\). Consequently, \(u\) is also statistically stationary and independent of \(i\).
To establish (4.18) we must bound \(\sum _{i} \mathbb E \left[ H_i^2\right] \), and we will use (4.19). First, observe that there is \(p > 2\) such that \(v \in (L^p(D_L))^d\) almost surely. This is a consequence of Lemma 4.3. Therefore, since \(u\) satisfies (4.20), Lemma 3.4 immediately implies the following: \(\square \)
Corollary 4.5
There is an exponent \(p^{*} > 2\) and a constant \(C\) such that, with probability one,

holds for all \(p \in [2,p^*]\) and \(L > 1\) and \(\beta \ge 0\), where \(v\) is the vector field defined by (4.23).
Now we proceed with the proof of (4.18). From (4.19) we have
where
First we bound \(S_1\). Let \(p \in (2,p^*)\) be as in Corollary 4.5, and let \(r = p/2 > 1\) and \(q = r/(r-1)\). By Hölder’s inequality and the fact that \(\tau \) is independent of \(L\),
In this last step we have used the stationarity of both \(u\) and \(\phi \). Now, as in the proof of Lemma 4.2, the stationarity of \(u\) implies
So, by Jensen’s inequality, this implies

Combining this with Corollary 4.5 \((p = 2r)\), we obtain

By definition of \(v\) and the bound \(|s_j| \le C \Phi _j^{\prime }(\tilde{a}_t)\),

where \(n > 2\) and \(m = n/(n-1)\). In the last step we have used Lemma 4.2. Observe that \(\Phi _0(\tilde{a}_t)\) has the same law as \(\Phi _0(a)\), so \(\mathbb E \left[ (\Phi _0(\tilde{a}_t))^{np} \right] ^{1/n} = \mathbb E \left[ (\Phi _0(a))^{np} \right] ^{1/n}\).Since \(p < p^*\), we may choose \(n\) large so that \(pm < p^*\). Then Jensen’s inequality and Lemma 4.3 imply that
so that

Combining the above computations applying Lemma 4.2, we conclude that
By choosing \(p > 2\) smaller, if necessary, we may assume that \(q \ge np\). Therefore, by Jensen’s inequality, \(S_1 \le C L^d \mathbb E \left[ \Phi _0(a)^q \right] ^{3/q}\).
Bounding \(S_2\) involves similar arguments. By Minkowski’s inequality and Hölder’s inequality
Now we conclude that
\(\square \)
Having proved (4.18), the bound \(\kappa _3 \le C L^{-3d/2} \mathbb E [\Phi _0^q]^{3/(2q)} + C L^{-3d/2} \mathbb E \left[ (\Phi _0)^4 \right] ^{1/2}\) now follows immediately from the definition of \(\kappa _3\) in Theorem 1.2 and the fact that the right side of (4.18) is independent of \(t \in [0,1]\). This completes the proof of Theorem 1.1.
5 Stochastic moment estimates
We close with some estimates on the moments of the random variable \(\Phi _0\) which appears in Theorem 1.1. First, we have an estimate which is Lemma 2.7 from [13]:
Lemma 5.1
Let \(d \ge 1\). Let \(n \ge 0\) be an even integer. Then
holds for any cube \(Q_j,\,j \in D_L\).
This is proved by using the test function \(v = \phi ^{n+1} \in H^1_{per}(D_L)\) in the variational equality (1.6) satisfied by \(\phi \). By (1.6) and the Cauchy–Schwarz inequality one obtains
Therefore, since \(a_* \le a(x) \le a^*\), we conclude that
Then (5.1) follows by the stationarity of \(\phi \) and \(\nabla \phi \).
Corollary 5.2
Let \(d \ge 1\) and let \(m\) be a positive integer. Then
holds for all \(\beta > 0\) and \(L > 1\).
Proof of Corollary 5.2
Observe that the final product over \(k=1,\dots ,m\) is bounded by \(2^m (m!)\). By (5.1) with \(n=0\), we have
So, (5.4) holds for \(m = 1\). Now, arguing inductively, suppose that (5.4) holds for some integer \(m \ge 1\). Then by (5.1) and the induction hypothesis
So, (5.4) also holds for \(m+1\) and by induction on \(m\) it holds for all \(m \ge 1\). \(\square \)
Proposition 5.3
Let \(d \ge 1\). For each even integer \(n \ge 0\), there is a constant \(C_n\) such that
holds for all \(L > 1\) and \(\beta \ge 0\).
Proof of Proposition 5.3
By Caccioppoli’s inequality (Lemma 3.1) we know that
holds with probability one, where \(b\) is the random constant
Therefore, with probability one, we have
Then, by (5.7) and Lemma 5.4 below, we have
Now by applying the Poincaré inequality in \(3Q_0\) to the last integral, we conclude that
Consider the term \(\beta ^{n+1} b^{2(n+1)}\). By Jensen’s inequality, the stationarity of \(\phi \), and Corollary 5.2 we know that
holds for all \(L > 1,\,\beta \ge 0\). Also, by Lemma 5.1,
So, returning to (5.8), we conclude that for a constant \(C_n\) independent of \(L > 1\) and \(\beta \ge 0\),
By the De Giorgi-Nash-Moser theory (e.g. [11], Theorem 8.24), \(\phi \) is Hölder continuous with
for some deterministic constants \(\alpha > 0\) and \(C > 0\), which depend on \(a^*\) and \(a_*\) but not on \(L\) or \(\beta \ge 0\). (Recall \(\rho _{3,0}\) defined at (3.2).) There must be a point \(x_0 \in Q_0\) such that \(|\phi (x_0)| \le ||\phi (\cdot ) ||_{L^2(Q_0)}\). Therefore, if \(|\phi |_{C^\alpha (3Q)} = h\),
Now returning to (5.10), we conclude that
The last inequality follows from the stationarity of \(\phi \).
By the triangle inequality,
where \(\rho = \int _{Q_0} \phi (x) \,dx\). Combining this with the Poincaré inequality in \(Q_0\), we obtain
Therefore, by (5.12) we have
The bound (5.6) now follows from Young’s inequality. \(\square \)
The following fact was used in the proof of Proposition 5.3:
Lemma 5.4
Let \(n \ge 2\) be an even integer. For all \(z \in \mathbb{R }\) and \(m \in \mathbb{R }\)
Proof
If \(m = 0\), the bound (5.13) obviously holds. If \(m \ne 0\), then \(m^{2(n+1)} > 0\) and we see that the bound is equivalent to
where \(\hat{z} = z/m\). Let \(f(z) = (z^{n+1} - 1)^2\) and \(g(z) = (z- 1)^{2(n+1)}\). Both of these polynomials are nonnegative for \(z \in \mathbb{R }\) and \(f(1) = g(1) = 0\). If \(z \in [-2,0]\), we observe that \(g(z) \le 3^{2(n+1)}\) and \(f(z) \ge 1\), so \(g(z) \le 3^{2(n+1)}f(z)\) holds for \(z \in [-2,0]\). For other \(z \in \mathbb{R }\), consider the factorization
where \(\omega _k = e^{i2\pi k/(n+1)}\) is a \((n+1)^{th}\) root of unity. The products \((z - w_k)(z - \bar{w}_k)\) are real and positive for \(z \in \mathbb{R },\,k = 1,\dots ,n\). For \(z \le - 2\), it is easy to see that \((z - 1)^2 \le 3^2(z - w_k)(z - \bar{w}_k)\). It follows that \(g(z) \le 3^{2n} f(z)\) for \(z \le -2\). For \(z \ge 0\), we also have \((z - 1)^2 \le (z - w_k)(z - w_k)\) for each \(k = 1,2,\dots ,n\). Hence \(g(z) \le f(z)\) for \(z \ge 0\). We have shown that \(g(z) \le 3^{2(n+1)} f(z)\) for all \(z \in \mathbb{R }\). Hence (5.14) holds. \(\square \)
From Corollary 5.2 and Proposition 5.3 we immediately obtain the following:
Corollary 5.5
Let \(d \ge 1\). For all positive odd integers \(m\), there is a constant \(C_m > 0\) such that
holds for all \(L > 1,\,\beta > 0\). Hence, \(\mathbb E [\Phi _0^m] \le C (1 + \beta ^{1 - m})\).
Observe that the bound in Proposition 5.3 is better than what is immediately implied by the Caccioppoli inequality and the stationarity of \(\phi \), since the homogeneity of the integral term on the right side of (5.6) is less than that of the term on the left side. This fact plays an important role in the method of Gloria and Otto [13] to bound moments of \(\Phi _0\), independently of \(\beta \ge 0\) and \(L > 1\). Although [13] pertains to the discrete setting on all of \(\mathbb{Z }^d\) (rather than continuum, periodic), that method can still be applied here. In view of Proposition 5.3, the moments of \(\Phi _0\) are bounded by \(\mathbb E [ \Phi _0^{n+1} ] \le C_n (1 + \mathbb E [\rho ^{2n}])\), where the random variable
has zero mean. In the discrete setting of [13], \(\phi (0)\) is analogous to this \(\rho \).
Let us briefly sketch the method of [13] to bound moments of \(\rho \). For integers \(m \ge 1\), define \(E_m = \mathbb E [\rho ^m]\) and \(V_m = \text {Var}[\rho ^m]\). Therefore,
for all \(m\). Because \(E_1 = \mathbb E [\rho ] = 0\) and \(E_{2} = V_1\), the equality (5.16) can be iterated to obtain
for any integer \(\ell > 2\), where \(C_0 = 1\) and \(C_q = 2^1 2^2 \cdots 2^{2^q}\) for \(q > 0\). Of course, this bound is very general. However, in the discrete case on \(\mathbb{Z }^d\), Gloria and Otto proved that for \(\ell \) sufficiently large,
holds for all \(q = 0,\dots ,\ell \), for some power \(r_\ell < 1\) and constant \(K_\ell \). For \(d \ge 3\), the constant \(K_\ell \) is independent of \(L>1\) and \(\beta > 0\). Applying this fact and Young’s inequality at (5.17), we conclude that \(E_{2 \cdot 2^{\ell }}\) must be bounded independently of \(L > 1\) and \(\beta \ge 0\), for \(d \ge 3\); this is the first bound in (1.11). For \(d=2\), the constant \(K_\ell \) is independent of \(L\), but it depends on \(\log \beta \). So, in the \(d = 2\) case, one obtains \(E_{2 \cdot 2^{\ell }} \le C |\log \beta |^{\gamma _\ell }\) for some \(\gamma _\ell > 0\). This is second bound in (1.11).
The bound (5.18) on the variances \(V_m\) is obtained from a spectral-gap estimate (e.g. the Efron-Stein inequality [22]). If \(F(\zeta )\) is a function of the random vector \(\zeta = (\zeta _j),\,j\in D_L\), this inequality is
where
and \(\zeta _j^{\prime }\) is an independent copy of \(\zeta _j\). By the mean value theorem, this implies that
Therefore, since
we have
where
(Recall Lemma 4.1.) Hence,
Now, suppose \(1 \le m \le n\). By applying Hölder’s inequality with \(p = (n+1)\) and \(p^{\prime } = (n+1)/n\) to each term in (5.22) we have
Consider the first term in the right side of (5.23). By Lemma 2.2, the stationarity of \(\phi \), Lemma 4.2, and then Proposition 5.3 we have
Therefore,
To estimate the sum remaining in (5.25), one needs to control the random variables \(\int _{Q_0} |\hat{w}_j(x)|^2 \,dx\), which we expect to be small if \(dist(B_\tau (j), Q_0)\) is large. In fact, it was observed in [13] that the function \(\hat{w}_j\) is related to the gradient of the Green’s function \(G(x,y)\) for the operator \(u \mapsto - \nabla \cdot a \nabla u + \beta u\) on \(D_L\). As has been pointed out in [12], in the periodic setting it is important to choose the Green’s function that respects the normalization \(\int _{D} u(x) \,dx = 0\) in order to obtain the optimal estimates on \(\nabla G\), uniformly in \(\beta \ge 0\). For each \(y \in D_L\), this function \(G(\cdot ,y)\) is periodic over \(D_L\), and for each \(r > 0,\,G(\cdot ,y) \in H^1_{loc}(D_L \setminus B_r(y))\). Also,
holds for all smooth, periodic functions \(\varphi \). That is,
In the present setting, the connection between \(\hat{w}_j(x)\) and \(G(x,y)\) is as follows:
Lemma 5.6
Let \(d \ge 1\). Suppose \(A \subset D_L\) is an open set for which \(dist(A,B_\tau (j)) > 0\). Then we have
In particular, if \(dist(Q_0,B_\tau (j)) > 0\),
When \(n\) is large, the exponent \(q = 2(n+1)/n\) is only slightly larger than 2, and Meyers’ estimate implies that \(|\nabla _x G| \in L^q_{loc}\) if \(q - 2 > 0\) is small enough (away from the singularity at \(x=y\)). As shown in [13], this fact, the Caccioppoli inequality, and uniform decay estimates on \(G(x,y)\) can be used to obtain optimal bounds on the decay of \(|\nabla G|^q\) away from the singularity. This leads to the optimal estimate of (5.25). Extension of the Green’s function estimates of [13] and of the moment estimates on \(\Phi _0\) to the periodic setting is being carried out in [12].
Proof of Lemma 5.6
Let \(v \in H^1_{per}(D_L)\) satisfy
By applying Lemma 4.1 to \(\partial _j \phi = (\Phi _j^{\prime })^{1/2} \hat{w}_j\) and using \(\int _{D_L} \partial _j \phi (x) \,dx = 0\), we have
since \(\xi _j\) is supported in \(B_\tau (j)\). On the other hand,
hold for almost every \(x\) outside \(A\). Therefore, by Cauchy–Schwarz we have
for almost every \(x\) in \(B_\tau (j)\). Also, \(\int _{B_\tau (j)} |\xi _j|^2 \,dx \le C_2^2 \Phi _j^{\prime }\), by (1.3). Combining this with (5.27) we obtain (5.26). \(\square \)
Proof of Proposition 1.4
This also follows from the inequality (5.19). Specifically, using (5.19) and (4.10), we obtain
By Lemma 2.2, \(\mathbb E [ \sup _{\zeta _j} (\Phi _j^{\prime })^2] \le C \mathbb E [ (\Phi _j^{\prime })^2]\). By stationarity of \(\phi \) and Lemma 4.2, \(\mathbb E [ (\Phi _j^{\prime })^2] = \mathbb E [ (\Phi _0^{\prime })^2] \le C \mathbb E [ (\Phi _0)^2]\). Hence
\(\square \)
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Acknowledgments
I am grateful to Felix Otto whose insight and helpful comments led to improvement of the main argument. I also thank Jan Wehr, Sourav Chatterjee, and Jonathan Mattingly for stimulating discussion in the early stages of this work. The anonymous referees also provided very helpful comments. The author’s research is partially funded by grant DMS-1007572 from the US National Science Foundation.
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Nolen, J. Normal approximation for a random elliptic equation. Probab. Theory Relat. Fields 159, 661–700 (2014). https://doi.org/10.1007/s00440-013-0517-9
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DOI: https://doi.org/10.1007/s00440-013-0517-9
Mathematics Subject Classification
- 35B27
- 35J15
- 60F05
- 60H25