Strong Convergence to the homogenized limit of elliptic equations with random coefficients

Consider a discrete uniformly elliptic divergence form equation on the $d$ dimensional lattice $\Z^d$ with random coefficients. It has previously been shown that if the random environment is translational invariant, then the averaged Green's function together with its first and second differences, are bounded by the corresponding quantities for the constant coefficient discrete elliptic equation. It has also been shown that if the random environment is ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized elliptic PDE on $\R^d$. In this paper point-wise estimates are obtained on the difference between the averaged Green's function and the homogenized Green's function for certain random environments which are strongly mixing.


Introduction.
Let (Ω, F , P ) be a probability space and denote by · expectation w.r. to the measure P . We assume that the d dimensional integer lattice Z d acts on Ω by translation operators τ x : Ω → Ω, x ∈ Z d , which are measure preserving and satisfy the properties τ x τ y = τ x+y , τ 0 = identity, x, y ∈ Z d . Consider a bounded measurable function a : Ω → R d(d+1)/2 from Ω to the space of symmetric d × d matrices which satisfies the quadratic form inequality where I d is the identity matrix in d dimensions and Λ, λ are positive constants. We shall be interested in solutions u(x, η, ω) to the discrete elliptic equation (1.2) ηu(x, η, ω) + ∇ * a(τ x ω)∇u(x, η, ω) = h(x), x ∈ Z d , ω ∈ Ω.
In (1.2) we take η ≥ 0 and ∇ the discrete gradient operator, which has adjoint ∇ * . Thus ∇ is a d dimensional column operator and ∇ * a d dimensional row operator, which act on functions φ : Z d → R by . In (1.3) the vector e i ∈ Z d has 1 as the ith coordinate and 0 for the other coordinates, 1 ≤ i ≤ d.
It is well known [13,18,23] that if the translation operators τ x , x ∈ Z d , are ergodic on Ω then solutions to the random equation (1.2) converge to solutions of a constant coefficient equation under suitable scaling. Thus suppose f : R d → R is a C ∞ function with compact support and for ε satisfying 0 < ε ≤ 1 set h(x) = ε 2 f (εx), x ∈ Z d , in (1.2). Then u(x/ε, ε 2 η, ω) converges with probability 1 as ε → 0 to a function u(x, η), x ∈ R d , which is the solution to the constant coefficient elliptic PDE (1.4) ηu(x, η) + ∇ * a hom ∇u(x, where the d× d symmetric matrix a hom satisfies the quadratic form inequality (1.1). This homogenization result can be viewed as a kind of central limit theorem, and our purpose here will be to show that the theorem can be strengthened for certain probability spaces (Ω, F , P ). We consider what the homogenization result says about the expectation of the Green's function for equation (1.2). By translation invariance of the measure we have that (1.5) u(x, η, where G a,η (x) is the expected value of the Green's function. Setting h(x) = ε 2 f (εx), x ∈ Z d , then (1.5) may be written as where integration over εZ d is defined by Let G a hom ,η (x), x ∈ R d , be the Greens function for the PDE (1.4). One easily sees that G a hom ,η (·) satisfies the scaling property (1.8) ε 2−d G a hom ,ε 2 η (x/ε) = G a hom ,η (x), ε, η > 0, x ∈ R d − {0}.
From (1.6), (1.8) we see that homogenization implies that the function ε 2−d G a,ε 2 η (x/ε), x ∈ εZ d , converges in an averaged sense to the Greens function G a hom ,η (x), x ∈ R d . A consequence of our results here will be that for certain probability spaces (Ω, F , P ) and functions a : Ω → R d(d+1)/2 this convergence is point-wise in x. In particular for some α satisfying 0 < α ≤ 1, there are positive constants C, γ such that (1.9) We shall also denote by G a hom ,η (x), x ∈ Z d , the Greens function for the difference equation (1.4) on Z d . Evidently the Z d Green's function has the property that G a hom ,η (0) is finite, unlike the corresponding R d Green's function. It is also clear that the inequality (1.9) for ε < 1 follows from the same inequality for ε = 1 i.e.
The limit as m → 0 of the measure (1.13) is a probability measure on gradient fields ω : Z d → R d , where formally ω(x) = ∇(φ(x)), x ∈ Z d . This massless field theory measure is ergodic with respect to translation operators [3,8] for all d ≥ 1. In the case d = 1 it has a simple structure since then the variables ω(x), x ∈ Z, are i.i.d. Note that in the probability space (Ω, F , P ) for the massless field theory, the Borel algebra F is generated by the intersection of finite dimensional rectangles and the hyperplanes imposing the gradient constraints for ω(·). For d ≥ 3 the gradient field theory measure induces a measure on fields φ : Z d → R which is simply the limit of the measures (1.13) as m → 0. For d = 1, 2 the m → 0 limit of the measures (1.13) on fields φ : Z d → R does not exist.
Our method of proof for Theorems 1.1-1.3 combine methods used to prove regularity of averaged Green's functions for pde with random coefficients with methods for obtaining rates of convergence in homogenization. Regularity of averaged Green's functions was first proved in [5]. The results of that paper imply that for any probability space (Ω, F , P ) with translation invariant operators τ x : Ω → Ω, x ∈ Z d , the inequalities (1.10), (1.11) hold for α = 0 and (1.12) for any α < 0. The approach of the paper is to obtain good control on the Fourier transformĜ a,η (ξ), ξ ∈ [−π, π] d , of G a,η (x), x ∈ Z d , for |ξ| close to 0. Using the Fourier inversion formula, one then obtains the inequalities (1.10)-(1.12). In [6] the inequality (1.12) is proven with α = 0, and in fact Hölder continuity of the second difference of G a,η (x), x ∈ Z d , is also established. In contrast to [5], the approach of [6] is local in configuration space, and uses results from harmonic analysis which are deeper than those used in [5]. In particular, the Harnack inequality [11] for uniformly elliptic equations in divergence form is needed to prove (1.12) with α = 0, whereas the proof of (1.12) with α < 0 in [5] follows from interpolation inequalities.
We have already observed that the inequality (1.10) with α > 0 implies (1.9), which gives a rate of convergence of ε α in homogenization. The first results establishing a rate of convergence for homogenization of elliptic PDE in divergence form were obtained in [22]. These results require much stronger assumptions on the translation operators τ x : Ω → Ω, x ∈ Z d , than ergodicity. One needs to assume that the variables a(τ x ·), x ∈ Z d , are independent, or at least are very weakly correlated. Rates of convergence in homogenization when (Ω, F , P ) is either the massive field theory of Theorem 1.2 or the massless field theory of Theorem 1.3 were obtained in [16]. The main tools used to prove these results are the Brascamp-Lieb (BL) inequality [1] and Meyer's theorem [14]. Meyer's theorem is a consequence of the continuity in p of the norms of Calderon-Zygmund operators acting on the spaces L p (Z d ) of functions whose pth powers are summable. In [4] it was shown that Meyer's theorem could also be used to obtain a rate of convergence in the independent variable case of Theorem 1.1. Recently [9,10] the independent variable case was taken up again, showing that the method of [16], which uses a BL inequality plus Meyer's theorem, could also be directly implemented in this case. Because of the perturbative nature of Meyer's theorem, this method alone does not yield optimal rates of convergence to homogenization. However by combining the method with some deterministic estimates on Green's functions, optimal rates of convergence to homogenization are obtained in [9,10] .
In the present paper we follow the methodology of [5] to obtain estimates on G a,η (ξ), ξ ∈ [−π, π] d , which imply (1.10), (1.11) with α = 0. The estimates on G a,η (ξ) are improved by the use of Meyer's theorem for the independent variable environment, and by the BL inequality plus Meyer's theorem in the field theory case. The inequalities (1.10), (1.11) for some α > 0 follow then upon using the Fourier inversion formula. To prove (1.12) we also have to use the results of [6] to estimate the contribution of high Fourier modes. These estimates are a consequence of the Hölder continuity of the second difference of G a,η (x), x ∈ Z d , already mentioned.

Fourier Space Representation and Estimates
In this section we summarize relevant results of previous work [4,5] which were used to prove pointwise bounds on the Green's function G a,η (x), x ∈ Z d , defined by (1.5). The starting point for this is the Fourier representation where the d × d matrix function q(ξ, η), ξ ∈ R d , η > 0, is a complex Hermitian positive definite function of (ξ, η), periodic in ξ with fundamental region [−π, π] d , which satisfies the quadratic form inequality The d dimensional column vector e(ξ) in (2.1) has jth entry e j (ξ) = e −iej ·ξ − 1, 1 ≤ j ≤ d.
It will be useful later to express the operator T ξ,η in its Fourier representation. To do this we use the standard notation for the Fourier transformĥ(ζ), ζ ∈ [−π, π] d , of a function h : Z d → C. Thus and the Fourier inversion formula yields (2.11) h Now the action of the translation group τ x , x ∈ Z d , on Ω can be described by a set A 1 , ..., A d of commuting self-adjoint operators on L 2 (Ω), so that where A = (A 1 , .., A d ). It follows then from (2.7) and (2.12) that It is easy to see that the function q(ξ, η) is C ∞ for ξ ∈ R d , η > 0. In [4,5] it was further shown that if the translation operators τ x , x ∈ Z d , are ergodic on (Ω, F , P ) then lim (ξ,η)→(0,0) q(ξ, η) = q(0, 0) exists. We can extend this result as follows: Proposition 2.1. Suppose that the operator τ ej is weak mixing on Ω for some j, 1 ≤ j ≤ d. Then q(ξ, η), ξ ∈ R d , η > 0, extends to a continuous function on ξ ∈ R d , η ≥ 0.
Proof. We first define an operator T ξ,η on H(Ω) for ξ ∈ R d and η = 0. To do this first observe from (2.6) that if the function g(·) ∈ H(Ω) satisfies ∂ * ξ g(·) = 0, we should set T ξ,0 g(·) = 0. Alternatively if g(·) = ∂ ξ h(·) for some h ∈ L 2 (Ω), then from (2.7) we should set T ξ,0 g(·) to be given by the formula, In view of the inequality for a constant C depending only on d, we see that the right hand side of (2.14) is in H(Ω). Since the orthogonal complement in H(Ω) of the null space of the operator ∂ * ξ is the closure of the linear space E ξ (Ω) = {∂ ξ h(·) : h ∈ L 2 (Ω)}, we have defined T ξ,0 g(·) for a dense set of functions g ∈ H(Ω) and Using the fact that T ξ,η ≤ 1 for all η > 0, one sees that the operator T ξ,0 , defined above on a dense linear subspace of H(Ω), extends to a bounded operator on H(Ω) with norm T ξ,0 ≤ 1, and the limit (2.16) holds for all g ∈ H(Ω).
Remark 1. Note that the projection operator P in the formula (2.17) plays a critical role in establishing continuity. For a constant function g(·) ≡ v ∈ C d , one has which does not extend to a continuous function of (ξ, η) on the set ξ ∈ R d , η ≥ 0.
Next we show that the function q(ξ, η) with domain ξ ∈ R d , η > 0, can be extended to complex ξ = ℜξ + iℑξ ∈ C d with small imaginary part.
Proof. The fact that there is an analytic continuation to the region {ξ ∈ C d : |ℑξ| < C 1 η/Λ} is a consequence of the bound on the function G η (·) of (2.8), where the constants C 3 , C 4 depend only on d. The bound on T ξ,η can be obtained from (2.6). Thus on multiplying (2.6) byψ(ξ, η, ω), taking the expectation and using the Schwarz inequality, we see that where the constants C 5 , C 6 depend only on d. Evidently (2.24) yields the bound on T ξ,η on taking C 1 sufficiently small, depending only on d.
There is a constant C 2 depending only on d such that for ξ in this region, Proof. The fact that q(ξ, η) has an analytic continuation follows from the representations (2.17), (2.18), Lemma 2.1 and the matrix norm bound b(ω) ≤ 1 − λ/Λ, ω ∈ Ω. On summing the perturbation series (2.18), we conclude that for ξ satisfying |ℑξ| < C 1 λη/Λ 2 , then q(ξ, η) ≤ C 2 Λ 2 /λ for a constant C 2 depending only on d, provided C 1 is chosen sufficiently small, depending only on d. By arguing as in Lemma 2.1 we also see that there are positive constants C 1 , C 2 such that The inequality (2.25) follows from (2.26).
We have seen in Corollary 2.1 that the periodic matrix function q(ξ + ia, η), ξ ∈ [−π, π] d , is bounded provided a ∈ R d satisfies |a| ≤ C 1 λη/Λ 2 . It was shown in [5] that derivatives of this function are in certain weak L p spaces.
The weak L p norm of f (·), f p,w is the minimum constant C such that (2.27) holds. From [5] we have the following: Then there exists a positive constant C 1 depending only on d such that if |a| ≤ C 1 λη/Λ 2 and |m| < d, the function is in the space L p w ([−π, π] d ) with p = d/|m| and its norm is bounded by CΛ, where the constant C depends only on d and Λ/λ ≥ 1.

Configuration space estimates from Fourier space estimates
In this section we shall show how to obtain configuration space estimates on G a,η (x), x ∈ Z d , from Fourier space estimates on the function q(ξ, η), ξ ∈ R d . In [5] it was shown that for d ≥ 3, Proposition 2.2 implies the inequality where the positive constants C, γ depend only on d and Λ/λ. It was also shown that for d ≥ 2, Proposition 2.2 implies a similar inequality for the gradient of G a,η (x), Finally for d ≥ 1, Proposition 2.2 implies Hölder continuity of ∇G a,η (x), for any δ, 0 < δ ≤ 1, where the constant C δ depends now on δ > 0 as well as on d and Λ/λ. In subsequent sections we shall establish a strengthened version of Proposition 2.2 for certain environments (Ω, F , P ) as follows: Theorem 3.1. There exist positive constants C 1 , C 2 and α ≤ 1 depending only on d and Λ/λ, such that With the same assumptions as in Proposition 2.2, the derivative (2.28) is in the and its norm is bounded by CΛ, where the constant C depends only on d and Λ/λ. The difference (2.29) is in the space and its norm is bounded by CΛ, where now α and C depend on δ as well as d and Λ/λ.
Note that the function G a hom ,η (·) of (3.5) is not the same as the Green's function for the PDE (1.4) restricted to Z d . One can however easily estimate the difference on Z d between these two functions.
Proof. LetG a hom ,η (x) be the function dξ , so (2.1) and Corollary 2.1 imply that where the function f (ξ) is given by the formula (3.11) It follows from (3.4) and Corollary 2.1 that there are positive constants C 1 , C 2 depending only on d and Λ/λ such that the function in (3.11) is bounded by Choosing a appropriately and α < 1 one sees already from (3.10), (3.12) Let ρ ∈ R d be the vector of minimum norm which satisfies e −iρ.x = −1. Then we have that (3.14) The integral on the RHS of (3.14) can be estimated by separately estimating the integral over the region |ξ| < C/ for a constant C 1 depending only on d and Λ/λ, provided C is sufficiently large. Hence the integral over |ξ| > C/[|x| + 1] is bounded by a constant times [|x| + 1] −α log[|x| + 2]. We have shown that |∇G a,η (x) − ∇G a hom ,η (x)| is bounded by the RHS of (3.7) for any α less than the Hölder constant in (3.4). We can similarly bound |∇G a hom ,η (x) − ∇G a hom ,η (x)| by the RHS of (3.7) on using the estimate q(0, η) − q(0, 0) ≤ CΛ(η/Λ) α/2 from (3.4). Hence (3.7) holds for d = 1, and by similar argument (3.8).
In order to prove (3.6) for d ≥ 2 we need to use the bounds on the derivatives of the function q(·, ·) given in Proposition 2.2. For d = 2 the first derivative estimate is sufficient. Thus we write where n(ξ) is the unit inward normal vector at ξ on the sphere {|ξ| = C/[|x| + 1]}. It follows from (3.12) that the first two integrals on the RHS of (3.16) are bounded by C 2 /Λ[|x| + 1] d−2+α for some constant C 2 depending only on d and Λ/λ. The third integral can be similarly bounded for d = 2 by using the fact from Theorem 3.1 that the function ∇ ξ q(ξ + ia, η) is in L p w ([−π, π] 2 ) with p = 2/(1 − α). To see this we note that for any measure space (X, B, µ) and 1 < p < ∞ one has that if g ∈ L p w (X) then where C p depends only on p. It follows that for n = 1, 2, .., where C 1 depends only on d = 2 and Λ/λ. Hence on summing over n ≥ 1 in (3.18) we see that for d = 2 the function |G a,η (x) −G a hom ,η (x)| is bounded by the RHS of (3.6). We can bound |G a hom ,η (x) − G a hom ,η (x)| using the Hölder continuity (3.4) of q(0, η), 0 ≤ η ≤ Λ, by the RHS of (3.6) for any α smaller than the Hölder constant in (3.4). We have therefore proven (3.6) for d = 2.
To prove (3.7) for d = 2 we need to use the fact from Theorem 3.1 that the difference [∇ ξ q(ξ+ρ+ia, η)−∇ ξ q(ξ+ia, η)]/|ρ| 1−δ is in L p w ([−π, π] 2 ) with p = 2/(2− δ − α) and norm bounded independent of |ρ| ≤ 1. Thus in bounding ∇G a,η (x) − ∇G a hom ,η (x) we write the integral over ξ as in (3.16). The first two terms on the RHS can be bounded in a straightforward way. To bound the third term we need to estimate the integral In (3.19) the vector ρ ∈ R d is as in (3.14) and C ′ is a universal constant. The error term in (3.19) is bounded by which can be appropriately estimated by using the fact that . Similarly to (3.18) we have for n = 1, 2, .., the bound (3.21) where C δ depends only on d = 2 and Λ/λ. Choosing now α < 1 − δ in (3.21), we conclude that |∇G a,η (x) − ∇G a hom ,η (x)| is bounded by the RHS of (3.7). As in the previous paragraph, we can bound |∇G a hom ,η (x) − ∇G a hom ,η (x)| using the Hölder continuity (3.4) of q(0, η), 0 ≤ η ≤ Λ, by the RHS of (3.7) for any α smaller than the Hölder constant in (3.4). We have therefore proven (3.7) for d = 2. We proceed similarly for the proof of (3.8) in the case d = 2. Thus we write where ρ ∈ R d is the vector of minimum norm such that e −iρ.x ′ = −1. Arguing as in (3.20) we see that where C 1 depends only on Λ/λ. From (3.21) we see that the first term on the RHS of (3.22) is bounded by the sum for any δ ′ satisfying 0 < δ ′ ≤ 1. Provided α < δ we may choose δ ′ > 0 so that the sum in (3.24) converges, whence the sum is bounded by C δ |x − x ′ | 1−δ /Λ[|x| + 1] α−δ for a constant C δ depending only on δ > 0 as well as Λ/λ. The second term on the RHS of (3.22) is [1 − e iρ.(x ′ −x) ] times an integral similar to the integral on the LHS of (3.19). Hence the second term is bounded in absolute value by C 1 |x − x ′ |/Λ[|x| + 1] α , where C 1 depends only on Λ/λ. We have therefore shown that |∇G a,η (x) − ∇G a hom ,η (x)| is bounded by the RHS of (3.8). Since we can argue as previously to bound |∇G a hom ,η (x) − ∇G a hom ,η (x)|, the proof of (3.8) is complete.
To prove the result for d ≥ 3 we use multiple integration by parts and the integrability properties of the higher derivatives of q(ξ + ia, η), ξ ∈ [−π, π] d , given in Theorem 3.1.

Independent Variable Environment
Our goal in this section will be to prove Theorem 3.1 in the case when the variables a(τ x ·), x ∈ Z d , are independent. Following [4] we first consider the case of a Bernoulli environment. Thus for each n ∈ Z d let Y n be independent Bernoulli variables, whence Y n = ±1 with equal probability. The probability space (Ω, F , P ) is then the space generated by all the variables Y n , n ∈ Z d . A point ω ∈ Ω is a set of configurations {(Y n , n) : n ∈ Z d }. For y ∈ Z d the translation operator τ y acts on Ω by taking the point ω = {(Y n , n) : n ∈ Z d } to τ y ω = {(Y n+y , n) : n ∈ Z d }. The random matrix a(·) is then defined by where 0 ≤ γ < 1. In [4] we defined for 1 ≤ p < ∞ Fock spaces F p (Z d ) of complex valued functions, and observed that F 2 (Z d ) is unitarily equivalent to L 2 (Ω). We can similarly define Fock spaces H p F (Z d ) of vector valued functions with domain C d , such that H 2 F (Z d ) is unitarily equivalent to H(Ω). Hence we can regard the operator T ξ,η of (2.7) as acting on H 2 F (Z d ), and by unitary equivalence it is a bounded operator satisfying T ξ,η ≤ 1 for ξ ∈ R d , η > 0. We may apply now the Calderon-Zygmund theorem [20] to conclude the following: Lemma 4.1. For ξ ∈ R d , 0 < η ≤ 1, and 1 < p < ∞, the operator T ξ,η is a bounded operator on H p F (Z d ) with norm T ξ,η p satisfying an inequality T ξ,η p ≤ 1 + δ(p), where lim p→2 δ(p) = 0.
Proof. We use the representation (2.17), (2.18) for q(ξ, η). From (2.17) we have that From (2.7) and the weak Young inequality we see that for 0 < α ≤ 1, the operator for a constant C depending only on d if d ≥ 3. In the case d ≤ 2 we need to take α < d/2, in which case C depends also on α. For the inequality (4.3) to hold it is necessary to include the projection P (see remark following Proposition 2.1). It follows now from Lemma 4.1 and (4.2), (4.3) that where p = 2d/(d + 2α). Note here we are using the fact that (4.1) implies that a column vector of b(·) is in H p F (Z d ) with norm less than 2γ/(1 + γ). The uniform Hölder continuity of the family of functions q(·, η), 0 < η ≤ Λ, follows from (4.4) and Lemma 4.1 by taking p sufficiently close to 2 so that (1 − λ/Λ)[1 + δ(p)] < 1.
The uniform Hölder continuity of the family of functions q(ξ, ·), ξ ∈ [−π, π] d , can be obtained in a similar way by observing that where C and p are as in (4.3).
It follows now from (4.9), (4.10) and the Riesz convexity theorem [21] that We can use (4.11) to obtain an improvement on Proposition 2.2 in the case |m| = 1. Proof. Observe from (2.13) and (2.17) that for certain d × d matrix valued functions g j,r (x), h j,r (x), x ∈ Z d . The functions g j,r (·), h j,r (·) are determined from their Fourier transforms (2.10) by the formula which follows from (2.13). Evidently one can choose the g j,r (·), h j,r (·) satisfying (4.13) so that they also satisfy the inequality (4.14) for positive constants C, γ depending only on d ≥ 1. Hence we may estimate the RHS of (4.12) by using (4.11) for any p > d/(d − 1/2). Since we require p ≤ 2 in (4.11) it is only possible to do this when d ≥ 2. The result follows.
Lemma 4.4. Suppose (Ω, F , P ) is the independent random variable environment corresponding to (4.1). Then there exists p 0 (Λ/λ) with 1 < p 0 (Λ/λ) ≤ 2 depending only on Λ/λ and d such that if 2 ≤ q ≤ ∞ and Proof. We have already proved the lemma for k = 1 so we consider the case k = 2.
Observe that (4.19) holds if p 2 = 1 in a similar way to the proof of (4.11). Evidently Lemma 4.3 implies that (4.19) also holds if q = 2. Hence by an application of the Riesz convexity theorem we conclude that the result holds for the case k = 2. To prove the result for k = 3 we proceed similarly, using the fact that we have proved it for k = 2, and Lemma 4.3 with k = 3 and q = 2.
Proof of Theorem 3.1. We argue just as in [5] to show that for a = 0, Lemma 4.4 implies the derivative (2.28) is in the space L p ([−π, π] d ) with p = d/(|m| − α), and hence in L p w ([−π, π] d ). A similar argument holds for the difference (2.29). Since one can easily show that the proofs of Proposition 4.1 and Lemma 4.4 continue to hold for |a| ≤ C 1 λη/Λ 2 , we have proven Theorem 3.1 for the environment corresponding to (4.1). It is shown in [4] how to extend the argument for the Bernoulli environment corresponding to (4.1) to general i.i.d. environments a(τ x ·), x ∈ Z d . We have therefore proven Theorem 3.1 for a(τ x ·), x ∈ Z d , i.i.d. such that (1.1) holds.

Massive Field Theory Environment
In this section we shall show that Theorem 3.1 holds if (Ω, F , P ) is given by the massive field theory environment determined by (1.13). The main tool we use to prove the theorem is the Brascamp-Lieb (BL) inequality [1]. This is perhaps natural to expect since the BL inequality is needed to prove that the operators τ ej , 1 ≤ j ≤ d, on Ω are strong mixing, which by Proposition 2.1 implies the continuity of the function q(ξ, η) in the region ξ ∈ R d , η ≥ 0.
We recall the main features of the construction of the measure (1.13). Let L be a positive even integer and Q = Q L ⊂ Z d be the integer lattice points in the cube centered at the origin with side of length L. By a periodic function φ : Q → R we mean a function φ on Q with the property that φ(x) = φ(y) for all x, y ∈ Q such that x − y = Le k for some k, 1 ≤ k ≤ d. Let Ω Q be the space of all periodic functions φ : Q → R, whence Ω Q with Q = Q L can be identified with R N where N = L d . Let F Q be the Borel algebra for Ω Q which is generated by the open sets of R N . For m > 0, we define a probability measure P Q on (Ω Q , F Q ) as follows: for some constants C, A. Differentiating the probability density in (5.1) we see that for any f ∈ Ω Q (5.2) ∇f (·), V ′ (∇φ(·)) + m 2 f (·), φ(·) ΩQ = 0, where (·, ·) denotes the Euclidean inner product on L 2 (Q). Hence by translation invariance of the measure (5.1) we conclude that (f, φ) ΩQ = 0 for all f (·). The BL inequality [1] applied to (5.1) and function F (φ(·)) = exp[(f, φ)] then yields the inequality The probability space (Ω, F , P ) on fields φ : Z d → R is obtained as the limit of the spaces (Ω Q , F Q , P Q ) as |Q| → ∞. In particular one has from Lemma 2.2 of [3] the following result: Proposition 5.1. Assume m > 0 and let F : R k → R be a C 1 function which satisfies the inequality for some constants A, B. Then for any x 1 , ....x k ∈ Z d , the limit exists and is finite.
From (5.3) and the Helly-Bray theorem [2,7] one sees that Proposition 5.1 implies the existence of a unique Borel probability measure on R m corresponding to the probability distribution of the variables (φ(x 1 ), .., φ(x m )) ∈ R m , and this measure satisfies (5.5). The Kolmogorov construction [2,7] then implies the existence of a Borel measure on fields φ : Z d → R with finite dimensional distribution functions satisfying (5.5). This is the measure (1.13), which we have formally written as having a density with respect to Lebesgue measure. Note however that we do not have a proof of this fact. In particular, we do not know if the distribution measure for the one dimensional variable φ(x) ∈ R is absolutely continuous with respect to Lebesgue measure.
Proposition 5.2. Let (Ω, F , P ) be the probability space corresponding to the massive field theory with measure (1.13). Then the operators τ ej , 1 ≤ j ≤ d, on Ω are strong mixing.
Proof. It will be sufficient for us to show [19] that for any m ≥ 1 and x 1 , .., x m ∈ Z d , for all C ∞ functions f, g : R m → R with compact support. We shall just consider the case m = 1 since the general case follows from this in a straightforward manner. We define the function h : Z → R by Then Proposition 5.1 implies that for any function k : Z → R of finite support, where h Q (·) is given by We assume that Q = Q L with L large enough so that the support of k(·) is contained in the interval [−L/2 + 1, L/2 − 1]. Hence both k(·) and h Q (·) are periodic functions on I L = Z ∩ [−L/2, L/2]. We may therefore write the sum on the LHS of (5.8) in its Fourier representation. Thus the Fourier transform of a periodic function F : I L → C is the periodic functionF :Î L → C defined by whereÎ L is the set of lattice points of (2π/L)Z which lie in the interval [−π, π]. Then with integration onÎ L defined by We can estimateĥ Q (ζ) by using translation invariance of the measure (5.1) and the BL inequality. Thus translation invariance implies that where a(f, ζ, φ(·)) is given by the formula The BL inequality implies then that Hence there is a constant C independent of Q such that |ĥ Q (ζ)| ≤ C, ζ ∈Î L . Applying then the Schwarz inequality in (5.11) and using (5.8) we conclude that It follows that h(·) ∈ L 2 (Z), and consequently lim n→∞ h(n) = 0.
We shall show how the BL inequality can be used to improve the most elementary of the inequalities contained in §2. Thus let us consider an equation which differs from (2.4) only in that the projection operator P has been omitted, ∈ Ω. For any v ∈ C d we multiply the row vector (5.17) on the right by the column vector v and by the function Φ(ξ, η, ω)v on the left. Taking the expectation we see that where · denotes the norm in H(Ω). Let g : with norm given by (4.6). If p = 1 then (5.18) implies that The BL inequality enables us to improve (5.19) to allow g ∈ L p (Z d , C d ⊗ C d ) for some p > 1. Then there exists p 0 (Λ/λ) depending only on d and Λ/λ and satisfying 1 < p 0 (Λ/λ) < 2, such that for g ∈ L p (Z d , C d ⊗ C d ) with 1 ≤ p ≤ p 0 (Λ/λ) and v ∈ C d , where Λ 1 is the constant in Theorem 1.2 and C depends only on d and Λ/λ.
Consider now the Hilbert space H(Z d × Ω) of functions g : Z d × Ω → C d with norm g 2 given by where g(y, φ(·)) is the norm of g(y, φ(·)) ∈ H(Ω). Evidently the function h of (5.38) is in H(Z d × Ω) and Since T ξ,η ≤ 1 and B(·, φ(·)) ≤ 1 − λ/Λ, we conclude from (5.40) on summing the Neumann series for (5.37) that (5.32) holds for Q → Z d . We may define for any q ≥ 1 the Banach space L q (Z d × Ω, C d ) of functions g : Z d × Ω → C d with norm g q given by (5.41) g q q = y∈Z d g(y, φ(·)) q .
As in Lemma 4.1 the operator T ξ,η is bounded on L q (Z d × Ω, C d ) for q > 1 with norm T ξ,η q ≤ 1 + δ(q), where lim q→2 δ(q) = 0. Noting that h q is bounded by the RHS of (5.40) for all q ≥ 1, we conclude then from (5.37) that there exists q 0 (Λ/λ) < 2 depending only on d and Λ/λ such that where the constant C 1 depends only on d and Λ/λ. The result follows from (5.34), (5.42) and Young's inequality.
We proceed now to establish Theorem 3.1 for the massive field theory environment (Ω, F , P ) along the same lines followed in §4 for the i.i.d. environment.
Then from (5.46) we see that the G r (y, φ(·)) satisfy the equations . From (5.45) and BL we have that Just as in (5.44) we see that (5.50) for any p satisfying 1 ≤ p ≤ 2. The second term in the last expression in (5.49) can be bounded using an inequality similar to (5.42). It is clear from (5.48) that Applying the Calderon-Zygmund theorem [20] to (5.48) we see that there exists q 0 (Λ/λ) < 2 depending only on d and Λ/λ such that provided q 0 (Λ/λ) ≤ q ≤ 2. The result follows from (5.49), (5.50), (5.52) and Young's inequality.
Proof. We proceed as in the proof of Proposition 4.1. Instead of (4.3) we use the fact that Evidently for 0 ≤ α ≤ 1 one has that g ∈ L p (Z d , C d ⊗ C d ) for any p > d/(d − α) and g p ≤ C p |ξ ′ − ξ| α for a constant C p depending only on p and d. The Hölder continuity of q(ξ, η) in ξ follows then from Lemma 5.1. The Hölder continuity of q(ξ, η) in η can be obtained in a similar way.
To complete the proof of Theorem 3.1 for the massive field theory environment we need to prove a version of Lemma 4.4 and also that one can do analytic continuation in the variable ξ ∈ R d . The proof of this follows along the same lines as in §4.

Massless Field Theory Environment
In this section we shall prove Theorem 3.1 for the massless field theory environment (Ω, F , P ) with measure given by the m → 0 limit of the massive field theory measure (1.13). The measure is constructed by means of the following result proved in [3]: Proposition 6.1. Let F : R kd → R be a C 1 function which satisfies the inequality for some constants A, B, and · m denote the massive field theory expectation with measure (1.13). Then for any x 1 , ....x k ∈ Z d , the limit exists and is finite.
As for the massive case, Proposition 6.1 defines a unique Borel probability measure on gradient fields ω : Z d → R d by using the inequality derived from (5.3), for any function f : Z d → R d of finite support. This can most easily be seen by using a simple identity. For a function G(ω(·)) of vector fields ω : Z d → R d we define its gradient d ω G(·, ω(·)) similarly to (5.22) by Thus d ω G(z, ω(·)), z ∈ Z d , is for fixed ω(·) a vector field from Z d to R d , and hence we may compute its divergence ∇ * d ω G(z, ω(·)), z ∈ Z d . Then with d defined as in (5.22) we have the identity (6.5) dG(z, ∇φ(·)) = ∇ * d ω G(z, ω(·)), z ∈ Z d .
Proof. We proceed as in the proof of Proposition 5.3. Thus from (6.5) and BL we see that (6.10) The remainder of the proof is exactly as in Proposition 5.3.
Proof of Theorem 3.1. This follows the same lines as the proof of Theorem 3.1 in §5.