Strong Convergence to the Homogenized Limit of Parabolic Equations with Random Coefficients

This paper is concerned with the study of solutions to discrete parabolic equations in divergence form with random coefficients, and their convergence to solutions of a homogenized equation. It has previously been shown that if the random environment is translational invariant and ergodic, then solutions of the random equation converge under diffusive scaling to solutions of a homogenized parabolic PDE. In this paper point-wise estimates are obtained on the difference between the averaged solution to the random equation and the solution to the homogenized equation 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 space translation operators τ x,0 : Ω → Ω, x ∈ Z d , which are measure preserving and satisfy the properties τ x,0 τ y,0 = τ x+y,0 , τ 0,0 = identity, x, y ∈ Z d .We assume also that either the integers Z or the real line R acts on Ω by time translation operators τ 0,t : Ω → Ω, where t ∈ Z in the former case and t ∈ R in the latter.In either case we assume that for all t, s, one has τ 0,t τ 0,s = τ 0,t+s , and that the operators τ 0,t commute with the operators τ x,0 , so we may set τ x,t = τ x,0 τ 0,t = τ 0,t τ x,0 .
One expects that if the translation operators τ x,t are ergodic on Ω then solutions to the random equation (1.2) or (1.4) converge to solutions of a constant coefficient homogenized equation under diffusive scaling.Thus suppose f : R d → R is a C ∞ function with compact support and for ε satisfying 0 < ε ≤ 1 set h(x) = f (εx), x ∈ Z d , in (1.3), and let u ε (x, t, ω) denote the corresponding solution to (1.2) or (1.4) with this initial data.It has been shown in [21], just assuming ergodicity of the translation operators, that u ε (x/ε, t/ε 2 , ω) converges in probability as ε → 0 to a function u hom (x, t), x ∈ R d , t > 0, which is the solution to a constant coefficient parabolic PDE (1.6) ∂u hom (x, t) ∂t = −∇ * a hom ∇u hom (x, t) , x ∈ R d , t > 0, with initial condition The d × d symmetric matrix a hom in (1.6) satisfies the quadratic form inequality (1.1).Similar results under various ergodic type assumptions on Ω can be found in [1,6,12,27].In time-independent environments the corresponding results for elliptic equations in divergence form have been proven much earlier -see [19,20,25,32].
It has been shown in the case of homogenization of elliptic equations in divergence form with random coefficients, that a rate of convergence in homogenization can be obtained provided the random environment satisfies some quantitative strong mixing property.The first results in this direction were proven in the 1980's by Yurinski [31], but there have been several papers more recently extending his work.In particular, Caffarelli and Souganidis [10] have obtained rates of convergence results in homogenization of fully nonlinear PDE.In recent work of Gloria and Otto [15] an optimal rate of convergence result was obtained for linear elliptic equations in divergence form.Following an idea of Naddaf and Spencer [23], they express the quantitative strong mixing assumption as a Poincaré inequality.This formulation of the strong mixing assumption is very useful when the random environment is a Euclidean field theory with uniformly convex Lagrangian.
For the case of parabolic equations in divergence form with random coefficients, we were unable to find in the literature any results on rate of convergence in homogenization.Here we shall obtain a rate of convergence, but only for the averaged solution to the parabolic equation as in Theorem 1.1, and for two particular environments.For the discrete time problem (1.2), (1.3) we assume the environment is i.i.d.That is we assume the variables a(τ x,t •), (x, t) ∈ Z d+1 , are i.i.d.For the continuous time problem (1.3), (1.4) we assume the environment is the stationary process associated with a massive Euclidean field theory.
This Euclidean field theory is determined by a potential V : R d → R, which is a C 2 uniformly convex function, and a mass m > 0. Thus the second derivative a(•) = V ′′ (•) of V (•) is assumed to satisfy the inequality (1.1).Consider functions φ : Z d × R → R which we denote as φ(x, t) where x lies on the integer lattice Z d and t on the real line R. Let Ω be the space of all such functions which have the property that for each x ∈ Z d the function t → φ(x, t) on R is continuous, and F be the Borel algebra generated by finite dimensional rectangles {φ(•, •) ∈ Ω : |φ(x i , t i ) − a i | < r i , i = 1, ..., N }, where (x i , t i ) ∈ Z d × R, a i ∈ R, r i > 0, i = 1, ..., N, N ≥ 1.The translation operators τ x,t : Ω → Ω, (x, t) ∈ Z d × R, are defined by τ x,t φ(z, s) = φ(x + z, t + s), z ∈ Z d , s ∈ R.
For any d ≥ 1 and m > 0 one can define [7,14] a unique ergodic translation invariant probability measure P on (Ω, F ) which depends on the function V and m.In this measure the variables φ(x, t), x ∈ Z d , t > 0, conditioned on the variables φ(x, 0), x ∈ Z d , are determined as solutions of the infinite dimensional stochastic differential equation (1.10) dφ(x, t) = − ∂ ∂φ(x, t) {V (∇φ(x ′ , t))+m 2 φ(x ′ , t) 2 /2} dt+dB(x, t) , x ∈ Z d , t > 0, where B(x, •), x ∈ Z d , are independent copies of Brownian motion.Formally the invariant measure for the Markov process (1.10) is the Euclidean field theory measure (1.11) exp Hence if the variables φ(x, 0), x ∈ Z d , have distribution determined by (1.11), then φ(•, t), t > 0, is a stationary process and so can be extended to all t ∈ R to yield a measure P on (Ω, F ).For this measure the translation operators τ x,t , (x, t) ∈ Z d × R, form a group of measure preserving transformations on (Ω, F , P ).
Theorem 1.2.Let f : R d → R be a C ∞ function of compact support and set h(x) = f (εx), x ∈ Z d in (1.3).Then if 4dΛ ≤ 1 and the variables a(τ x,t •), (x, t) ∈ Z d+1 , are i.i.d., the solution u ε (x, t, ω) of (1.2) with initial data (1.3) has the property (1.12) sup where α > 0 is a constant depending only on d, Λ/λ and C is a constant depending only on d, Λ, λ and the function f (•).
We consider what Theorem 1.2 tells us about the expectation of the Green's function for the equations (1.2) and (1.4).By translation invariance of the measure we have that where G a (x, t) is the expected value of the Green's function.Setting h(x) = f (εx), x ∈ Z d , then (1.14) may be written as where integration over εZ d is defined by (1.16) Let G a hom (x, t), x ∈ R d , t > 0, be the Greens function for the PDE (1.6).One easily sees that G a hom (•, •) satisfies the scaling property with respect to x ∈ εZ d are bounded by Cε α for some constant C. Conversely Theorem 1.2 is implied by the point-wise estimate on Green's functions: (1.18) It is clear that the inequality (1.18) for ε < 1 follows from the same inequality for ε = 1: provided Λt ≥ 1 and x ∈ Z d .We shall prove such an inequality and also similar inequalities for the derivatives of the expectation of the Green's function, (1.20)The proofs of Theorem 1.2 and Theorem 1.3 follow the same lines as the proofs of the corresponding results for elliptic equations in [10].One begins with a Fourier representation for the average of the solution to the random parabolic equation, which was obtained in [8].Then for the i.i.d.environment the generalization by Jones [18] of the Calderon-Zygmund theorem [5] to parabolic multipliers, together with some interpolation inequalities, yields Theorem 1.2 and the inequalities (1.19), (1.20) of Theorem 1.3 in the discrete time case.Similarly to [10] we need to use the result of Delmotte and Deuschel [11] on the Hölder continuity of the second difference ∇∇G a (x, t) in order to prove (1.21).In the continuous time case we need in addition to prove some Poincaré inequalities for time dependent fields.To do this we follow the methodology of Gourcy-Wu [16] by using the Clark-Ocone formula [24].

Fourier Space Representation and Homogenization
In this section we shall prove the homogenization result Theorem 1.1.The proof of this is based on a Fourier representation for the solutions of (1.2), (1.4) which was given in [8].
We begin by summarizing relevant results from [8] for the discrete time equation (1.2).Thus we are assuming a probability space (Ω, F , P ) and a set of translation operators τ x,t , x ∈ Z d , t ∈ Z, acting on Ω.For ξ ∈ R d and ψ : Ω → C a measurable function we define the ξ derivative of ψ(•) in the j direction ∂ j,ξ , and its adjoint Similarly to (1.5) its adjoint ∂ * ξ is given by the row operator ).Let P : L 2 (Ω) → L 2 (Ω) be the projection orthogonal to the constant function and η ∈ C with ℜη > 0. Then there is a unique square integrable solution Φ(ξ, η, ω) to the equation be the d × d matrix function given in terms of the solution to (2.2) by the formula We define the d dimensional periodic column vector e(ξ) ∈ C d to have jth entry given by the formula e j (ξ) = e −iej •ξ − 1, 1 ≤ j ≤ d.It was shown in [8] that the solution u(x, t, ω), x ∈ Z d , t = 0, 1, 2, .., ω ∈ Ω, to the initial value problem (1.2), (1.3) has the representation (2.4) where ĥ(•) is the Fourier transform of h(•), It follows from (2.4) that the Fourier transform Ĝa (ξ, η) of the averaged Green's function G a (•, •) for (1.2), (1.3) given by has the representation The solution to (2.4) can be generated by a convergent perturbation expansion.Let H(Ω) be the Hilbert space of measurable functions ψ : Ω → C d with norm ψ given by ψ 2 = |ψ(•)| 2 .We define an operator T ξ,η on H(Ω) as follows: For any g ∈ H, let ψ(ξ, η, ω) be the solution to the equation where G Λ (•) is the solution to the initial value problem Equation (2.10) has a unique solution provided 4dΛ ≤ 1, and the function G Λ (x, t) satisfies an inequality Since T ξ,η ≤ 1 and b(ω) ≤ 1−λ/Λ, ω ∈ Ω, the Neumann series for the solution to (2.12) converges in H(Ω).
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 transform of a function h : The Fourier inversion formula yields Now the action of the translation group τ x,0 , 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 ).Similarly the action of the translation group τ 0,t , t ∈ Z, on Ω can be described by a self-adjoint operator B on L 2 (Ω) which commutes with A 1 , .., A d , so that It follows then from (2.9),(2.15),(2.16) that (2.17) The Neumann series for the solution to (2.12) yields a convergent perturbation expansion for the function q(ξ, η) of (2.3).Thus for m = 1, 2..., let the matrix function h m (ξ, η) be defined for ℜη > 0, ξ ∈ R d , by (2.18) It is easy to see that the function As in [8,10] we can extend this result as follows: Proposition 2.1.Suppose that 4dΛ ≤ 1 and any of the translation operators τ ej ,0 , 1 ≤ j ≤ d, or τ 0,1 is ergodic on Ω.Then the limit lim (ξ,η)→(0,0) q(ξ, η) = q(0, 0) exists.If any of the translation operators is weak mixing [26] on Ω then q(ξ, η), ξ ∈ R d , ℜη > 0, extends to a continuous function on ξ ∈ R d , ℜη ≥ 0.
Proof.We follow the same argument as in Lemma 2.5 of [8]  which does not extend to a continuous function of (ξ, η) on the set ξ ∈ R d , ℜη ≥ 0.
In the continuous time case there is a similar development to the above.The solution u(x, t, ω) to (1.3), (1.4) has the representation (2.40) where now the d dimensional row vector Φ(ξ, η, ω) is the solution to the equation . In (2.41) the operator ∂ is the infinitesimal generator of the time translation group τ 0,t , t ∈ R. The d × d matrix function q(ξ, η) in (2.40) is given in terms of the solution to (2.41) by the formula (2.3).It follows from (2.40) that the Fourier transform Ĝa (ξ, η) of the averaged Green's function G a (•, •) for (1.4) defined by has the representation Let G(x, t), x ∈ Z d , t > 0, be the solution to the initial value problem Then the equation (2.41) is equivalent to (2.12) where the operator T ξ,η is given by the formula Note that in the continuous time case there is no restriction on the value of Λ > 0. The operator T ξ,η of (2.45) is bounded on H(Ω) with T ξ,η ≤ 1, provided ξ ∈ R d , ℜη > 0, and hence the Neumann series for the solution of (2.12) converges in H(Ω).
As in the discrete time case 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 transform of a function h : The Fourier inversion formula yields Now the action of the translation group τ x,0 , 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 ).Similarly the action of the translation group τ 0,t , t ∈ R, on Ω can be described by a self-adjoint operator B on L 2 (Ω) which commutes with A 1 , .., A d , so that The Neumann series for the solution to (2.12)-with the operator T ξ,η given now by (2.50)-yields a convergent perturbation expansion (2.18), (2.19) for the function q(ξ, η).It is easy to see that the analogues of Proposition 2.1, Lemma 2.1 and Corollary 2.1 continue to hold for the continuous time case.In the continuous time analogue of Corollary 2.2 the inequality (2.27) is replaced by The inequality (2.51) follows from Lemma 5.3 of [8].
Proof of Theorem 1.1-continuous time case.We proceed as in the discrete time case replacing (2.4) by (2.40) and using Lemma 5.4, Lemma 5.5 of [8] in place of Lemma 2.9, Lemma 2.10 of [8].

Rate of Convergence in Homogenization
In this section we shall prove Theorem 1.2 under the assumption that the solutions Φ(ξ, η, ω) of (2.2), (2.41) satisfy a certain property which we describe below.In §5 we shall show that this property holds for the independent variable environment, and in §6 for the massive field theory environment.We first consider the discrete time case, whence Φ(ξ, η, ω) is a solution to (2.2). For where |g(x, t)v| is the Euclidean norm of the vector g(x, t)v ∈ C d .We assume the following: Hypothesis 3.1.There exists p 0 (Λ/λ) > 1 depending only on d, Λ/λ and a constant C such that for 2) holds for p = 1.Hence if (3.2) holds for p = p 0 (Λ/λ), by the Riesz convexity theorem [30] it also holds for any p satisfying 1 ≤ p ≤ p 0 (Λ/λ).
We show that if Hypothesis 3.1 holds then the function q(ξ, η) defined by (2.3) is Hölder continuous with exponent depending on d, Λ/λ.Lemma 3.1.Assume Hypothesis 3.1 holds.Then there exists α > 0 depending only on d, Λ/λ and a constant C α such that the d × d matrix function q(ξ, η) of (2.3) satisfies the inequality Proof.It follows from (2.18) that (3.4) Hence we conclude from (2.12), (2.19) and (3.4) upon using the inequality From (2.9) we see that the RHS of (3.5) is the same as the LHS of (3.2) with the function g(•, •) given by the formula It is easy to see that for 0 , and with g(•, •) p satisfying the inequality where the constant C p depends only on d, p.The Hölder continuity (3.3) for sufficiently small α > 0 follows from (3.5) and (3.7).
Proof of Theorem 1.2-discrete time case.We follow the proof of Theorem 1.1 using the Hölder continuity of the function q(•, •).
For the continuous time case we prove Theorem 1.2 assuming a hypothesis analogous to Hypothesis 3.1.For 1 where |g(x, t)v| is the Euclidean norm of the vector g(x, t)v ∈ C d .
Proof.We assume first that p 2 = p 3 = • • • = p k = 1, in which case Hypothesis 4.1 and Lemma 4.1 imply respectively that (4.7) holds for 1/p ′ 1 ≤ 1−1/p 0 (Λ/λ), q = ∞, and for p 1 = q = 2.The Riesz convexity theorem then implies that (4.7) holds if p ′ 1 , q satisfy (4.6) with Next assume for induction that we have proved (4.7) in the case when (4.6) holds with , where the functions g r+1 , .., g k are fixed with , with the same functions g r+1 , .., g k .Now we fix the functions g 1 , .., g r , g r+2 , .., g k with Applying the Riesz convexity theorem to the functions g r+1 , we conclude that (4.7) holds if p 1 , .., p r+1 satisfies (4.6) with ) be the space of functions g : [−π, π] d+1 → C which are weakly p integrable.The norm g p,w of g is defined to be the minimum number satisfying the inequality ) and its norm is bounded by CΛ 1−m+1/p for some constant C.
If m is the largest integer strictly less than 1 + d/2 and 0 ≤ δ < 1 + d/2 − m, then for any ρ ∈ R satisfying |ρ| ≤ 1, the function ) and its norm is bounded by C p Λ 1−m−δ+1/p , where the constant C p can diverge as p → 1.
Proof.The Hölder continuity (4.9) of the function q(•, •) has already been proven in Lemma 3.1.We first prove that the derivative (4.10 Denoting by [•, •] the inner product for H(Ω), we therefore have for •) are determined from their Fourier transforms (2.13) by the formula which follows from (2.17).We take ĥ(•, •) to be given by the formula , where 1 d is the d dimensional column vector with all entries equal to 1. From (2.10) and (4.15) it follows that (4.16) Observe now that by the Hunt interpolation theorem [28] the inequality (4.7) also holds for the operator T 1,ℑξ,ℜη as a mapping from ) provided q satisfies the inequality in (4.6) with p 1 = (d + 2)/(d + 1).Evidently we can choose q so that q/2 > 1 + d/2.Since we can make an exactly similar argument for the function g(x, t) and T1,ℑξ,ℜη g, we conclude from (4.13) that ∂q r,r ′ (ξ, η)/∂η is in the space L q/2 w ([−π, π] d+1 ) with norm bounded by Λ 2−2/p times a constant.We have proved for m = 1 that the derivative (4.10) is in the appropriate weak L p space.
We proceed similarly to estimate the higher derivatives (4.10) and the fractional derivative (4.11).
Remark 5. Proposition 4.1 with α = 0 was proven in [8].In that case the constant C in the statement of the Proposition depends only on d, Λ/λ.Proposition 4.1 enables us to compare the averaged Green's function If δ satisfies 0 < δ ≤ 1 then there exists α, γ > 0 depending only on d, Λ/λ, δ and a constant C δ such that the following inequality holds: The constant α in (4.20) must satisfy α < δ.
Proof.From (2.7), (4.17) and Corollary 2.1 there is a constant C depending only on Λ/λ such that for a ∈ R d with |a| ≤ 1, (4.21) where the function f a (ξ, ℑη) is given by the formula .
The exponential decay in the inequalities (4. To complete the proof of the theorem we need to obtain the polynomial decay in [Λt + 1] in (4.18)-(4.20),whence we may assume that Λt ≥ 1.We divide the torus [−π, π] d+1 into various regions, the first of which is It follows then from (4.24) that there is a constant C 2 such that (4.26) Next we consider for k = 1, 2, .., regions From (4.24) we see that if |a| ≤ 2/Λt there is a constant C 3 such that (4.28) In general a = O(1), so we need to take advantage of the oscillatory nature of the integral in (4.28).Let ρ = π/(t + 1) so that e iρ(t+1) = −1, and It follows again from (4.24) that the last two integrals on the RHS of (4.29) are bounded by the RHS of (4.28).In order to bound the first integral we observe from the Hölder continuity (4.9) of the function q(•, •) that there are constants C 4 , C 5 and Since we are assuming |a| ≥ 2/Λt it follows from (4.30) that (4.31) for some constant C 6 .We therefore conclude from (4.26)-(4.31) that there is a constant C 7 and (4.32) The inequality (4.32) can also be derived by using the fact from Theorem 3.1 that the derivative ∂q(ξ where ∂E 0,k is the union of sets {(ξ, ℑη) : Λt|e(ξ)| 2 ≤ 1, ℑη = constant} with the constant given by ±2 k /t or ±2 k−1 /t.It follows from (4.24) that the first integral on the RHS of (4.33) is bounded by the RHS of (4.28).To bound the second integral we use the inequality where C 8 , C 9 are constants.We can bound the integral of the first term on the RHS of (4.34) just as we did with the second term on the RHS of (4.30).To bound the integral of the second term we use the well known fact that if f ∈ L p w ([−π, π] d+1 ) with 1 < p < ∞, then for any measurable set F , one has where the constant C p depends only on p. Taking p = (1 + d/2)/(1 − α/2) we conclude from Proposition 4.1 that 1/(t + 1) times the integral over E 0,k of the second term on the RHS of (4.34) is bounded by for some constant C 10 .Summing (4.36) over k ≥ 1 we obtain the inequality (4.32) again.
In order to prove (4.19) we follow the previous argument, replacing the function f a (ξ, ℑη) by the function e(ξ)f a (ξ, ℑη).To prove (4.20) we use the inequality Proof.Taking a hom = q(0, 0) in (1.6), we see from (4.17) that G lattice a hom (•, •) is the Green's function for the discrete parabolic equation corresponding to (1.6), To prove the theorem we follow a standard method of numerical analysis for estimating error between the solution of a continuous problem and its approximating discrete problems.The method is to regard the solution of the continuous problem as an approximate solution to the discrete problem.An alternative approach based on comparison of the Fourier representation (4.17) of the lattice Green's function G lattice a hom (•, •) to the Fourier representation of the continuous Green's function G a hom (•, •) is pursued in [22] for the case of elliptic equations.
Let f : R d → R be a nonnegative C ∞ function with support contained in the ball {x ∈ R d : |x| < 1} and u(x, t) = u hom (x, t) be the solution to the initial value problem (1.6), (1.7).With ∇ x , ∇ * x denoting the discrete operators (1.5), we have that where the d × d matrix A(y, t) = [A i,j (y, t)], y ∈ R d , t > 0 is given by the formula (4.52) G lattice a hom (x−y, t−r)h 1 (y+z, r−1) Let Q 0 ⊂ R d be the unit cube centered at the origin.Then we have that (4.57) We also have that there are constants γ, C depending only on d such that (4.61) Proof.Let χ : R d+1 → R be a C ∞ function with compact support such that the integral of χ(•) over R d+1 equals 1.We write where There also exists for positive integers n constants C n such that We assume now that R < √ Λt + 1 < 2R and choose L = R 1−δ for some δ > 0. Then from (2.4) we see that (4.66) where a is given by (4.23) and f a (ζ, θ) is defined by (4.67) If |a|L < 1 we also have from Corollary 2.2 and (4.65) that (4.69) for some constant C 1 .
We can essentially repeat the foregoing arguments for the continuous time averaged Green's function G a (x, t), x ∈ Z d , t ≥ 0, for (1.4).In the continuius time case our hypothesis is: Assuming Hypothesis 4.2 holds, we can prove the analogues of Proposition 4.1, Theorem 4.1 and Theorem 4.2 for the continuous case.Theorem 1.3 therefore follows in the continuous time case once we are able to establish Hypothesis 4.2.

Independent Variable Environment
Our goal in this section will be to prove Hypothesis 3.1 and its generalized form Hypothesis 4.1 in the case when the variables a(τ x,t •), x ∈ Z d , t ∈ Z, are independent.Following [9] we first consider the case of a Bernoulli environment.Thus for each x ∈ Z d , t ∈ Z, let Y x,t be independent Bernoulli variables, whence Y x,t = ±1 with equal probability.The probability space (Ω, F , P ) is then the space generated by all the variables Y x,t , (x, t) ∈ Z d+1 .A point ω ∈ Ω is a set of configurations {(Y n , n) : n ∈ Z d+1 }.For (x, t) ∈ Z d+1 the translation operator τ x,t acts on Ω by taking the point where 0 ≤ γ < 1.
In [9] we defined for 1 ≤ p < ∞ Fock spaces F p (Z d+1 ) of complex valued functions, and observed that F 2 (Z d+1 ) is unitarily equivalent to L 2 (Ω).We can similarly define Fock spaces H p F (Z d+1 ) of vector valued functions with range C d , such that H 2 F (Z d+1 ) is unitarily equivalent to H(Ω).Hence we can regard the operator T ξ,η of (2.9) as acting on H 2 F (Z d+1 ), and by unitary equivalence it is a bounded operator satisfying T ξ,η ≤ 1 for ξ ∈ R d , ℜη > 0. From (2.9) we see that T ξ,η acts as a convolution operator on N particle wave functions Note that for all N particle wave functions, T ξ,η acts as a convolution operator on functions on Z d+1 .Hence its action is determined by its action on 1 particle wave functions be the Fourier transform (2.13) of the 1 particle wave function ψ 1 (x, t), x ∈ Z d , t ∈ Z.We see from (5.2) that for ξ ∈ C d , ℜη > 0, the action of T ξ,η in Fourier space is given by (5.3) Hence the result of Lemma 2.1 for the Bernoulli case follows from: Lemma 5.1.Assume 4dΛ ≤ 1.Then there exist positive constants C 1 , C 2 depending only on d such that for (ξ, η) in the region {(ξ, η) ∈ C d+1 : 0 < ℜη < Λ, |ℑξ| < C 1 ℜη/Λ}, there is the inequality Proof.We have that Proof.It will be sufficient for us to prove the theorem on the space of 1 particle wave functions.To do this we follow the argument of Jones [18], which adapts the methodology of Calderon-Zygmund [5] to Fourier multipliers associated with parabolic PDE.A more general theory of Fourier multipliers can be found in Chapter IV of [29], but because of the generality it is hard to estimate the values of constants using this theory.
For a set E ⊂ Z d+1 , we denote by |E| the number of lattice points of Z d+1 contained in E. Let ψ(x, t), x ∈ Z d , t ∈ Z, be a 1 particle wave function with finite support.We shall show that for any γ > 0, the set where C 2 is the constant of Lemma 5.1 and C 4 , C 5 depend only on d.The function β ψ (•) is defined in [5,18] in terms of the distribution function of ψ(•, •).Once (5.9) is proved the result follows from the argument of [5], which shows that T ξ,η p is simply bounded in terms of the constants occurring in (5.9).
We use a Calderon-Zygmund decomposition to prove (5.9).Recalling that 1/Λ ≥ 4d, let N 0 ≥ 2 be the integer which satisfies 2 N0 ≤ 1/Λ < 2 N0+1 .We choose a 1 , ..a d , b ∈ Z and sufficiently large integer N 1 such that the rectangle R = {(x, t) = (x 1 , ., x d , t) ∈ R d+1 : a j + 1/2 ≤ x j ≤ 2 N1 + a j + 1/2, j = 1, .., d, and b + 1/2 ≤ t ≤ 2 2N1+N0 + b + 1/2 } contains the support of ψ(•, •) and Note that the length of the side of R in the t direction is 2 N0 times the square of the length of a side in an x j direction for all 1 ≤ j ≤ d.We subdivide R into 2 d × 4 sub-rectangles with the same property and continue to similarly subdivide until we reach a set of disjoint rectangles R m , m = 1, .., M 1 , with side in the x j , 1 ≤ j ≤ d, direction a non-negative power of 2, which satisfy the inequality together with a set of rectangles R ′ m , m = 1, 2, ...M 2 , with side in the x j , 1 ≤ j ≤ d, direction equal to 1 and equal to 2 N0 in the t direction which satisfy (5.12) We subdivide the rectangles R ′ m , m = 1, .., M 2 , into 2 rectangles with side in the t direction of length 2 N0−1 , and continue to subdivide until we reach a set of disjoint rectangles R m , m = M 1 + 1, .., M, with side in the t direction a non-negative power of 2, which satisfy the inequality together with a set of unit cubes centered at lattice points of Z d+1 .Setting D γ = ∪ M m=1 R m , one sees that R d+1 − D γ is a union of unit cubes centered at lattice points of Z d+1 , whence We consider the distribution function  We write ψ(•, , where the function ψ 1 (•, •) is defined by (5.16) From Lemma 5.1 and (5.14) we have then that (5.17 To bound the distribution function of ψ 2 (•, •) which has support contained in D γ , we consider a rectangle R m , 1 ≤ m ≤ M, with center (x m , t m ) ∈ Z d+1 and let Rm be the double of R m .We observe that similarly to (2.11) there is a constant C d depending only on d such that the function ∇∇ * G Λ (x, t) satisfies inequalities (5.18) It follows from (5.11), (5.13), (5.20) that if Dγ = ∪ M m=1 Rm , then (5.21) for some constant C d depending only on d.Hence we have that The inequality (5.9) follows from (5.17 Proof.The result follows from Lemma 5.1, Lemma 5.2 and the Riesz-Thorin interpolation theorem [30]. Proof of Hypothesis 4.1.We choose q 0 = q 0 (Λ/λ) with 1 < q 0 < 2 so that δ(q 0 ) ≤ λ/2Λ, where δ(•) is the function in the statement of Corollary 5.1.It follows then from Young's inequality that Hypothesis 4.1 holds if we choose p 0 = p 0 (Λ/λ) > 1 with 1/p 0 + 1/q 0 = 3/2.It is shown in [9] how to extend the argument for the Bernoulli environment corresponding to (5.1) to general i.i.d.environments a(τ x,t •), (x, t) ∈ Z d+1 .We have therefore proven Hypothesis 4.1 for a(τ x,t •), (x, t) ∈ Z d+1 , i.i.d.such that (1.1) holds.

Massive Field Theory Environment
In this section we show that Hypothesis 3.2 and its generalization Hypothesis 4.2 holds if (Ω, F , P ) is given by the massive field theory environment determined by (1.10), (1.11).We recall the main features of the construction of this measure.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 → R we mean a function φ on Q × R with the property that φ(x, t) = φ(y, t) for all x, y ∈ Q, t ∈ R, such that x − y = Le k for some k, 1 ≤ k ≤ d.Let Ω Q be the space of continuous in time periodic functions φ : Q × R → R and F Q be the Borel algebra generated by the requirement that the functions φ(•, •) → φ(x, t) from Ω Q → R are Borel measurable for all x ∈ Q and t rational.For m > 0 we define a probability measure P Q on (Ω Q , F Q ) by first defining expectations of functions of the variables φ(x, 0), x ∈ Q, as follows: where , for some constants C, A. By translation invariance of the measure (6.1) we see that φ(x, 0) ΩQ = 0 for all x ∈ Q and hence the Brascamp-Lieb inequality [2] applied to (6.1) and function where (•, •) is the Euclidean inner product for periodic functions on Q, yields the inequality The variables φ(x, t), x ∈ Q, t > 0, are determined from the variables φ(x, 0), x ∈ Q, by solving the stochastic differential equation (6.3) where B(x, •), x ∈ Q, are independent copies of Brownian motion modulo the periodicity constraint on Q.Since (6.1) is the invariant measure for the stochastic process φ(•, t), t ≥ 0, it follows that (6.1), (6.3) determine a stationary process for t ≥ 0, which therefore can be extended to all t ∈ R. Furthermore the functions t → φ(x, t) on R are continuous with probability 1 for all x ∈ Q.The probability measure P Q on (Ω Q , F Q ) is the measure induced by the stationary process φ(•, t), t ∈ R.
The probability space (Ω, F , P ) on continuous in time fields φ : Z d × R → R is obtained as the limit of the spaces (Ω Q , F Q , P Q ) as |Q| → ∞.In particular one has from Lemma 2.4 of [7] the following result: Proposition 6.1.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 , and t 1 , .., t k ∈ R, the limit (6.5) lim exists and is finite.
From (6.2) and the Helly-Bray theorem [3,13] one sees that Proposition 6.1 implies the existence of a unique Borel probability measure on R k corresponding to the probability distribution of the variables (φ(x 1 , t 1 ), .., φ(x k , t k )) ∈ R k , and this measure satisfies (6.5).The Kolmogorov construction [3,13] then implies the existence of a Borel measure on fields φ : Z d × R → R with finite dimensional distribution functions satisfying (6.5).We have constructed the probability space (Ω, F , P ) corresponding to (1.10), (1.11) for which Ω is the set of continuous in time functions φ : Z d × R → R, and it is clear that the translation operators τ x,t , x ∈ Z d , t ∈ R, are measure preserving and form a group.
Our first goal here will be to prove strong mixing of the operator τ e1,0 on (Ω, F , P ).In order to do this we will need a Poincaré inequality for the measure (Ω Q , F Q , P Q ), in particular a generalization of (6.6) to functions F (φ(•, t 1 ), .., φ(•, t k )) depending on values of the field φ(•, •) at different times.To do this we follow the development of Gourcy-Wu [16] who make use of the Malliavin calculus [24] to prove a log-Sobolev inequality for such measures.The basic insight of the Malliavin calculus is that the Wiener space generated by independent Brownian motions B(x, t), x ∈ Q, t > 0, can be identified with a probability space whose set of configurations is the Hilbert space L 2 (Q × R + ), where R + is the open interval (0, ∞).We denote the Euclidean inner product on The measure on L 2 (Q × R + ) is uniquely determined by the requirement that the variables ψ → [ψ, ψ j ], j = 1, .., k, are i.i.d.standard normal for any set of orthonormal vectors ψ j , j = 1, .., k.We denote this Malliavin probability space by The identification of the Wiener space with (Ω Q,Mal , F Q,Mal , P Q,Mal ) follows from the fact that the expectation of a function is the same as the expectation of F (W (•, •)) with respect to Wiener space, where W (•, •) is the white noise process corresponding to B(•, •) in (6.3).Hence the identification may be summarized as follows: For t > 0 let F t be the σ−field generated by the Brownian motions B(x, s), x ∈ Q, s < t, of (6.3), so from (6.8) we can regard F t as a sub σ−field of F Q,Mal .We consider next vector fields G : The Martingale representation theorem [24] implies that for any function F ∈ L 2 (Ω Q,Mal ) there is a predictable vector field The Clark-Ocone formula [24] states that the vector field G(ψ(•, •)) in (6.9) can be expressed in terms of the Malliavin derivative We show how the Clark-Ocone formula (6.9), (6.10) implies the HS formula (6.7).Let φ(•, T ) be the solution at time T > 0 of (6.3) with initial data φ(•, 0) = 0 and f : Q → R. We can find an expression for the Malliavin derivative of the function F (ψ(•, •)) = (f (•), φ(•, T )) by analyzing the first variation equation for (6.3).Evidently one has that D Mal F (x, t; ψ(•, •)) = 0 for x ∈ Q, t > T .To get an expression for D Mal F (x, t; ψ(•, •)) when t ≤ T we first note from (6.3) that (6.11) It follows from (6.11) that for From (6.12) we see that ξ(x, t), x ∈ Q, t > 0, is the solution to the initial value problem for the parabolic PDE ξ(x, 0) = 0.

Consider now the terminal value problem for the backwards in time parabolic
Then the solution to (6.13) is given by the formula We conclude from (6.16) that (6.17 for some constants C, A. Then from (6.17) it follows that (6.18) Next we observe from (6.14), (6.18) that the conditional expectation (6.10) is given by the formula (6.19) where the operator H is as in (6.7), so Since for any fixed s ≥ 0 the distribution of φ(•, T − s) converges as T → ∞ to the distribution of φ(•) for the invariant measure (6.1), it follows that (6.20) lim Now (6.7) for F 1 = F 2 follows from (6.9), (6.20) on letting T → ∞.The identity (6.7) for general F 1 , F 2 is then a consequence of the symmetry of the LHS of (6.7) in F 1 , F 2 .Proposition 6.2.Let (Ω, F , P ) be the massive field theory probability space defined by Proposition 6.1.Then the operators τ ej ,0 , 1 ≤ j ≤ d, on Ω are strong mixing.
Proof.We proceed as in the proof of Proposition 5.2 of [10].It will be sufficient to prove that for k ≥ 1 and (x j , t j ) ∈ Z d × R, j = 1, ..., k, ( for all C ∞ functions f, g : R k → R with compact support.Let Q ⊂ Z d be a large cube centered at the origin with side of length an even integer L. We define h Q,T (n) for n ∈ Z and T > 0 large by The function h Q,T : Z → R is periodic on the interval . We shall show that there is a constant C independent of L, T as L, T → ∞ such that (6.23) Then (6.21) follows from (6.23) and Proposition 6.1 as in Proposition 5.2 .of[10].
To proceed further we need to obtain a more general Poincaré inequality than was used in Proposition 6.2.In order to do this we consider functions For the functions F (φ(•, •)) we shall be interested in, the directional derivative (6.31) can be written as (6.32) We shall call dF (•, •; φ(•, •)) the field derivative of F (φ(•, •)).One can use the HS formula (6.7) to obtain a Poincaré inequality for functions F (φ(•, •)) of the form (6.33) where g : R → C is a continuous function of compact support and G(φ(•)) is a complex valued C 1 function of fields φ : Then the variance of F (φ(•, •)) is given in terms of the Fourier transforms of g(•) and h(•) by Note that the function ĥ(•) is real and non-negative.Observe next that h(t) can be written as an expectation with respect to the measure (6.1) by using the operator d * d which occurs in (6.7).Thus we have that (6.37) with a similar formula for t < 0. For ζ ∈ R let u(ζ, φ(•)) be the solution to the elliptic PDE We conclude from (6.37), (6.38) that If we apply the gradient operator d to (6.38) we obtain the equation (6.40) Hence (6.39), (6.40) and the HS formula (6.7) imply that (6.41) ĥ(ζ) = 4 × real part of Just as (6.6) follows from (6.7), we see from (6.41) that It follows from (6.36), (6.42) that Since from (6.34) the inequality (6.43) can be rewritten as we have obtained a Poincaré inequality for functions F (φ(•, •)) of time dependent fields which are of the form (6.33).We generalize this as follows: Proof.Let T > 0 be large and consider F (τ 0,T φ(•, •)) as a function of solutions φ(x, t), x ∈ Q, t > 0, to the stochastic equation (6.3).By the chain rule we have that (6.45) It follows then from (6.17 and so we conclude from (6.47) that (6.50) Hence (6.9), (6.10) imply that The result follows now by observing that the limit of the LHS of (6.51) as T → ∞ is equal to the LHS of (6.44).Similarly the RHS of (6.51) converges to the RHS of (6.44).
We shall show how the Poincaré inequality (6.44) can be used to improve the most elementary of the inequalities contained in §2.Thus let us consider an equation which differs from (2.41) only in that the projection operator P has been omitted, (6.52) ηΦ(ξ, η, ω) + ∂Φ(ξ, η, ω) For any v ∈ C d we multiply the row vector (6.52) 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 (6.53) where • denotes the norm in H(Ω).Let g : with norm given by (3.8).If p = 1 then (6.53) implies that (6.54) The Poincaré inequality (6.44) enables us to improve (6.54) to allow g Proposition 6.3.Suppose a(•) in (6.52) is as in the statement of Theorem 1.2.
We can also show that the RHS of (6.61) converges as Q → Z d by generating the function ∇G Q from a perturbation expansion.Thus let B : It follows from (6.69), (6.72) that ∇v Q is the solution to the equation follows by the uniform in Q estimates of the previous paragraph that it is sufficient to prove convergence as Q → Z d for any finite number of terms in the Neumann series expansion of (6.73).The convergence for a finite number of terms follows from Proposition 6.1 using the fact that the function g(•, •) in (6.61) has compact support and that ℜη > 0. We have shown now that (6.74) and ∇v is the solution to the equation We can now easily extend the previous argument by using the continuous time version of the Calderon-Zygmund theorem, Corollary 5.1, to prove (6.55) for a range of p > 1. Define for q ≥ 1 the Banach space L q (Z d × R × Ω, C d ) of functions g : Z d × R × Ω → C d with norm g q given by (6.77) where g(y, r; φ(•, •)) is the norm of g(y, r; φ(•, •)) ∈ H(Ω).By following the argument of Lemma 5.2, we see that T η is bounded on L q (Z d × R × Ω, C d ) for q > 1 with norm T η q ≤ 1 + δ(q), where lim q→2 δ(q) = 0. Noting that h q ≤ C q Λ 1−1/q Λ 1 |v|/λ for a constant C q depending only on d, q, we conclude from (6.76) and the Calderon-Zygmund theorem that there exists q 0 (Λ/λ) < 2 depending only on d, Λ/λ, such that ∇G is in L q (Z d × R × Ω, C d ) for q 0 (Λ/λ) ≤ q ≤ 2, and ∇G q ≤ CΛ −1/q Λ 1 |v| where the constant C depends only on d, Λ/λ.The inequality (6.55) with p = 2q/(3q − 2) follows from (6.74) and Young's inequality.
When k > 1 we write (6.85) a k (g, ξ, η)v = P Instead of estimating the norm of the function a k (g, ξ, η)v of (6.85) directly by using the Poincaré inequality as in (6.61), we begin with the Clark-Okone formula (6.9).Let φ(•, t), t > 0, be the solution of (6.3) with initial condition φ(•, 0) = 0. We extend the function φ(•, t) to t < 0 by setting φ(•, t) = 0 for t < 0. It is then easy to see that We have now from (6.45) that for x ∈ Q, t > 0, the Malliavin derivative D To estimate the first term on the RHS of (6.93) we argue as in Lemma 6.1.Thus from (6.91) we have that Hence on using the fact that for any s ∈ R, one has (6.100) lim where the constant C depends only on d, we conclude as in the argument showing (6.80) that for 1 ≤ p ≤ 2 there is a constant C depending only on d such that (6.101) lim From (6.86) we see that for ξ ∈ R d (6.102) and so (6.101) implies that for ξ ∈ R d the first term on the RHS of (6.93) is bounded as (6.103) lim where the p norm of g(•, •) is given by (3.8).
We can estimate the second term on the RHS of (6.93) by following the argument of Proposition 6.3.Thus we have that