Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

Transcription

1 Moment bounds on the corrector of stochastic homogenization of non-symmetric ellitic finite difference equations Jonathan Ben-Artzi Daniel Marahrens Stefan Neukamm January 8, 07 Abstract. We consider the corrector equation from the stochastic homogenization of uniformly ellitic finite-difference equations with random, ossibly non-symmetric coefficients. Under the assumtion that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality LSI), we obtain otimal bounds on the corrector and its gradient in dimensions d. Similar estimates have recently been obtained in the secial case of diagonal coefficients making etensive use of the maimum rincile and scalar techniques. Our new method only invokes arguments that are also available for ellitic systems and does not use the maimum rincile. In articular, our roof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green s function in weighted saces. In the critical case d = our argument for the estimate on the gradient of the ellitic Green s function uses a Calderón-Zygmund estimate in discrete weighted saces, which we state and rove. As alications, we rovide a quantitative two-scale eansion and a quantitative aroimation of the homogenized coefficients. Keywords: stochastic homogenization, corrector equation, two-scale eansion, variance estimate. 00 Mathematics Subject Classification: 35B7, 39A70, 60H5, 60F99. Contents Introduction j.ben-artzi@imerial.ac.uk, Deartment of Mathematics, South Kensington Camus, Imerial College London, London SW7 AZ, United Kingdom daniel.marahrens@mis.mg.de, Ma-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstraße, 0403 Leizig, Germany stefan.neukamm@tu-dresden.de, Technische Universität Dresden, Fachrichtung Mathematik, 006 Dresden, Germany

2 Main results and sketch of roof 7. General framework Main results Sketch of roof of Theorem Sketch of roof of Theorem Auiliary results and roofs 3. Well-osedness of the modified corrector Oscillations and Green s function estimates Logarithmic Sobolev inequality and sectral ga revisited Proof of Theorem Proof of Theorem A weighted Calderón-Zygmund estimate 7 5 Alications Quantitative two-scale eansion Quantitative aroimation of the homogenized coefficients Annealed Green s function estimates A Proof of Lemma 5 Introduction We study the modified corrector equation T φ T + a φ T ) = aξ) in Z d, d, ) which is a discrete ellitic finite-difference equation for the real valued function φ T, called the modified corrector. As we elain below, it arises in stochastic homogenization. The symbols and denote the discrete finite-difference) gradient and the negative divergence, see Section below for the recise definition. In the modified corrector equation T denotes a ositive cut-off arameter which we think of to be very large), and ξ R d is a vector, fied throughout this aer. We consider ) with a random, uniformly ellitic field of coefficients a : Z d R d d. To be recise, for a fied constant of elliticity λ > 0 we denote by Ω 0 those matrices a 0 R d d that are uniformly ellitic in the sense that v R d : v a 0 v λ v and a 0 v v, ) and define the set of admissible coefficient fields Ω := Ω Zd 0 = { a : Z d Ω 0 }. In this aer we derive otimal bounds for finite moments of the modified corrector and its gradient, under the assumtion that the coefficients are distributed according to a stationary and ergodic law on Ω, where ergodicity holds in the quantitative form

3 of a Logarithmic Sobolev Inequality LSI), see Definition below. The bounds in the symmetric case and in the non-symmetric discrete case are new. Below we shall discuss in more detail the novelties of this aer and their relationshi with eisting results. Our main results are resented in Theorems and below. For easy reference, let us state them already here, somewhat informally. Throughout the aer, we write for the eected value associated to the law on Ω. The first result concerns a bound on all moments of the gradient of the corrector. Under the assumtions of stationarity and LSI, we have for all < and T that φ T 0) + ξ C ξ, where the constant C is indeendent of T. Note that here and throughout the aer the constant in T has no secial meaning. In fact, since we are interested in the behavior T, we could relace with any number greater than ). The second result is a bound on the corrector itself. Under the same assumtions even under a slightly weaker assumtion than LSI, see Theorem below), we have that φ T 0) C { φ T 0) + ξ log T ) for d =, for d >. These estimates are otimal, even in dimension d = where we recover the otimal logarithmic rate of divergence of the moment of φ T. While the first result is relatively easy to rove, the argument for the second result is substantially harder and the main urose of our aer. Let us emhasize that the coefficients in ) are not assumed to be diagonal or even symmetric. Thus, equation ) in general does not enjoy a maimum rincile; this constitutes a major difference to revious works where the maimum rincile layed a major role and eclusively the case of diagonal coefficients was studied, see e.g. [4, 5, 0]. In fact, the method resented in this aer only relies on arguments that are also available in the case of ellitic systems. The etension of our findings to discrete systems, in articular a discrete version of linear elasticity, is work in rogress. Recently, Bella and Otto considered in [6] systems of ellitic equations on R d ) with eriodic but still random) coefficients. As a main result, they obtain moment bounds on the gradient of the corrector with hel of an argument that avoids the maimum rincile and even the use of Green s functions. Still, the derivation of moment bounds on the corrector itself which is the main urose of our aer remains oen. Alications and relation to stochastic homogenization. The modified corrector equation ) aears in stochastic homogenization: For ε > 0 and a Ω distributed according to we consider the equation u ε εa ε ) εu ε ) = f in εz d, 3) where ε and ε denote the discrete gradient and divergence see Section 5.). As shown in [39, 8, 9, 3], in the homogenization limit ε 0 the solution u ε a; ) converges for almost every a Ω to the unique solution u 0 H R d ) of the homogenized equation u hom diva hom u hom ) = f in R d. 3

4 Here a hom Ω 0 is deterministic and determined by the formula e i a hom e j = lim T e i + φ T,i 0)) a0)e j + φ T,j 0)), 4) where φ T,j is the solution to ) with ξ = e j. Let us comment on the aearance of the limit as T in this formula. Formally, and in analogy to eriodic homogenization, we eect that e i a hom e j = e i + φ i 0)) a0)e j + φ j 0)), where φ i is a solution to the corrector equation that is stationary in the sense of a φ i + e i )) = 0 in Z d, 5) φ i a; + z) = φ i a + z); ) -almost every a Ω and all, z Z d. 6) In the case of deterministic, eriodic homogenization, it suffices to solve 5) on the reference torus of eriodicity and eistence essentially follows from Poincaré s inequality on the torus. In the stochastic case, the corrector equation 5) has to be solved on the infinite sace Z d subject to the stationarity condition 6). Since this is not ossible in general, the corrector equation 5) is tyically regularized by adding the zeroth-order term T φ i with arameter T. In fact this was already done in the ioneering work of Paanicolaou and Varadhan [39] and leads to the modified corrector equation ), which in contrast to 5), admits for all a Ω a unique bounded solution φ T a; ) l Z d ) that automatically is stationary, see Lemma below. While simle energy bounds, cf. 50), make it relatively easy to ass to the regularization-limit T on the level of φ T and thus in the homogenization formula 4)), it is difficult, and in general even imossible, to do the same on the level of φ T itself. For similar reasons and in contrast to the eriodic case), it is difficult to quantify errors in stochastic homogenization, such as the homogenization error u ε u hom. In Section 5. we rovide an otimal H -estimate for the two-scale eansion u ε u hom + ε d j= φ j ε ) ju hom ). 7) in dimensions d 3 and obtain an estimate for the homogenization error as corollary. Previous quantitative results and novelty of the aer. For eriodic homogenization the quantitative behavior of 3) and the eansion 7) is reasonably well understood e.g. see [5,, 7]). In the stochastic case, due to the lack of comactness, the quantitative understanding of 3) is less develoed and in most cases only subotimal estimates are obtained, see [4, 38,, 3,, 9, 4]. In articular, the first quantitative result is due to Yurinskii [4] who roved an algebraic rate of convergence with an subotimal eonent) for the homogenization error u ε u hom in dimensions d > for algebraically miing coefficients. For refinements and etensions to dimensions d we refer to the insiring work by Naddaf and Sencer [38], and the recent works by Conlon and Naddaf [] and Conlon and Sencer [3]. Most recently, Armstrong and Smart [4] obtained 4

5 the first result on the homogenization error for the stochastic homogenization of conve minimization roblems. Their aroach, which builds u on ideas of Avellaneda and Lin [5], substantially differs from what has been done before in stochastic homogenization of divergence form equations. It in articular alies to the continuum version of 3) with symmetric coefficients, and otentially etends to symmetric systems at least under sufficiently strong elliticity assumtions). For results on non-divergence form ellitic equations see [0, 3]. While qualitative stochastic homogenization only requires to be stationary and ergodic, the derivation of error estimates requires a quantification of ergodicity. Pursuing otimal error bounds, in a series of aers [4, 5, 6, 0,, 34, 3, 37] initiated by Gloria and Otto) a quantitative theory for 3) is develoed based on Sectral Ga SG) and LSI as tools to quantify ergodicity. In contrast to earlier results, the estimates in the aers mentioned above are otimal: E.g. [0] contains a comlete and otimal analysis of the aroimation of a hom via eriodic reresentative volume elements and [] establishes otimal estimates for the homogenization error and the eansion in 7). A fundamental ste in the derivation of these results are otimal moment bounds for the corrector, see [4, 5, 0]. The etension to the continuum case has been discussed in recent aers: In [6] moment bounds on the corrector and its gradient have been obtained for scalar equations with ellitic coefficients. In the resent contribution we continue the theme of quantitative stochastic homogenization and resent a new aroach that relies on methods, that we believe etend with only few modifications to the case of systems satisfying sufficiently strong elliticity assumtions. In the works discussed above, arguments restricted to scalar equations are used at central laces. Most significantly, Green s function estimates are required and derived via De Giorgi-Nash-Moser regularity theory e.g. see [0, Theorem 3]). This method is based on the maimum rincile, which holds for diagonal coefficients, but not for general symmetric or ossibly non-symmetric coefficients as considered here. In fact, in our case the Green s function is not in general ositive everywhere. We derive the required estimates on the gradient of the Green s function from the corresonding estimate on the constant coefficient Green s function by a erturbation argument that invokes a Helmholtz rojection; this is insired by [4]. Secondly, revious works rely on a gain of stochastic integrability obtained by a nonlinear Cacciooli inequality see Lemma.7 in [4]). In the resent contribution we aeal to an alternative argument that invokes the LSI instead. While SG, which is weaker than LSI see [8]), has been introduced into the field of stochastic homogenization by Naddaf and Sencer [38, Theorem ] in form of the Brascam-Lieb inequality), the LSI has been used in [34] in the contet of stochastic homogenization to obtain otimal annealed estimates on the gradient of the Green s function and bounds on the random art of the homogenization error u ε u ε. Note that in the secial case of diagonal coefficients i.e. when the maimum rincile and the De Giorgi-Nash-Moser regularity theory is available) our results are not new: The T -indeendent results on φ T and φ T in d > dimensions have already been established in [4, 0] under the slightly weaker assumtion SG on the statistics see 0) below), and the estimate on the corrector in the otimal form of φ T Clog T ) with a constant indeendent of T is obtained in [0]. Relation to random walks in random environments. There is a strong link between 5

6 stochastic homogenization and random walks in random environments see [8] and [30] for recent surveys). Suose for a moment that concentrates on diagonal matrices. Then for each diagonal-matri-valued field a : Z d R d d, we may interret 3) as a conductance network, where each edge [, + e i ] Z d, i =,..., d) is endowed with the conductance a ii ). The ellitic oerator a ) generates a stochastic rocess, called the variable seed random walk X = X a t)) t 0 in a random environment with law. Using arguments from stochastic homogenization, Kinis and Varadhan [7] see also [3] for an earlier result) show that the law of the rescaled rocess εxεt) converges weakly to that of a Brownian motion with covariance a hom. This annealed invariance rincile for X has been ugraded to a quenched result by Sidoravicious and Sznitman [40]. The key ingredient in their argument is to rove that the anchored corrector i.e. the function ϕ introduced in Corollary a) below) satisfies a quenched sublinear growth roerty. The quantitative analysis derived in the resent aer is stronger. Indeed, our estimate on φ T almost immediately imlies that the anchored corrector grows sublinearly. On to of that in dimensions d > the moment bound on φ T imlies that the anchored corrector is almost bounded, in the sense that it grows slower than any rate, see Corollary and the subsequent remark. If the coefficients are not diagonal, then 3) is not any longer related to a random conductance model. As mentioned before, for non-symmetric a and even for certain symmetric coefficients) the maimum rincile for a ) generally fails to hold. In that case the semigrou generated by a ) is not a Markov rocess and there is no natural robabilistic interretation for 3). This may also be seen in terms of Dirichlet forms. While the non-symmetric) ellitic oerator diva hom ) acting on functions on R d generates a Dirichlet form R d u a hom vd in the sense of [33, Definition I.4.5] and a corresonding Markov rocess, the discrete oerator a ) with associated bilinear form Z d u a v defined on l Z d ) l Z d ) does not. Indeed, the contraction roerty 4.4) in [33] which encodes a maimum rincile) generally fails to hold in the non-diagonal discrete case. However, the limiting rocess can be aroimated by non-symmetric) Markov rocesses, see [6] for a recent construction. Let us finally remark that we do not use any ingredients from robability theory ecet for the quantification of ergodicity via SG and LSI in this aer. Furthermore, since we view our resent contribution as a first ste towards systems which certainly are unrelated to robability theory), we do not further investigate the connection to random walks in the resent aer. Outline of the aer. In Section, we resent the main results of our aer and give a brief sketch of our roof. The roof of the main results and auiliary lemmas are contained in Section 3. Let us mention that in the critical dimension d =, we invoke a Calderón-Zygmund estimate on weighted l -saces on Z d. We give a roof of this estimate, which may be of indeendent interest, in Section 4. Finally, in Section 5 we resent some alications, including a quantitative two-scale eansion and a variance estimate for a reresentative volume aroimation of the homogenized coefficients. Acknowledgements. The authors gratefully acknowledge Feli Otto for suggesting the roblem and for helful discussions. J. B.-A. and S. N. thank the Ma-Planck-Institute 6

7 for Mathematics in the Sciences, Leizig, for its hositality. S. N. was artially suorted by ERC-00-AdG no.6780 AnaMultiScale. Main results and sketch of roof. General framework Discrete functions and derivatives. Let {e i } d i= denote the canonical basis of R d. For a scalar function u : Z d R and a vector field g : Z d R d with comonents g = g,..., g d ) we define the discrete gradient u : Z d R d and negative divergence g : Z d R as follows: u := u,..., d u), g := i u) := u + e i ) u), d i g i, i= where i u) := u e i ) u). We denote by l Z d ),, the sace of functions u : Z d R with u l <, where u l := Z u) ) for < and u d l := su Z d u). Note that and are adjoint: We have the discrete integration by arts formula u) g) = u) g) Z d Z d for all eonents, q such that = + and all functions u q l Z d ) and g l q Z d, R d ). Random coefficients and quantitative ergodicity. In order to describe random coefficients, we endow Ω with the roduct toology induced by R d d and denote by C b Ω) the set of continuous functions ζ : Ω R that are uniformly bounded in the sense that ζ := su ζa) <. a Ω Throughout this work, we consider a robability measure on Ω with resect to the Borelσ-algebra. Following the convention in statistical mechanics, we call this robability measure an ensemble and write for the associated eected value, the ensemble average. We assume that is stationary w. r. t. translation on Z d, i.e. for all Z d, the maing τ : Ω Ω, a a + ) is measurable and measure reserving: ζ : Ω R : ζτ ) = ζ ). Our key assumtion is that is quantitatively ergodic where the ergodicity is quantified through either LSI or SG. To be recise, we make the following definitions: Definition Definition in [34]). We say that satisfies the LSI with constant ρ > 0 if ζ log ζ ) osc ζ ρ ζ. 8) a) for all ζ C b Ω). 7 Z d

8 Here the oscillation of a function ζ C b Ω) is defined by taking the oscillation over all ã Ω that coincide with a outside of Z d, i.e. osc ζa) := su{ζã) ã Ω s.t. ãy) = ay) y } a) inf{ζã) ã Ω s.t. ãy) = ay) y }. 9) The continuity assumtion on ζ ensures that the oscillation is well-defined. A weaker form of quantitative ergodicity is the SG which is defined as follows. Definition. We say that satisfies the SG with constant ρ > 0 if ϕ ϕ ) ) ϕ ρ for all ϕ C b Ω). The SG 0) is automatically satisfied if LSI 8) holds, which may be seen by eanding ζ = +ɛϕ in owers of ɛ. Moreover, LSI and SG are satisfied in the case of indeendently and identically distributed coefficients, i.e. when is the Z d -fold roduct of a robability measure on Ω 0, cf. [34, Lemma ]. We refer to [8] for a recent eosition on LSI and to [0] for a systematic alication of SG to stochastic homogenization. Z d osc a) 0). Main results Throughout this aer the modified corrector φ T is defined as the unique bounded solution to ), see Lemma below for details. Our first result yields boundedness of the finite moments of φ T. Theorem. Assume that is stationary and satisfies LSI 8) with constant ρ > 0. Then the modified corrector defined via ) satisfies φ T ) + ξ Cd, λ,, ρ) ξ ) for all Z d, < and T. Here and throughout this work, Cd, λ,, ρ) stands for a constant which may change from line to line and that only deends on the eonent, the LSI-constant ρ, the elliticity ratio λ and the dimension d. As already mentioned earlier, the lower bound for T is arbitrary and may be relaced by any other constant greater than. The second result establishes moment bounds on the corrector itself. More recisely, we establish control of moments of φ T by moments of φ T. As oosed to Theorem, we just need to assume that the ensemble satisfies SG, i.e. Definition. Theorem. Assume that is stationary and satisfies SG 0) with constant ρ > 0. There eists 0 = 0 d, λ) such that the the modified corrector defined via ) satisfies φ T ) Cd, λ,, ρ) { φ T ) + ξ log T ) for d =, ) for d >, for all Z d, 0 and T. 8

9 By letting T, we obtain the following estimate for the unmodified) corrector. Corollary. Assume that is stationary and satisfies LSI 8) with constant ρ > 0. Then: a) In dimensions d there eists a unique measurable function ϕ : Ω Z d R that solves 5) for -almost every a Ω and a) ϕ satisfies the anchoring condition ϕa, 0) = 0 for -almost every a Ω, a) ϕ is stationary in the sense of 6) and ϕ) = 0 for all Z d, a3) ϕ) < for all Z d and <. b) In dimensions d > there eists a unique measurable function φ : Ω Z d R that solves 5) for -almost every a Ω, and b) φ is stationary in the sense of 6), b) φ) < for all Z d and <. Remark. The anchored corrector ϕ defined in Corollary a) has already been considered in the seminal works by Paanicolaou and Varadhan [39] and Kozlov [8]. In fact, for eistence and uniqueness which can be roved by soft arguments only a) and a) are required. The new estimate a3) follows from Theorem in the limit T. Note that a3) imlies by a short ergodicity argument) sublinearity of the anchored corrector in the sense that for -almost every a Ω. lim ma ϕa, ) R R R Eistence, uniqueness and moment bounds of the stationary corrector φ defined in Corollary b) have been obtained in the case of diagonal coefficients in [4], see also [0]. Note that the anchored corrector ϕ can be obtained from φ via ϕ, a) := φa, ) φa, 0), and, as elained in the discussion below [3, Corollary ], the moment bound b) imlies that = 0 θ 0, ] : ϕa, ) lim ma = 0 R R R θ for -almost every a Ω. Remark. Instead of the modified corrector, one might consider the eriodic corrector which in the stochastic contet is defined as follows: For L N let Ω L := { a Ω : a + Lz) = a for all z Z d } denote the set of L-eriodic coefficient fields. In the L-eriodic case, one considers the corrector equation 5) together with an L-eriodic ensemble, i. e. a stationary robability measure on Ω L. In that case, equation 5) admits a unique solution φ L with [0,L) Z) φ d L ) = 0 for all a Ω L. The L-eriodic versions of LSI and SG are 9

10 obtained by relacing the sum Z in 8) and 0) by d [0,L) Z). With these mod- d ifications, Theorem and Theorem etend to the L-eriodic case with L = T since the cut-off term involving T effectively restricts the equation to a domain of side length T ). In articular, if the L-eriodic ensemble satisfies an L-eriodic LSI with constant ρ > 0, then the L-eriodic corrector satisfies for all < { φ log L) for d =, L otherwise. The roof follows along the same lines and can easily be adated. For estimates on the eriodic corrector φ L in the case of diagonal coefficients, see [0]..3 Sketch of roof of Theorem Theorem is relatively straight-forward to rove. We simly follow the aroach develoed in [34] and use the LSI 8) of Definition to ugrade a lower order L Ω)-bound to a bound in L Ω). Note that by stationarity of and φ T, see 6), it suffices to rove the estimates ) at = 0. The lower order bound φ T 0) + ξ Cd, λ) ξ, cf. 50), follows from a simle energy argument, i.e. an L -estimate obtained by testing the equation for φ T with φ T itself. The integral here is the ensemble average and not the sum over Z d ; this is ossible thanks to stationarity of φ T. For details, we refer to Ste in the roof of Theorem. This bound is then ugraded via the following consequence of LSI 8): ) φ T 0) + ξ Cd,, ρ, δ) φ T 0) + ξ + δ osc φ0) a) for all δ > 0, where we have imlicitly taken the oscillation of the vector φ T comonentwise. This reverse Jensen inequality is the content of Lemma 5 below. Net, we need an eression for osc a) φ T. In Lemma 3 we will show that the resonse to a variation at in the coefficient field is given via the Green s function G T as: Z d osc jφ T a; 0) + ξ j ) Cd, λ) G T a; 0, ) φ T a; ) + ξ, a) where G T is the Green s function associated to ), see Definition 3. Throughout this work, G T, y) = y G T, y) R d d denotes the mied derivative and we use the sectral norm on R d d. The above estimate on the oscillation then yields osc φ0) ) ) Cd, λ, ) G T a; 0, ) φ T a; ) + ξ a) Z d Z d Cd, λ, ) φ T a; 0) + ξ, 0

11 where in Ste of the roof of Theorem, we will obtain the last inequality from stationarity and the energy estimate 6), i.e. Z d G T, y) Cd, λ), which holds in any dimension d..4 Sketch of roof of Theorem By stationarity of and φ T, it suffices to rove ) at = 0. In contrast to Theorem, the roof of Theorem only requires the weaker ergodicity assumtion SG of Definition, which we will use in form of φt 0) ) ) C, ρ) osc φ T 0), a) see Lemma 6 below. Again, we require an estimate on the oscillation, which we shall obtain in Lemma 3 and which yields Z d osc φ T a; 0) Cd, λ) G T a; 0, ) φ T a; ) + ξ. a) This will be substituted into the above SG-tye inequality. In contrast to the roof of Theorem, where a simle l -estimate of G T sufficed, we will see that we require a bound on G T including weights: In Lemma 4, we show that { G T a; 0, ) q log T for d =, ω q ) Cd, λ, q) for d > Z d for all q close enough to, and weight ω q given by { + ) q ) + T q + ) 4q ) for d =, ω q ) := + ) dq ) for d >. The case d > is relatively straight-forward and follows by testing the equation with weights and alying Hardy s inequality. The case d = is critical for this estimate and we will rove it by reducing the roblem via a erturbation argument to the constantcoefficient case; this aroach involves a Helmholtz rojection and is insired by the work [4]. To make it rigorous, we require a Calderón-Zygmund estimate in discrete weighted saces which may be of indeendent interest and which is roved in Section 4. With this estimate at hand, we may smuggle in the weight ω q and aly Hölder s inequality with q and large dual eonent to obtain ) G T a; 0, ) φ T a; ) + ξ Z d as long as is large enough such that ω q ) <. Cd, λ, q) φ T a; ) + ξ { log T for d =, for d >

12 3 Auiliary results and roofs In this section we first resent and rove some auiliary results and then turn to the actual roofs of our main results. We start in Section 3. with the definition of the modified corrector and rove its eistence and some continuity roerties. This invokes the ellitic Green s function, which we introduce in the same section. Section 3. and Section 3.3 contain the two key ingredients of our aroach: In Section 3., we rove estimates on the oscillation of the corrector and estimates on the gradient of the Green s function; in Section 3.3, we revisit LSI and SG, which quantify ergodicity and are the only ingredients from robability theory in our aroach. Finally in Sections 3.4 and 3.5, we resent the roofs of Theorems and. 3. Well-osedness of the modified corrector We define the modified corrector φ T : Ω Z d R as the unique bounded solution to ), i.e. for each a Ω, we require φ T a, ) : Z d R to solve ) and to be bounded, see Lemma for details. Note that this definition is ointwise in a Ω and does not invoke any robability measure on Ω. This is in contrast to what is tyically done in stochastic homogenization e.g. in the seminal work [39], where φ T is unambigously defined through an equation on the robability sace L Ω)). We ot for the nonrobabilistic definition, since later we need to estimate the oscillation in a of φ T, which is most conveniently done when φ T is defined for all a Ω and not only -almost surely. However, since the right-hand side of ) is only in l Z d ), it is not clear a-riori whether ) admits a bounded solution. To settle this question we consider the ellitic Green s function G T : Ω Z d Z d R and rove integrability of G T in Lemma below. The latter then imlies eistence of φ T together with some continuity roerties, see Lemma below. Definition 3 Green s function). Given a Ω and y Z d, the Green s function G T a;, y) associated to equation ) is the unique solution in l Z d ) to T G T a;, y) + a G T a;, y)) = δ y) in Z d, 3) where δ : Z d {0, } denotes the Dirac function centered at 0. Equation 3) can also be eressed in its weak formulation: For all w l Z d ) we have that G T a;, y)w) + w) a) G T a;, y) = wy). 4) T Z d Z d It immediately follows from the unique characterization of G T Green s function is stationary: through 3) that the G T a, + z, y + z) = G T a + z),, y). 5) Furthermore it is symmetric in the sense that G T a; y, y) = G T a t ; y, y ), 6)

13 where a t denotes the transose of a in R d d. This can be seen from alying 4) to w) = G T a t ;, y ), yielding the reresentation G T a t ; y, y ) = T G T a t ;, y )G T a;, y) + G T a t ;, y ) a) G T a;, y). On the other hand, choosing w) = G T a;, y) in the definition for G T a t ;, ) shows G T a; y, y) = T G T a;, y)g T a t ;, y ) + G T a;, y) a t ) G T a t ;, y ). By definition of the transose a t, this shows G T a; y, y ) = G T a t ; y, y) and hence 6). The Green s function is useful since by linearity it encodes all the information for the solution u to the equation T u + a u) = f in Z d. 7) Indeed, testing ) with G T a;, y) and integrating by arts formally yields ua; ) = y Z d G T a;, y)fy). 8) Of course, to make sense of this for f = aξ) l Z d ), we need G T in l Z d ). On the other hand, the definition of the Green s function only yields G T, y) l Z d ) but this is not enough to establish well-osedness of ). It is not difficult to establish that G T, y) = T for all y Z d and a Ω but without the maimum rincile, G T may be negative and it does not follow that G T is in l Z d ). Therefore we need another argument to establish well-osedness of ). This is rovided by the following lemma, which shows eonential decay of G T and in articular that G T is in l Z d ). Lemma. There eist a large constant C = Cd, λ, T ) < and a small constant δ = δd, λ, T ) > 0, both only deending on d, λ and T, such that ) G T a;, y) + G T a;, y) e δd,λ,t ) y Cd, λ, T ) Z d for all a Ω and y Z d. Since we could not find a suitable reference for this estimate in the discrete, non-symmetric case, we resent a roof in the aendi. The roof is essentially done by testing with e δ this is also known as Agmon s ositivity method []). In the discrete setting this is insired by [9, Proof of Lemma 3]. With this result at hand, we can rovide wellosedness of the modified corrector φ T. In addition to well-osedness, Lemma allows us to deduce φ T 0) = φ T a; 0) C b Ω), which is necessary for the alication of LSI 8) and SG 0) to φ T. Lemma Modified corrector). For all a Ω the modified corrector equation ) admits a unique bounded solution φ T a; ) l Z d ). The so defined modified corrector φ T : Ω Z d R satisfies φ T, ) C b Ω) for all Z d, and φ T a; ) CT, λ, d) ξ for all a Ω and all Z d. 9) 3

14 Furthermore, φ T is stationary, i.e. φ T a; + z) = φ T a + z); ) for all a Ω and all, z Z d. 0) Proof. Ste. Eistence and uniqueness of φ T : In this ste, we argue that for arbitrary f l Z d ) equation 7) admits a unique solution u and u can be reresented as in 8). The eistence and uniqueness of φ T then follows by setting f := aξ). For the argument, note that by Lemma we have G T a;, y) l Z d ). Hence, for every f l Z d ), equation 8) defines a function ua; ) l Z d ) that solves 7). For the uniqueness, let ũ l Z d ) solve 7). Testing 7) with G T a t ;, ) yields G T a t ; y, )fy) = G T a t ; y, ) a ))) T + ũy) y Z d y Z d = ) T + a t ) G T a t ; y, )ũy) y Z d = y Z d δ y)ũy) = ũ). By symmetry the left-hand side is equal to y Z d G T a;, y)fy) = ua; ) and thus ua; ) = ũ ) follows. Ste. Argument for 9) and 0): The stationarity roerty 0) directly follows from uniqueness and the stationarity of the oerator and the right-hand side aξ). We turn to estimate 9). By the Green s reresentation 8), which is valid by Ste, and an integration by arts ossible since G T, ) l Z d )), we have φ T a; ) = y Z d y G T a;, y) ay)ξ. We smuggle in the eonential weight from Lemma, use uniform elliticity and the Cauchy-Schwarz inequality to get φ T a; ) ) ) y G T a;, y) e δ y ay)ξ e δ y y Z d y G T a;, y) e δ y e δ y y Z d y Z d where δ > 0 is given in Lemma. By symmetry, cf. 6), and Lemma, the right-hand side is bounded by Cd, λ, T ) ξ and 9) follows. Ste 3. Argument for φ T ; ) C b Ω): Thanks to 9), we only need to show that φ T a; ) is continuous in a. Furthermore, by stationarity, cf. 0), it suffices to consider φ T a; 0). Now, consider a sequence a n Ω that converges to some a Ω in the roduct toology. We need to show that φ T a n ; 0) φ T a; 0). To that end, consider the function ψ n ) := φ T a n ; ) φ T a; ), ξ, 4

15 which can be characterized as the unique bounded solution to Hence, by Ste we have T ψ n + a n ψ n ) = a a n ) φ T a, ) + ξ)) in Z d. ψ n 0) = y Z d y G T a n ; 0, y) ay) a n y)) φ T a, y) + ξ), and thus Lemma and the result of Ste yield ) ψ n 0) su su y Z d a Ω CT, λ, d) φ T a, y) + ξ y G T a n ; 0, y) e δ y y Z d e δ y ay) a n y). y Z d e δ y ay) a n y) y Z d Since a n a in the roduct toology, i.e. a n y) ay) for all y Z d, the right-hand side vanishes as n by dominated convergence. 3. Oscillations and Green s function estimates In this section, we estimate the oscillation of the corrector and its gradient, see Lemma 3 below, and establish estimates on the gradient of the ellitic Green s functions, see Lemma 4 below. These bounds are at the core of our analysis. Indeed, the roofs of Theorem and Theorem start with an alication of quantitative ergodicity: In Theorem, the LSI 8) in form of Lemma 5 is alied to ζ = j φ T 0)+ξ j, while in Theorem, the SG 0) in form of Lemma 6 is alied to ζ = φ T 0). Hence we require estimates for osc a) j φ T a; 0) + ξ j ) and osc a) φ T a; 0). Following [4], these eressions are related to the ellitic Green s function: Lemma 3. For all T > 0, a Ω, Z d and j =,..., d we have osc φ T a; 0) Cd, λ) G T a; 0, ) φ T a; ) + ξ, a) osc jφ T a; 0) + ξ j ) Cd, λ) G T a; 0, ) φ T a; ) + ξ. a) a) b) Proof. Let a Ω and Z d be fied. As in the definition of the oscillation, let ã Ω denote an arbitrary coefficient field that differs from a only at, i.e. ãy) = ay) for all y. We consider the difference φ T ã; ) φ T a; ). Equation ) yields T φ T ã; ) φ T a; )) + ã ) φ T ã; ) φ T a; ) ) = a ã) ) φ T a; ) + ξ) ) 5

16 and consequently the Green s function reresentation 4) yields φ T ã; y) φ T a; y) = G T ã; y, ) a) ã)) φ T a; ) + ξ) ) for all y Z d. In articular, taking the gradient w. r. t. y j and then setting y = yields j φ T ã; ) j φ T a; ) j G T ã;, ) φ T a; ) + ξ since a, ã Ω are uniformly bounded. In view of 6), the mied derivative of G T is bounded by λ and we obtain Echanging a and ã in ) yields j φ T ã; ) j φ T a; ) λ φ T a; ) + ξ. 3) φ T a; y) φ T ã; y) = G T a; y, ) ã) a)) φ T ã; ) + ξ). 4) We take the absolute value to obtain φ T a; 0) φ T ã; 0) G T a; 0, ) φ T ã; ) + ξ. On the right hand side, we lug in 3) to obtain φ T a; 0) φ T ã; 0) Cd, λ) G T a; 0, ) φ T a; ) + ξ. Since ã) was arbitrary, it follows that osc φ T a; 0) Cd, λ) G T a; 0, ) φ T a; ) + ξ, a) which is recisely the claimed identity a). in 4) yields Taking the gradient with resect to y j j φ T a; y) j φ T ã; y) = y,j G T a; y, ) ã) a)) φ T ã; ) + ξ). We take the absolute value and insert 3) to obtain j φ T a; y) j φ T ã; y) Cd, λ) y,j G T a; y, ) φ T a; ) + ξ. and b) follows. In view of a) and b) it is natural that integrability roerties of G T are required. Net to quantitative ergodicity, these Green s function estimates are the second key ingredient in our aroach. For Theorem, which invokes b), a standard l -energy estimate for G T suffices, see 6). For Theorem, which invokes a), some more regularity of the Green s function is required. We need a satially weighted estimate on the gradient G T that is uniform in a Ω. To this end, as announced in Section.4, we define a weight { + ) q ) + T q + ) 4q ) for d =, ω q ) := 5) + ) dq ) for d >, for every q and T. 6

17 Lemma 4. There eists q 0 > only deending on λ and d such that y,j G T, y) λ, j =,..., d, 6) Z d { G T a;, 0) q log T for d =, ω q ) Cd, λ) 7) for d > Z d for all q q 0. Lemma 4 establishes a weighted l q -estimate on the gradient G T of the Green s function. For the alication, it is crucial that the integrability eonent q is larger than. The weight is chosen in such a way that the estimate remains valid for the constant coefficient Green s function G 0 T ) := G T ;, 0) where we use the symbol to denote the identity in R d d ) whose gradient behaves as G 0 T ) Cd) + ) d + ) e c 0 T for some generic constant c 0 > 0, which can easily be deduced from the well-known heat kernel bounds on the gradient of the arabolic Green s function for lack of a better reference, we refer to [5, Theorem.] in the secial case of a measure concentrating on a) = ) along the lines of [36, Proosition 3.6]. With this bound at hand, the definition of the weight 5) yields { G 0 T ) q log T for d =, ω q ) Cd, q) 9) for d > Z d for all q >. Hence, Lemma 4 says that the variable-coefficient Green s function ehibits on a satially averaged level) the same decay roerties as the constant-coefficient Green s function. In the diagonal, scalar case, Lemma 4 is a consequence of [4, Lemma.9] and can also be derived from the weighted estimates on the arabolic Green s function in [0, Theorem 3]. Although the arguments in [4, 0] rely on scalar techniques, Lemma 4 also holds in the case of systems. Indeed, our roof relies only on techniques which are also available for systems. The roof will be slit into three arts: First we will rovide a simle argument for 6) valid in all dimensions. Then we will rove 7) in d > dimensions. The hardest art is the roof of 7) if d = since this is the critical dimension. Proof of 6). An alication of y,j to 4) yields the following characterization for y,j G T a;, y) T y,j G T a;, y)w) + Z d Z d w) a) y,j G T a;, y) = j wy) for all w l Z d ). Taking w ) := y,j G T, y) l Z d ) yields T y,j G T, y) + Z d Z d y,j G T, y) a) y,j G T, y) = j j G T y, y), 7 8)

18 where j j G T y, y) =,j y,j G T, y) =y. The first term on the l. h. s. is ositive and elliticity yields λ ) y,j G T, y) j j G T y, y) y,j G T, y). Z d Z d Thus 6) follows. Proof of 7) in d > dimensions. Ste. A riori estimate: We rove G T 0, 0) + G T, 0) Cd, λ). 30) The weak form of 4) with ζ = G T, 0) and elliticity immediately yield 0 λ G T, 0) G T 0, 0), in articular G T 0, 0) 0. Now a Sobolev embedding in d > with constant Cd) yields ) d G T 0, 0) G T, 0) d d d ) Cd) G T, 0) Cd, λ) GT 0, 0). The Sobolev embedding is readily obtained from its continuum version on R d via a linear interolation function on a triangulation subordinate to the lattice Z d. Hence G T 0, 0) Cd, λ) and 30) follows. Ste. A bound involving weights: In this ste we show that there eists α 0 d) > 0 such that + ) α G T, 0) Cd) + ) α G T, 0) 3) for all 0 < α α 0. Note that both sides are well-defined for G T.) We start by recalling Hardy s inequality in R d if d > : ˆRd f ) ˆ d f d d R d for all f H R d ). A discrete counterart can be derived by interolation w. r. t. a triangulation subordinate to the lattice and yields + ) α G T, 0) Cd) + ) α G T, 0)). 3) The discrete Leibniz rule i fg)) = f + e i ) i g) + g) i f) yields i + ) α G T, 0)) = + e i + ) α i G T, 0) + G T, 0) i + ) α. 8

19 By the mean value theorem we obtain the simle inequality a α b α αa α +b α ) a b for all a, b 0 and we trivially have that + ) + e + + ). The choice a = + e + and b = + thus yields i + ) α 3α + ) α for all 0 α. Summation over i =,..., d and the discrete Leibniz rule above consequently yield + ) α G T, 0) ) ) Cd) + ) α G T, 0) + α + ) α G T, 0) for any 0 α. We substitute this estimate in Hardy s inequality 3) and take α = α 0 d) small enough to absorb the last term into the l. h. s. to obtain 3), i.e. + ) α0 G T, 0) Cd) + ) α 0 G T, 0). Ste 3. Imrovement of Ste to include weights: Now we deduce the eistence of α 0 = α 0 d, λ) > 0 smaller than d and ossibly smaller than α 0 d) from Ste ) such that ) α0 + G T, 0) Cd, λ). 33) To this end, we set w) = + ) α G T, 0) and note that i w) = + ) α i G T, 0) + i + e i + ) α )G T + e i, 0). Hence, 4) yields for y = 0): T + ) α G T, 0) + + d G T + e i, 0) i + ei + ) α) a ij ) j G T, 0) i,j= + ) α G T, 0) a) G T, 0) = G T 0, 0). 34) As in Ste, we have that i + ) α ) 4α + ) α + e i + ) α. for all 0 α and i =,..., d. Thus 34), elliticity, and Hölder s inequality yield λ + ) α GT, 0) G T 0, 0) + ) Cd)α G T, 0) + ) α 9 G T, 0) + ) α ).

20 We aly the result of Ste with α α 0 d) and then ossibly decrease α further to absorb the second term on the r. h. s. This is ossible for α α 0 d, λ) for some α 0 d, λ) > 0. By Ste, we conclude 33). By the discrete l q l -inequality f l q Z d ) f l Z ), d it follows that ) qα0 + G T, 0) q Cd, λ) for all q >. Hence Lemma 4 holds for d > with ω q defined in 5) as long as dq ) qα 0, i.e. we may take q 0 = d d α 0. Proof of 7) in d = dimensions. Let us remark that the following roof is valid in all dimensions d. However, if d >, we have the simler roof above. Fi T > 0 and a Ω. For convenience, we set G) := G T a;, 0) and G 0 ) := G T ;, 0), 35) λ where denotes the identity in R d d and λ denotes the constant of elliticity from Assumtion. We first introduce some notation. For q < and γ > 0, we denote by l q γ the sace of vector fields g : Z d R d with Likewise we denote by l q ω q g l q γ := Z d g) q + ) γ the sace of vector fields with g l q ωq := Z d g) q ω q ) ) q ) q <. <, with ω q defined by 5). We write H BX) for the oerator norm of a linear oerator H : X X defined on a normed sace X. Ste. Helmholtz decomosition: We claim that the gradients of the variable coefficient Green s function G and of the constant coefficient Green s function G 0 from 35) are related by Id +Ha) G = λ G 0 36) where a = λa, H := L denotes the modified Helmholtz rojection, L := λ + T, and Id denotes the identity oerator. Here and in the following, we tacitly identify a with the multilication oerator that mas the vector field g : Z d R d to the vector field ag)) := a)g). Moreover, since G is integrable in the sense of Lemma, the oerators L, and thus H and Id +Ha) are bounded linear oerators on l Z d ) res. l Z d, R d )) and the weighted saces discussed in Ste below. Identity 36) may be seen by aealing to 3) satisfied by G and the equation LG 0 = δ satisfied by G 0 : Id +Ha) G = G + λ L a G L G = G + λ L δ ) T G L L λ ) G T = λ L δ = λ G 0. 0

21 Ste. Invertibility of Id +Ha) in a weighted sace: In this ste, we rove that there eists q 0 = q 0 d, λ) > such that the oerator Id +Ha) : l q ω q l q ω q is invertible and for all q q 0 We slit the roof into several sub-stes. Id +Ha) Bl q ωq ) Cd, λ) 37) Ste a. Reduction to an estimate for H: We claim that it suffices to rove the following statement. There eists q 0 = q 0 λ) > such that { } ma H Bl q ), H q Bl q 4q 4 ) λ λ) for all q q 0. Our argument is as follows: We only need to show that 38) imlies that 38) Ha Bl q ωq ) λ, 39) since then Id +Ha) can be inverted by a Neumann-series. Since the Bl q ωq )-norm is submultilicative, inequality 39) follows from H Bl q ωq ) λ λ) and a Bl q ωq ) λ. 40) We start with the argument for the second inequality in 40). Thanks to ), we have for all a 0 Ω 0 and v R d : λa 0 )v = v λa 0 ) t λa 0 ))v = λ a 0 v v a0 + a t 0 v + v = λ a 0 v v a 0 v + v ) λ v λ v + v = λ) v, which shows 40) by definition of the sectral) oerator norm. Regarding the first inequality in 40), we note that q l q ωq can been seen by recalling definition 5). Hence, H q Bl q ωq ) = and 40) follows. su g l q ωq ma Hg q l q q + T q Hg q l q 4q 4 { H q Bl q H q q ), Bl q 4q 4 ) { = ma H q Bl q H q q ), Bl q 4q 4 ) } ) su g l q ωq } 38) < = q l q q g q l q q λ λ) ) q, + T q q, as l q 4q 4 ) + T q g q l q 4q 4

22 Ste b. Proof of 38): A standard energy estimate yields H Bl R d,z d )). 4) Indeed, given g [l Z d )] d, we have that Hg = u where u solves λ u + T u = g. Testing with u yields u l Z d ) g l Z d ) which is just another way of writing 4). In the following we rove the desired inequality 38) by comle interolation of Bl R d, Z d )) = Bl 0) with Bl γ) for suitable and γ. In Proosition below in Section 4) we rove a Calderón-Zygmund-tye estimate for H in weighted saces and obtain H Bl γ ) < for all and 0 γ < min{ ), }. 4) Fi such and γ and 0 < θ <. A theorem due to Stein and Weiss [7, Theorem 5.5.] that also holds in the discrete setting yields H Bl γ ) H θ Bl γ) H θ Bl ), if γ = θ)γ. 43) Likewise the classical Riesz-Thorin theorem [7, Theorem..] yields H Bl γ ) H θ Bl γ) H θ Bl γ), if = θ + θ. 44) In articular, the ma, γ) H Bl γ ) is continuous at, 0): Given ɛ > 0, we use 44) with γ = 0 to find > such that H Bl ) + ɛ. Then we aly 43) to find γ > 0 such that ma{ H Bl, H γ ) Bl )} + ɛ. Hence, we have H Bl γ) + ɛ for γ the corner oints, γ) of the square [, ] [0, γ ]. By 44) res. 43), we may always decrease either res. γ while achieving the same bound. Consequently we have that H Bl γ ) + ɛ for all, γ) [, ] [0, γ ]. In articular, letting ɛ = λ > 0, λ) there eists q 0 > such that H q λ and the same bound for H Bl 0 q 0 ) λ) Bl q 0 ). 4q 0 4 By monotonicity in the eonent, estimate 38) follows for all q q 0. This comletes the argument of Ste. Ste 3. In this last ste, we fi d = and derive the bound G) q ω q ) = G q Cλ, q) log T 45) l q ωq for q and ω q as in Ste. The relation 36) and the estimate 37) yield G l q ωq Cλ) G0 l q ωq so that it is enough to consider the constant coefficient Green s function whose behaviour is well-known and is given by cf. 8)) G 0 ) C + ) e λ C T ),

23 where C is a universal constant. Hence by slitting G 0 q into its contributions l q ωq coming from T and > T and using the definition of the weight ω q, we have G 0 q l q ωq = G 0 ) q + ) q + T q + ) 4q 4) C + ) q e q λ C T + ) q + T q + ) 4q 4) Cλ, q) T + ) + Cλ, q) Cλ, q) log T + Cλ, q) Cλ, q) log T + Cλ, q), where we have used that q >. > T > T T ) q 4e λ C T T T q + ) q 4 e q λ C T 3.3 Logarithmic Sobolev inequality and sectral ga revisited The LSI only enters the roof of Theorem in form of the following lemma borrowed from [34]. Lemma 5 Lemma 4 in [34]). Let statisfy LSI 8) with constant ρ > 0. Then we have that ) ) ζ Cδ,, ρ) ζ + δ osc ζ 46) a) for any δ > 0, < and ζ C b Ω). This inequality eresses a reverse Jensen inequality and allows to bound high moments of ζ to the eense of some control on the oscillations of ζ. The difference to SG lies in the fact that the imroved integrability roerties of LSI allow us to choose δ > 0 arbitrarily small. In the roof of Theorem, we will aly 46) to the random variables ζ = i φ T 0) + ξ i for i =,..., d. The second moment of i φ T 0) + ξ i will be controlled below, whereas the oscillation was already estimated Lemma 3 and involves the second mied derivatives of G T. In the roof of Theorem, we just require the weaker statement of SG. To be recise, we will use an L -version of SG which is the content of the following lemma. Lemma 6 cf. Lemma in [0]). Let statisfy SG 0) with constant ρ > 0. Then for arbitrary < and ζ C b Ω) it holds that ζ ζ C, ρ) ) ) osc ζ. 47) a) The roof is a combination of the roofs of [0, Lemma ] and [34, Lemma 4]. We resent it here for the convenience of the reader. Z d Z d 3

24 Proof. Without loss of generality assume that ζ C b Ω) satisfies ζ = 0. The triangle inequality and SG 0) yield ζ ζ ζ ) + ζ ρ ) osc a) ζ + ζ ζ. By Young s inequality, we may absorb ζ on the l. h. s. and we obtain that ζ 4 ) osc ρ a) ζ + C) ζ. 48) We insert SG 0), note ζ = 0 and aly Jensen s inequality to obtain that ζ ) ) ) ρ osc ζ ρ osc ζ. 49) a) a) In order to deal with the first term in 48), we note that the elementary inequality t s C)t t s + t s ) for all t, s 0 yields for every two coefficient fields a, ã Ω: ζa) ζã) C) ζa) ζa) ζã) + ζa) ζã) ), where we have in addition used the triangle inequality in form of ζa) ζã) ζa) ζã). Letting ã Ω run over the coefficient fields that coincide with a outside of Z d yields ) ) osc a) ζ C) ζ osc ζ + osc ζ a) a) Consequently we obtain ) osc a) ζ C) ζ ) ζ C) ) osc ζ + C) a) ) ) osc ζ + a) ) ) osc ζ a) ) ) ) osc ζ a) by Hölder s inequality and the discrete l l -inequality. Inserting this estimate as well as 49) into 48) yields ζ C, ρ) ζ ) ) osc ζ + a) ) ) ) osc ζ. a) Again, we may absorb the factor ζ on the l. h. s. using Young s inequality and thus conclude the roof of Lemma 6. 4

25 3.4 Proof of Theorem Ste. We claim the following energy estimate: φt 0) + ξ Cλ) ξ. 50) To see this, we multily ) with φ T 0) and take the eectation: T φt 0) + φ T 0) a φ T )0) = φ T 0) aξ)0). Thanks to the stationarity of and the stationarity of φ T, cf. 0), we have that φ T 0) w) = = d φ T 0) w i e i ) w i ) ) i= d φ T e i ) φ T 0) ) w i ) = φ T 0) w) i= for all stationary vector fields w : Z d R d. This integration by arts roerty then yields φt 0) + φ T 0) a0) φ T 0) = φ T 0) a0)ξ. T Since the first term on the left-hand side is non-negative, uniform elliticity, cf. ), yields φt 0) λ ξ, and 50) follows from the triangle inequality. Ste. We claim that ) G T 0, ) φ T ) + ξ λ φ T 0) + ξ. 5) We start by alying Hölder s inequality with eonent in sace: ) G T 0, ) φ T ) + ξ We now aly to obtain ) G T 0, ) G T 0, ) φ T ) + ξ. ) G T 0, ) φ T ) + ξ su a Ω ) G T 0, ) G T 0, ) φ T ) + ξ. 5

26 At this stage, we aeal to the stationarity of G T, cf. 5), the stationarity of φ T, cf. 0), and the stationarity of in form of which yields G T 0, ) φ T ) + ξ = G T, 0) φ T 0) + ξ, ) G T 0, ) φ T ) + ξ su a Ω su a Ω ) G T 0, ) G T, 0) φ T 0) + ξ G T 0, ) ) su a Ω G T, 0) ) φ T 0) + ξ. We conclude by aealing to symmetry, cf. 6), and 6). Note that the transosed coefficient field a t satisfies a t Ω. Ste 3. Conclusion: The combination of 5) and b) yields ) ) osc iφ T 0) + ξ i ) Cd, λ) φt 0) + ξ 5) a) for i =,..., d. We now aeal to Lemma 5 with ζ = i φ T 0) + ξ i, i.e. i φ T 0) + ξ i Cδ,, ρ) i φ T 0) + ξ i + δ Z d ) ) osc iφ T 0) + ξ i ). a) On the r. h. s. we insert the estimates 50) and 5) and sum in i =,..., d to obtain after redefining δ) d i= i φ T 0) + ξ i Cd, λ, δ,, ρ) ξ + δ φt 0) + ξ By the equivalence of finite-dimensional norms, it follows again, after redefining δ) φ T 0) + ξ Cd, λ, δ,, ρ) ξ + δ φt 0) + ξ. By choosing δ =, we may absorb the second term on the r. h. s. into the l. h. s. which comletes the roof. 3.5 Proof of Theorem As a starting oint, we aly SG in its -version Lemma 6: We aly this inequality with ζ = φ T 0). Since φ T 0) = 0 as can be seen by taking the eectation of ) and using the stationarity of and φ T ), estimate 47) yields φ T 0) ρ 6 ) ) osc φ T 0). a).