Mitia Duerinckx

Antoine Gloria

Felix Otto

ROBUSTNESS OF THE PATHWISE STRUCTURE OF FLUCTUATIONS IN STOCHASTIC HOMOGENIZATION

We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the large-scale fluctuations of the field and the flux of the solution operator can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as the pathwise structure of fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treat continuum elliptic (possibly non-symmetric) systems and allow for strongly correlated coefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations.

error estimates and convergence rates) in the simplified framework of the random conductance model. We proved three main results: the pathwise structure of fluctuations, their asymptotic normality, and the identification of the limiting covariance structure. In the present contribution, we focus on the fundamental pathwise aspect of the theory, that is, the accuracy of the two-scale expansion for large-scale fluctuations of the so-called homogenization commutator, and we extend its validity to continuum (non-symmetric) systems with strongly correlated coefficient fields. More precisely, we cover the general setting of coefficient fields satisfying multiscale functional inequalities as introduced in [START_REF] Duerinckx | Multiscale functional inequalities in probability: Concentration properties[END_REF][START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF], and therefore treat all the models considered in the reference textbook [20] on heterogenous materials. We take this as a sign of the robustness of the pathwise structure. Questions regarding the scaling limit of the standard homogenization commutator require more careful probabilistic assumptions and are addressed in the forthcoming contribution [START_REF] Duerinckx | Scaling limit of the homogenization commutator for correlated Gaussian coefficient fields[END_REF] in the case of correlated Gaussian fields (see below for an informal discussion of these results). In [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF], we further explain how this whole theory of fluctuations naturally extends to higher orders. We refer to the introduction of the companion article [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] for a general discussion of the literature on fluctuations in stochastic homogenization (a short discussion of the key pathwise structure is given at the end of this introduction).

Let a be a stationary and ergodic random coefficient field on R d that is bounded in the sense of

|a(x)ξ| ≤ |ξ| for all ξ ∈ R d and x ∈ R d , (1.1)
and satisfies the ellipticity property

ˆRd ∇u • a∇u ≥ λ ˆRd |∇u| 2 for all u ∈ C ∞ c (R d ), (1.2)
for some λ > 0; this notion of functional coercivity is weaker than pointwise ellipticity for systems. Throughout the article we use scalar notation, but no iota in the proofs would change for systems under assumptions (1.1) and (1.2). For all ε > 0, we set a ε := a( • ε ), and for a deterministic vector field f ∈ C ∞ c (R d ) d we consider the random family (u ε ) ε>0 of unique Lax-Milgram solutions in R d (which henceforth means the unique weak solutions in Ḣ1 (R d )) of the rescaled problems

-∇ • a ε ∇u ε = ∇ • f. (1.3)
(The choice of considering an equation on the whole space rather than on a bounded set allows us to focus on fluctuations in the bulk, and avoid effects of boundary layers. The choice of taking a right-hand side (RHS) in divergence form allows to treat all dimensions at once.) It is known since the pioneering work of Papanicolaou and Varadhan [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF] and Kozlov [START_REF] Kozlov | The averaging of random operators[END_REF] that, almost surely, u ε converges weakly (in Ḣ1 (R d )) as ε ↓ 0 to the unique Lax-Milgram solution ū in R d of

-∇ • ā∇ū = ∇ • f, (1.4)
where ā is a deterministic and constant matrix that only depends on the law of a. More precisely, for any direction e ∈ R d , the projection āe is the expectation of the flux of the corrector in the direction e, āe = E [a(∇φ e + e)] , (1.5) where the corrector φ e is the unique (up to a random additive constant) almost-sure solution of the corrector equation in R d , -∇ • a(∇φ e + e) = 0, in the class of functions the gradient of which is stationary, has finite second moment, and has zero expectation. We denote by φ = (φ i ) d i=1 the vector field the entries of which are the correctors φ i in the canonical directions e i of R d .

In [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF], we developed a complete theory of fluctuations in stochastic homogenization for the random conductance model (see also [START_REF] Gu | Scaling limit of fluctuations in stochastic homogenization[END_REF] for heuristic arguments). The key in our theory is to focus on the so-called homogenization commutator of the solution,

a ε ∇u ε -ā∇u ε , (1.6)
and to study its relation to the standard homogenization commutator Ξ := (Ξ i ) d i=1 , where the solution u ε is replaced by a-harmonic coordinates x → x i + φ i (x), Ξ i := a(∇φ i + e i ) -ā(∇φ i + e i ), Ξ ij := (Ξ i ) j .

(1.7)

In the framework of [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF], we showed the following three crucial properties (which we reformulate here in the non-symmetric continuum setting): (I) First and most importantly, the two-scale expansion of the homogenization commutator of the solution

a ε ∇u ε -ā∇u ε -E [a ε ∇u ε -ā∇u ε ] ≈ Ξ i ( • ε )∇ i ū (1.8)
is accurate in the fluctuation scaling in the sense that for all g ∈ C ∞ c (R d ) d and q < ∞,

E ˆRd g • a ε ∇u ε -ā∇u ε -E [a ε ∇u ε -ā∇u ε ] - ˆRd g • Ξ i ( • ε )∇ i ū q 1 q f,g,q ε E ˆRd g • Ξ i ( • ε )∇ i ū q 1 q
, (1.9) up to a |log ε| factor in the critical dimension d = 2. This property is highly nontrivial and is due to the special form of the commutator (1.6). (II) Second, both the fluctuations of the field ∇u ε and of the flux a ε ∇u ε can be recovered through deterministic projections of the fluctuations of the homogenization commutator (1.6), which shows that no information is lost by passing to the commutator. More precisely, the following elementary identities are easily checked, ˆRd g • (∇u ε -∇ū) = - where ā * denotes the transpose of ā. Similarly, the fluctuations of the field ∇φ and of the flux a∇φ of the corrector are also determined by those of the standard commutator Ξ itself: indeed, the definition of Ξ yields -∇ • ā∇φ i = ∇ • Ξ i and a(∇φ i +e i )-āe i = Ξ i + ā∇φ i , to the effect of ∇φ i = -PH Ξ i and a(∇φ i +e i )-āe i = (Id -ā PH )Ξ i in the stationary sense, hence formally,

ˆRd ( P * H g) • (a ε ∇u ε -ā∇u ε ), (1.10
ˆRd F : ∇φ( • ε ) = - ˆRd P * H F : Ξ( • ε ), ˆRd F : a ε (∇φ( • ε ) + Id) -ā = ˆRd P * L F : Ξ( • ε ), (1.12)
where P * H and P * L act on the second index of the tensor field F ; a suitable sense to these identities is given in Corollary 1.

(III) Third, the standard homogenization commutator Ξ is an approximately local function of the coefficients a, which allows to infer the large-scale behavior of Ξ from the large-scale behavior of a itself. This locality property is best seen when formally computing the so-called "vertical" derivatives of Ξ with respect to a: Letting φ * denote the corrector associated with the pointwise transpose field a * , and letting σ * denote the corresponding flux corrector (cf. (2.10)), we obtain (cf. [9, equation (1.10)] and (2.15) below)

∂ ∂a(x) Ξ ij = (∇φ * j + e j ) • ∂a ∂a(x) (∇φ i + e i ) -∇ • φ * j ∂a ∂a(x) (∇φ i + e i ) -∇ • (φ * j a + σ * j ) ∂∇φ i ∂a(x) .
In view of ∂a ∂a(x) = δ(•x), the first right-hand side term reveals an exactly local dependence upon a. The second term is exactly local as well, but since it is written in divergence form its contribution is negligible when integrating on large scales. The only non-local effect comes from the last term due to ∂∇φ ∂a , which is given by the mixed derivative of the Green's function for -∇ • a∇ and thus is expected to have only borderline integrable decay. However, it also appears inside a divergence, hence it is negligible when integrated on large scales.

Let us comment on the structure of fluctuations revealed in (I)-(II). Together with the two-scale expansion (1.9) of commutators, identities (1.10) and (1.12) imply that the fluctuations of ∇u ε , a ε ∇u ε , ∇φ( • ε ), and a ε ∇φ( • ε ) are determined to leading order by those of Ξ( • ε ), with error estimated in a strong norm in probability. We chose to refer to this key property as the "pathwise" structure of fluctuations in analogy with the language of SPDEs in order to emphasize that this result does not only compare probability laws of different objects (possibly constructed on different probability spaces), but compares these objects for the same realizations of the randomness (for the same "paths"), here in form of an error estimate at the level of stretched exponential moments. As emphasized in [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF], besides its theoretical importance, this pathwise structure is bound to affect multi-scale computing and uncertainty quantification in an essential way. This result is indeed of the complexity-reducing type of the central results in homogenization, as it provides a description of fluctuations of a general solution by means of an off-line procedure using the standard commutator Ξ in form of a two-scale expansion. Next, in case of a weakly correlated coefficient field a, we expect from property (III) that Ξ( • ε ) displays the CLT scaling and that ε -d/2 Ξ( • ε ) converges to a white noise; the pathwise structure (I)-(II) then allows to recover the known scaling limit results for the different quantities of interest in stochastic homogenization, as indeed shown in [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] for the random conductance model.

In the present contribution, we focus on the pathwise structure (I)-(II). More precisely, we mainly consider the class of Gaussian coefficient fields with a covariance function that decays at infinity at some fixed (yet arbitrary) algebraic rate (1 + |x|) -β parametrized by β > 0, and we show that properties (I)-(II) still hold for this whole Gaussian class, which illustrates the surprising robustness of the pathwise structure with respect to the large-scale behavior of the homogenization commutator. Indeed, in dimension d = 1 (in which case the quantities under investigation are simpler and explicit1 ), two typical behaviors have been identified in terms of the scaling limit of the standard homogenization commutator Ξ, depending on the parameter β (cf. [START_REF] Bal | Random integrals and correctors in homogenization[END_REF]),

• For β > d = 1: The standard commutator Ξ displays the CLT scaling and its rescaling ε -1 2 Ξ( • ε ) converges in law to a non-degenerate white noise (Gaussian fluctuations, local limiting covariance structure), but the convergence rate is arbitrarily slow as β gets closer to d = 1.

• For 0 < β < d = 1: The suitable rescaling ε -β 2 Ξ( • ε
) converges along a subsequence to a fractional Gaussian field (Gaussian fluctuations, nonlocal limiting covariance structure, potentially no uniqueness of the limit). Note that a different, non-Gaussian behavior may also occur in degenerate cases (cf. [START_REF] Gu | Random homogenization and convergence to integrals with respect to the Rosenblatt process[END_REF][START_REF] Lechiheb | Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization[END_REF] and second item in Remark 2.1). In particular, the pathwise result is shown to hold in both examples whereas the rescaled standard commutator does not necessarily converge to white noise or may even not converge at all. The identification of the scaling limit of the standard commutator is thus a separate question and is addressed in [START_REF] Duerinckx | Scaling limit of the homogenization commutator for correlated Gaussian coefficient fields[END_REF] in all dimensions for the whole range of values of β > 0, combining Malliavin calculus with techniques developed in [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF]. More precisely, this work extends [START_REF] Bal | Random integrals and correctors in homogenization[END_REF] to dimensions d > 1 in the following sense,

• For β > d: The rescaled commutator ε -d 2 Ξ( • ε )
converges in law to a generically nondegenerate white noise.

• For β < d: The rescaled commutator ε -β 2 Ξ( • ε ) converges along a subsequence to a generically non-degenerate fractional Gaussian field. Different limits can indeed be reached in general, unless the covariance function has a self-similar profile at infinity. These results illustrate the fact that the standard commutator Ξ is an approximately local function of the random coefficient field a (cf. (III) above), which essentially allows to relate the scaling limit of the commutator with the scaling limit of the coefficient field itself (as in dimension d = 1). Interestingly, this also shows that the pathwise structure of fluctuations can in general not be reduced to a quantitative joint convergence in law since there might not even be any convergence in law to talk about in the first place.

Although we focus here for shortness on the model case of Gaussian coefficient fields, the arguments that we provide in this contribution are robust enough to cover the general setting of mutiscale functional inequalities introduced and studied in [START_REF] Duerinckx | Multiscale functional inequalities in probability: Concentration properties[END_REF][START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF], and therefore to treat all the models of random coefficient fields considered in the reference textbook [20] on heterogenous materials (see indeed third item of Remark 2.1). This makes the results of this contribution not only of theoretical but also of practical interest.

Let us conclude this introduction with a short discussion of the recent literature concerning (I)-(III); we refer to [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF]Section 1.4] for more detail. The pathwise structure (I)-(II) of fluctuations, which we extend here to the continuum setting with long-range correlations, was first formulated and proved by us in [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] for the random conductance model. A related form of (I)-(II) was conjectured in [START_REF] Gu | Scaling limit of fluctuations in stochastic homogenization[END_REF] within the variational and renormalization framework of [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF][START_REF] Armstrong | The additive structure of elliptic homogenization[END_REF][START_REF] Armstrong | Quantitative stochastic homogenization and large-scale regularity[END_REF], but it has not been made rigorous yet (nor does it not appear in the textbook [START_REF] Armstrong | Quantitative stochastic homogenization and large-scale regularity[END_REF]). A variational quantity related to the standard commutator can be first traced back to [START_REF] Armstrong | Quantitative stochastic homogenization of convex integral functionals[END_REF], whereas its canonical form (1.7) used here was independently introduced in [1, 2] and [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] (there motivated by the seminal works of Murat and Tartar). The locality property of the standard commutator Ξ and its convergence to white noise were first established in [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF] for the random conductance model, and in [START_REF] Armstrong | The additive structure of elliptic homogenization[END_REF][START_REF] Gloria | The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations[END_REF] for the continuum setting with a finite range of dependence assumption, while the case of long-range correlations is first considered in our companion article [START_REF] Duerinckx | Scaling limit of the homogenization commutator for correlated Gaussian coefficient fields[END_REF].

2.

Main results and structure of the proof 2.1. Notation and statement of the main results. For some k ≥ 1 let a be an R k -valued Gaussian random field, constructed on a probability space (Ω, P) (with expectation E), which is stationary and centered, and thus characterized by its covariance function

c(x) := E [a(x) ⊗ a(0)] , c : R d → R k×k .
We assume that the covariance function decays algebraically at infinity in the sense that there exist β, C 0 > 0 such that for all x ∈ R d ,

1 C 0 (1 + |x|) -β ≤ |c(x)| ≤ C 0 (1 + |x|) -β . (2.1)
Given a map h ∈ C 1 b (R k ) d×d , we define a : R d → R d×d by a(x) = h(a(x)), and assume that it satisfies the conditions (1.1) and (1.2) almost surely. We then (abusively) call the coefficient field a Gaussian with parameter β > 0. If a is Gaussian with parameter β, then a is ergodic, hence we have existence and uniqueness of correctors φ and of the homogenized coefficients ā (cf. Lemma 2.2 below). From a technical point of view, we shall rely on (and frequently refer to) results and methods developed in [START_REF] Duerinckx | Multiscale functional inequalities in probability: Concentration properties[END_REF][START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Gloria | Quantitative stochastic homogenization for correlated fields[END_REF].

Throughout the article, we use the notation (... ) (resp.

(... ) ) for ≤ C× (resp. ≥ C×), where the multiplicative constant C depends on d, λ, β, ∇h L ∞ , on the constant C 0 in (2.1), and on the additional parameters "(. . . )" if any. We write ≃ (... ) when both (... ) and (... ) hold. In an assumption, we use the notation ≪ (... ) for ≤ 1 C × for some (large enough) constant C ≃ (... ) 1.

We now define a string of random functionals that encode the fluctuations of the different objects of interest. The notation I is reserved to functionals involving the solution operator, and the notation J to functionals involving correctors; the subscript 0 is reserved to commutators, the subscript 1 to fields, and the subscript 2 to fluxes. We consider the fluctuations of the commutator a ε ∇u ε -ā∇u ε , of the field ∇u ε , and of the flux a ε ∇u ε of the solution to (1.3), as encoded by the (centered) random bilinear functionals

I ε 0 : (f, g) → I ε 0 (f, g), I ε 1 : (f, g) → I ε 1 (f, g), and I ε 2 : (f, g) → I ε 2 (f, g) defined for all f, g ∈ C ∞ c (R d ) d by I ε 0 (f, g) := ˆRd g • (a ε ∇u ε -ā∇u ε -E [a ε ∇u ε -ā∇u ε ]), I ε 1 (f, g) := ˆRd g • ∇(u ε -E [u ε ]), I ε 2 (f, g) := ˆRd g • a ε ∇u ε -E [a ε ∇u ε ] .
Likewise, we consider the fluctuations of the standard commutator Ξ = (a -ā)(∇φ + Id), of the corrector field ∇φ, and of the corrector flux a(∇φ + Id) as encoded by the (centered) random linear functionals J ε 0 :

F → J ε 0 (F ), J ε 1 : F → J ε 1 (F ), and J ε 2 : F → J ε 2 (F ) defined for all F ∈ C ∞ c (R d ) d×d by J ε 0 (F ) := ˆRd F (x) : Ξ( x ε ) dx, J ε 1 (F ) := ˆRd F (x) : ∇φ( x ε ) dx, J ε 2 (F ) := ˆRd F (x) : a ε (x)(∇φ( x ε ) + Id) -ā dx.
We first prove the following boundedness result for J ε 0 , establishing the suitable βdependent scaling for the fluctuations of the homogenization commutator (see also [START_REF] Gloria | Quantitative stochastic homogenization for correlated fields[END_REF]Theorem 1]). More precisely, in the spirit of (III), this shows that large-scale averages of the standard commutator have the same scaling π * ( 1 ε ) -1/2 as large-scale averages of the coefficient field a itself (cf. [7, Proposition 1.5]); in the case of integrable correlations, this is the CLT scaling ε d/2 . Proposition 1 (Fluctuation scaling). Let d ≥ 1, assume that the coefficient field a is Gaussian with parameter β > 0, define π * : R + → R + by

π * (t) :=    (1 + t) β : β < d, (1 + t) d 1 log(2+t) : β = d, (1 + t) d : β > d, (2.2)
and define the rescaled functional

J ε 0 := π * ( 1 ε ) 1 2 J ε 0 . For all 0 < ε ≤ 1, F ∈ C ∞ c (R d ) d×d , 0 < p -1 ≪ 1, and α > d-(β∧d) 2 + d p-1 2p , we have | J ε 0 (F )| ≤ C ε,F α,p w α 1 F L 2p + 1 β≤d [F ] 2 L p , (2.3)
where w 1 (z

) := 1 + |z|, [F ] 2 (x) := ( ffl B(x) |F | 2 ) 1 2
, and where C ε,F α,p is a random variable with stretched exponential moments: there exists γ 1 ≃ 1 such that

sup 0<ε<1 E exp 1 C α,p (C ε,F α,p ) γ 1 ≤ 2
for some (deterministic) constant C α,p ≃ α,p 1. ♦

Our next main result establishes the accuracy of the two-scale expansion error for largescale averages of the homogenization commutator in the suitable fluctuation scaling. This error is encoded by the following (centered) random bilinear functional,

E ε (f, g) := ˆRd g • a ε ∇u ε -ā∇u ε -E [a ε ∇u ε -ā∇u ε ] - ˆRd g • Ξ( • ε )∇ū = I ε 0 (f, g) -J ε 0 (∇ū ⊗ g) = I ε 0 (f, g) -J ε 0 ( PH f ⊗ g).
(2.4) More precisely, we show that the typical scaling of this error

I ε 0 (f, g) -J ε 0 ( PH f ⊗ g) is an order εµ * ( 1 ε
) smaller than the typical scaling of large-scale averages of the commutator

J ε 0 ( PH f ⊗g) itself.
In view of the generic non-degeneracy result in [START_REF] Duerinckx | Scaling limit of the homogenization commutator for correlated Gaussian coefficient fields[END_REF], this can be viewed as a relative error estimate. This property summarizes the pathwise structure of fluctuations and is the key part of our theory.

Theorem 1 (Pathwise structure of fluctuations). Let d ≥ 1, assume that the coefficient field a is Gaussian with parameter β > 0, let π * be defined in (2.2), and define µ * :

R + → R + by µ * (r) :=                      1 : β > 2, d > 2, log 1 2 (2 + r) : β > 2, d = 2, √ 1 + r : β > 1, d = 1, log 1 2 (2 + r) : β = 2, d > 2, log(2 + r) : β = 2, d = 2, √ 1 + r log 1 2 (2 + r) : β = 1, d = 1, (1 + r) 1-β 2 : β < 2, d ≥ 2, or β < 1, d = 1.
ε := π * ( 1 ε ) 1 2 E ε . For all 0 < ε ≤ 1, f, g ∈ C ∞ c (R d ) d , 0 < p -1 ≪ 1, and α > d-(β∧d) 2 + d p-1
4p , we have

| E ε (f, g)| ≤ εµ * ( 1 ε ) C ε,f,g α,p µ * ∇f L 4 w α 1 g L 4p + µ * ∇g L 4 w α 1 f L 4p + 1 β≤d µ * ∇f L 2 g L 2 ∩ L 2p + µ * ∇g L 2 f L 2 ∩ L 2p , (2.6)
where C ε,f,g α,p is a random variable with stretched exponential moments: there exists γ 2 ≃ 1 such that

sup 0<ε<1 E exp 1 C α,p (C ε,f,g α,p ) γ 2 ≤ 2 for some (deterministic) constant C α,p ≃ α,p 1. ♦ Remark 2.1.
• The exponents γ 1 and γ 2 in the above results can be made explicit; we do not pursue this direction since the values obtained in the proofs are not expected to be optimal. • The ε-scaling in the above results is believed to be optimal. The rescaling in the definition of J ε 0 and E ε is natural since it precisely coincides with the scaling of largescale averages of the coefficient field a itself. For some non-generic examples, the bound (2.3) may however overestimate the variance. In dimension d = 1, one may indeed construct explicit Gaussian coefficient fields a such that fluctuations of the homogenization commutator J ε 0 are of smaller order than what (2.3) predicts [START_REF] Taqqu | Convergence of integrated processes of arbitrary Hermite rank[END_REF][START_REF] Gu | Random homogenization and convergence to integrals with respect to the Rosenblatt process[END_REF][START_REF] Lechiheb | Convergence of random oscillatory integrals in the presence of long-range dependence and application to homogenization[END_REF], in which case the suitable rescaling of J ε 0 has a non-Gaussian limit. In such situations, the pathwise property (2.6) (or its higher-order pathwise version as in [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]) might still provide relevant information. General necessary and sufficient conditions for the sharpness of (2.3) are provided in [START_REF] Duerinckx | Scaling limit of the homogenization commutator for correlated Gaussian coefficient fields[END_REF].

• The proofs of the above results are robust enough to cover the general setting of multiscale functional inequalities introduced in [START_REF] Duerinckx | Multiscale functional inequalities in probability: Concentration properties[END_REF][START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF]. In the case of functional inequalities with oscillation, we may indeed use Cauchy-Schwarz' inequality and an energy estimate to replace the perturbed functions φ and ∇ũ appearing in the representation formula (3.3) below by their unperturbed versions φ and ∇u. This allows to conclude whenever the weight has a superalgebraic decay (see indeed [START_REF] Gloria | Quantitative stochastic homogenization for correlated fields[END_REF]proof of Theorem 4]).

If one is only interested in Gaussian coefficient fields, one may replace the use of functional inequalities by a direct use of the Brascamp-Lieb inequality in terms of Malliavin calculus, which allows to shorten some of the proofs (and improve the norms of the test functions F, f, g), cf. [START_REF] Duerinckx | Higher-order pathwise theory of fluctuations in stochastic homogenization[END_REF]. ♦

In view of the identities (1.10) and (1.12), the above pathwise result implies that the large-scale fluctuations of I ε 0 , I ε 1 , I ε 2 , J ε 1 , and J ε 2 are driven by the fluctuations of J ε 0 in a pathwise sense (see [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF]Corollary 2.4] for details).

Corollary 1 ([9]

). Let d ≥ 2, assume that the coefficient field a is Gaussian with parameter β > 0, let π * and µ * be defined by (2.2) and (2.5), let PH , P * H , and P * L be as in (1.11), and recall the rescaled functionals

I ε i := π * ( 1 ε ) 1 2 I ε i , J ε i := π * ( 1 ε ) 1 2 J ε i , i = 0, 1, 2.
For all ε > 0 and f, g

∈ C ∞ c (R d ) d , we have for all 0 < p -1 ≪ 1 and α > d-(β∧d) 2 + d p-1 4p , | I ε 0 (f, g) -J ε 0 ( PH f ⊗ g)| + | I ε 1 (f, g) -J ε 0 ( PH f ⊗ P * H g)| + | I ε 2 (f, g) + J ε 0 ( PH f ⊗ P * L g)| ≤ εµ * ( 1 ε ) C ε,f,g α,p µ * ∇f L 4 w α 1 g L 4p + µ * ∇g L 4 w α 1 f L 4p + 1 β≤d µ * ∇f L 2 g L 2 ∩ L 2p + µ * ∇g L 2 f L 2 ∩ L 2p ,
where C ε,f,g α,p is a random variable with stretched exponential moments independent of ε as in the statement of Theorem 1. In addition, for all ε > 0 and F ∈ C ∞ c (R d ) d×d , we have almost surely

J ε 1 (F ) = -J ε 0 ( P * H F ), J ε 2 (F ) = J ε 0 ( P * L F ),
where in particular we may give an almost sure meaning to J ε 0 ( P * H F ) and [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF]). Let the coefficient field a be stationary and ergodic. Then there exist two random tensor fields (φ i ) 1≤i≤d and (σ ijk ) 1≤i,j,k≤d with the following properties: The gradient fields ∇φ i and ∇σ ijk are stationary 2 and have finite second moments and vanishing expectations:

J ε 0 ( P * L F ) for all F ∈ C ∞ c (R d ) d×d ,
E |∇φ i | 2 ≤ 1 λ 2 , d j,k=1 E |∇σ ijk | 2 ≤ 4d 1 λ 2 + 1 , E [∇φ i ] = E [∇σ ijk ] = 0. (2.7)
Moreover, for all i, the field σ i := (σ ijk ) 1≤j,k≤d is skew-symmetric, that is,

σ ijk = -σ ikj . (2.8)
Finally, the following equations are satisfied a.s. in the distributional sense on R d ,

-∇ • a(∇φ i + e i ) = 0, (2.9) ∇ • σ i = q i -E [q i ] ,
(2.10)

-△σ ijk = ∂ j q ik -∂ k q ij ,
where q i = (q ij ) 1≤j≤d is given by q i := a(∇φ i +e i ), and where the (distributional) divergence of a tensor field is defined as

(∇ • σ i ) j := d k=1 ∇ k σ ijk . ♦
The proofs of Proposition 1 and Theorem 1 are based on the combination of three main ingredients:

• A sensitivity calculus combined with functional inequalities for Gaussian ensembles [START_REF] Duerinckx | Multiscale functional inequalities in probability: Concentration properties[END_REF][START_REF] Duerinckx | Multiscale functional inequalities in probability: Constructive approach[END_REF]; • The bounds on correctors proved in [START_REF] Gloria | Quantitative stochastic homogenization for correlated fields[END_REF]; • A duality argument combined with the large-scale (weighted) Calderón-Zygmund estimates of [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF]. In the case when the coefficients satisfy a finite range of dependence assumption rather than a functional inequality, we do not have a convenient sensitivity calculus at our disposal, and this first ingredient can be replaced by a semi-group approach that provides a convenient disintegration of scales, cf. [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization II: The case of finite range of dependence[END_REF].

The sensitivity calculus measures the influence of changes of the coefficient field a on random variables X = X(a) via the functional (or Malliavin-type) derivative ∂ fct X(x) = ∂X ∂a (a, x), that is, the L 2 (R d ) d×d -gradient of X wrt a. We recall that this functional derivative is characterized as follows, for any compactly supported perturbation

b ∈ L ∞ (R d ) d×d , ˆRd ∂ fct X(a, x) : b(x) dx := lim t↓0 1 t X(a + tb) -X(a) . (2.11)
This quantity measures the sensitivity of the random variable X = X(a) wrt changes in the coefficient field. This sensitivity calculus is a building block to control the variance and the entropy of X via functional inequalities in the probability space [START_REF] Duerinckx | Multiscale functional inequalities in probability: Concentration properties[END_REF]. A crucial role is played by the parameter β > 0 that characterizes the decay of the covariance function of a, and we define as follows a weighted norm ||| • ||| 2 β on random fields G, depending on β > 0,

|||G||| 2 β := ˆ∞ 1 G 2 ℓ ℓ -β-1 dℓ, (2.12) 2 That is, ∇φi(a; • + z) = ∇φi(a(• + z); •) and ∇σ ijk (a; • + z) = ∇σ ijk (a(• + z); •) a.e. in R d , for all shift vectors z ∈ R d .
where for all ℓ ≥ 1

G 2 ℓ := ℓ -d ˆRd ˆBℓ (z) |G| 2 dz. (2.13)
As shown in [7, Proposition 2.4], in the integrable case β > d, we can drop the integral over ℓ, in which case

|||G||| 2 β ≃ |||G||| 2 := G 2 1 . (2.14)
In these terms, we may formulate the following multiscale logarithmic Sobolev inequality for the Gaussian coefficient field a. In view of (2.14), for β > d, this reduces to the standard logarithmic Sobolev inequality (LSI). The proof is based on a corresponding Brascamp-Lieb inequality (cf. [8, Theorem 3.1]).

Lemma 2.3 ([8]

). Assume that the coefficient field a is Gaussian with parameter β > 0.

Then for all random variables X = X(a),

Ent X 2 := E X 2 log X 2 -E X 2 E log X 2 E |||∂ fct X||| 2 β . ♦
Our general strategy for the proof of Proposition 1 and Theorem 1 consists in estimating the weighted norm (2.12) of the functional derivatives of J ε 0 (F ) and of E ε (f, g). The following lemma provides a useful representation formula for these functional derivatives. This is a continuum version of [START_REF] Duerinckx | The structure of fluctuations in stochastic homogenization[END_REF]Lemma 3.2]. By scaling, it is enough to consider ε = 1, and we write for simplicity J 0 := J 1 0 and E := E 1 . Lemma 2.4 (Representation formulas). Let the coefficient field a be Gaussian with parameter β > 0. For all f ∈ C ∞ c (R d ) d , let u := u 1 denote the solution of (1.3) (with ε = 1), let ū denote the solution of (1.4), and define the two-scale expansion error

w f := u-(1+φ i ∇ i )ū. Then, for all F ∈ C ∞ c (R d ) d×d , ∂ fct J 0 (F ) = (F ij e j + ∇S i ) ⊗ (∇φ i + e i ), (2.15
)

and for all g ∈ C ∞ c (R d ) d , ∂ fct E(f, g) = g j (∇φ * j + e j ) ⊗ (∇w f + φ i ∇∇ i ū) + (φ * j ∇g j + ∇r) ⊗ ∇ū -φ * j ∇(g j ∇ i ū) + ∇R i ⊗ (∇φ i + e i ), (2.16
) where the auxiliary fields S = (S i ) d i=1 , r, and

R = (R i ) d i=1 are the Lax-Milgram solutions in R d of -∇ • a * ∇S i = ∇ • F ij (a * -ā * )e j ,
(2.17)

-∇ • a * ∇r = ∇ • (φ * j a * -σ * j )∇g j , (2.18)
-∇ • a * ∇R i = ∇ • (φ * j a * -σ * j )∇(g j ∇ i ū) , (2.19
-∇ • ã∇z = ∇ • Z =⇒ |∇z(x)| |Z(x)| for all x ∈ R d ,
with ã = a or ā (which even in the homogeneous case ã = ā would only hold after taking suitable Lebesgue norms in view of the Calderón-Zygmund theory). For J 0 (F ), equation (2.17) would then yield the pointwise bound

|∂ fct J 0 (F )| |F |, hence |||∂ fct J 0 (F )||| 2 β ˆ∞ 1 ℓ -d ˆRd ˆBℓ (z) |F | 2 dz ℓ -β-1 dℓ.
To estimate the RHS, assume that F is compactly supported in B R for some R > 0, so that

ℓ -d ˆRd ˆBℓ (z) |F | 2 dz (ℓ d 1 ℓ≤R + R d 1 ℓ>R ) ˆRd |F | 2 ,
which after integration yields, in view of (2.2),

|||∂ fct J 0 (F )||| 2 β ˆR 1 ℓ d-β-1 dℓ + R d ˆ∞ R ℓ -β-1 dℓ ˆRd |F | 2 R d π -1 * (R) ˆRd |F | 2 . Replacing F by ε d F ( • ε ), hence replacing R by 1 ε R, we conclude by LSI, Var π * ( 1 ε ) 1 2 J ε 0 (F ) R F 2 L 2 (R d )
, as claimed in Proposition 1 (with a slightly stronger norm of the test function F ).

We now turn to the two-scale expansion error, for which (2.18) and (2.19) would yield the following pointwise bound, under the simplifying assumptions,

|∂ fct E(f, g)| |g|(|∇w f | + µ * |∇ 2 ū|) + µ * |∇g||∇ū|.
|∂ fct E(f, g)| µ * (|g||∇f | + |∇g||f |).
As above, after rescaling, and using µ

* (| x ε |) µ * (|x|)µ * ( 1 ε ), we conclude by LSI, Var π * ( 1 ε ) 1 2 E ε (f, g) ε 2 µ 2 * ( 1 ε ) µ * ∇g 2 L 4 (R d ) f 2 L 4 (R d ) + µ * ∇f 2 L 4 (R d ) g 2 L 4 (R d )
, as claimed in Theorem 1 (with slightly stronger norms of the test functions). Note that the additional factor ε 2 comes from the gradients ∇f and ∇g in the bound of the functional derivative (which indeed both yield an ε factor by rescaling).

The rigorous proof of Proposition 1 and Theorem 1 amounts to taking care of the fact that the above simplifying assumptions only hold in weaker forms. More precisely:

• Bounds on the correctors only hold for stretched exponential moments and not pointwise (cf. Lemma 2.8 below), so that the bounds on |∂ fct J 0 (F ) and |∂ fct E(f, g)| will not hold pointwise but for stretched exponential moments.

• More importantly, the Helmholtz projection never enjoys pointwise bounds, which must be weakened in two ways. First, for the homogeneous operator -∇ • ā∇, we must resort to the boundedness of the Helmholtz projection in L p spaces for 1 < p < ∞ (Calderón-Zygmund estimates). Second, for the heterogeneous operator -∇ • a∇, regularity theory can only hold on large scales [START_REF] Armstrong | Quantitative stochastic homogenization and large-scale regularity[END_REF][START_REF] Gloria | A regularity theory for random elliptic operators[END_REF], so that Calderón-Zygmund estimates must be locally averaged at some random scale r * (cf. Lemma 2.7); we will have to get rid of this random local average at some point using Hölder's inequality and a small weight (see e.g. the second RHS factor in (2.20)). Finally, when the corrector does not have uniformly bounded moments (that is, when it grows at infinity), we further need to resort to weighted Calderón-Zygmund estimates (cf. Lemma 2.7(c)); see e.g. the weight µ * in the third RHS factor in (2.22).

Estimates on |||∂ fct J 0 (F )||| β and |||∂ fct E(f, g)||| β are obtained in the following two technical propositions. As above, we prove the estimate for ε = 1 and then argue by scaling. Since we need some flexibility in the weights, some estimates involve a parameter R ≥ 1. This parameter is arbitrary and should be thought of as being R = 1 ε for the proof of the main results (similarly as in the above informal discussion). Henceforth we write ´instead of ´Rd for simplicity. (2.20)

(ii) If β ≤ d, we have for all R ≥ 1, 0 < γ < β, 0 < α -d ≪ 1, and 0 < p -1 ≪ γ,α 1, |||∂ fct J 0 (F )||| 2 β α,p R d π * (R) -1 [RHS(2.20)] + R 2d-β-2d p r * (0) 2d p-1 p ˆrd(1-p 2 ) * [F ] p 2 2 p + R d-β r * (0) d-γ+α p-1 p ˆrγ p p-1 * w -α R p-1 p ˆw(d-γ)p+α(p-1) R |F | 2p
< p -1 ≪ α 1, |||∂ fct E(f, g)||| 2
α,p r * (0)

α p-1 2p ˆr2d 2p p-1 * w -α R p-1 2p × ˆC4 µ 4 * [∇ 2 ū] 4 ∞ 1 2 ˆwα(p-1) R [g] 4p ∞ 1 2p + ˆC4 µ 4 * [∇g] 4 ∞ 1 2 ˆwα(p-1) R (|f | + [∇ū] ∞ ) 4p 1 2p
.

(2.22)

(ii) If β ≤ d, we have for all R ≥ 1, 0 ≤ γ < β, 0 < α -d ≪ 1, and 0 < p -1 ≪ α 1, |||∂ fct E(f, g)||| 2 β α,p R d π * (R) -1 [RHS(2.22)] + R -β ˆC2 µ 2 * [∇g] 2 ∞ ˆ|f | 2 + (Rr * (0)) d p-1 p ˆ[∇ū] 2p ∞ 1 p + R -β ˆC2 µ 2 * [∇ 2 ū] 2 ∞ ˆrd * [g] 2 ∞ + (Rr * (0)) d p-1 p ˆ[g] 2p ∞ 1 p + R d-β r * (0) d-γ+α p-1 2p ˆrγ 2p p-1 * w -α R p-1 2p × ˆC4 µ 4 * [∇g] 4 ∞ 1 2 ˆw2p(d-γ)+α(p-1) R (|f | + [∇ū] ∞ ) 4p 1 2p + ˆC4 µ 4 * [∇ 2 ū] 4 ∞ 1 2 ˆr2pd * w 2p(d-γ)+ α(p-1) R [g] 4p ∞ 1 2p
, (2.23)

where we use the short-hand notation [RHS(2.22)] for the RHS of (2.22). ♦

The proofs of Propositions 2.5 and 2.6 rely on two further ingredients: large-scale weighted Calderón-Zygmund estimates and moment bounds on the extended corrector (φ, σ) (which are at the origin of the scaling µ * in the estimates). We start by recalling the former, which follows from [11, Theorem 1, Corollary 4, and Corollary 5] (see also [START_REF] Armstrong | Quantitative stochastic homogenization and large-scale regularity[END_REF]Section 7]).

Lemma 2.7 ([11]

). Assume that the coefficient field a is Gaussian with parameter β > 0, and let π * be as in (2.2). There exists a stationary, 1 8 -Lipschitz continuous random field r * ≥ 1 (the so-called minimal radius), satisfying for some (deterministic) constant C ≃ 1,

E exp 1 C π * (r * ) ≤ 2, (2.24)
such that the following properties hold a.s., (a) Mean-value property:

For any a-harmonic function u in B R (that is, -∇ • a∇u = 0 in B R ), we have for all radii r * (0) ≤ r ≤ R, Br |∇u| 2 B R |∇u| 2 . ( 2

.25)

Applied to the extended corrector of Lemma 2.2, this yields for all ℓ ≥ 1 and x ∈ R d ,

ˆBℓ (x) |∇(φ, σ)| 2 ℓ + r * (x) d . ( 2
w(r) ≤ w(r ′ ) ≤ r ′ r γ w(r) for all 0 ≤ r ≤ r ′ ,
we have for all u and g as in (b) above,

ˆ B * (x) |∇u| 2 p 2 w * (x) dx p,γ ˆ B * (x) |g| 2 p 2 w * (x) dx, (2.28)
where w * (x) := w(|x| + r * (0)). ♦

Whereas the minimal radius r * quantifies the sublinearity of the extended corrector at infinity [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF], the precise growth of the latter is estimated as follows (cf. [START_REF] Gloria | Quantitative stochastic homogenization for correlated fields[END_REF]Theorem 2]).

Lemma 2.8 ( [START_REF] Gloria | A regularity theory for random elliptic operators[END_REF][START_REF] Gloria | Quantitative stochastic homogenization for correlated fields[END_REF]). Assume that the coefficient field a is Gaussian with parameter β > 0, let µ * be as in (2.5), and let r * be as in Lemma 2.7. Then the extended corrector (φ, σ) defined in Lemma 2.2 satisfies for all x ∈ R d ,

B(x) |(φ, σ)| 2 1 2 ≤ C(x)µ * (x), (2.29)
where C ≥ 1 is a 1-Lipschitz continuous random field with stretched exponential moments: there exist γ ≃ β 1 and C γ ≃ γ 1 such that

E exp 1 C γ C γ ≤ 2.
(2.30) ♦

In order to reformulate integrals in a form well-suited to apply (weighted) large-scale Calderón-Zygmund estimates, we display below an auxiliary lemma that takes advantage of the Lipschitz continuity of r * . Lemma 2.9. Let • ℓ be defined in (2.13) and let r * be as in Lemma 2.7. For all U, V and ℓ ≥ 1, we have

U V 2 ℓ ˆ ˆB2ℓ * (x) |U | 2 B * (x) |V | 2 dx, (2.31)
and the refined estimate

U V 2 ℓ ˆ|x|≥ℓ ˆB2ℓ * (x) |U | 2 B * (x) |V | 2 dx + ˆB7ℓ * (0) B * (x) |U | 2 1 2 B * (x)
|V | 2 We first introduce some notation. Let a and ã be two (admissible) coefficient fields, and set δa := ãa. For all random variables (or fields) F = F (a), we set F := F (ã) and δF := F -F . We then denote by (φ, σ), (φ * , σ * ), ( φ, σ), and ( φ * , σ * ) the extended correctors associated with a, a * , ã, and ã * , respectively.

Step 1. Proof of identity (2.15). The definition (1.7) of Ξ ij yields

δJ 0 (F ) = ˆFij δΞ ij = ˆFij e j • (a -ā)∇δφ i + ˆFij e j • δa(∇ φi + e i ).
Using the definition (2.17) of the auxiliary field S as well as the corrector equation (2.9) for φ i and φi in the form

∇ • a∇δφ i = -∇ • δa(∇ φi + e i ), (3.1)
the first RHS term above can be rewritten as

ˆFij e j • (a -ā)∇δφ i = -ˆ∇S i • a∇δφ i = ˆ∇S i • δa(∇ φi + e i ),
and the conclusion (2.15) follows from the definition (2.11) of the functional derivative.

Step 2. Proof of identity (2.16). We start by giving a suitable representation formula for the functional derivative of the homogenization commutator of the solution (a -ā)∇u. By the property (2.10) of the flux corrector σ * j in the form (a * -ā * )e j = -a * ∇φ * j + ∇ • σ * j , and by its skew-symmetry (2.8) in the form

(∇ • σ * j ) • ∇δu = -∇ • (σ * j ∇δu), we find δ e j • (a -ā)∇u = e j • δa∇ũ + e j • (a -ā)∇δu = e j • δa∇ũ -∇ • (σ * j ∇δu) -∇φ * j • a∇δu.
Equation (1.3) for u and ũ in the form

-∇ • a∇δu = ∇ • δa∇ũ (3.2)
allows us to rewrite the last RHS term as

-∇φ * j • a∇δu = -∇ • (φ * j a∇δu) + φ * j ∇ • a∇δu = -∇ • (φ * j a∇δu) -φ * j ∇ • δa∇ũ = -∇ • (φ * j a∇δu) -∇ • (φ * j δa∇ũ) + ∇φ * j • δa∇ũ.
Hence, we conclude

δ e j • (a -ā)∇u = (∇φ * j + e j ) • δa∇ũ -∇ • (φ * j a + σ * j )∇δu -∇ • (φ * j δa∇ũ),
and similarly, replacing x → u(x) by x → φ i (x) + x i ,

δΞ ij = (∇φ * j + e j ) • δa(∇ φi + e i ) -∇ • (φ * j a + σ * j )∇δφ i -∇ • φ * j δa(∇ φi + e i ) .
Considering δ e j • (a -ā)∇u -∇ i ū δΞ ij and multiplying by g j , we are led to

δE(f, g) = ˆgj (∇φ * j + e j ) • δa ∇ũ -(∇ φi + e i )∇ i ū + ˆφ * j ∇g j • δa∇ũ -ˆφ * j ∇(g j ∇ i ū) • δa(∇ φi + e i ) + ˆ∇g j • (φ * j a + σ * j )∇δu -ˆ∇(g j ∇ i ū) • (φ * j a + σ * j )∇δφ i .
For the first RHS term we use the definition of w f in form of ∇u -

(∇φ i + e i )∇ i ū = ∇w f +φ i ∇∇ i ū,
δE(f, g) = ˆgj (∇φ * j + e j ) • δa(∇ wf + φi ∇∇ i ū) + ˆφ * j ∇g j • δa∇ũ -ˆφ * j ∇(g j ∇ i ū) • δa(∇ φi + e i ) + ˆ∇r • δa∇ũ -ˆ∇R i • δa(∇ φi + e i ), (3.3
U V 2 ℓ = ℓ -d ˆ ˆBℓ (x) |U V | 2 dx ≤ ˆ ˆBℓ (x) |U | 2 B ℓ (x) |V | 2 dx ≤ ˆ B ℓ (x) |V (y)| 2 ˆB2ℓ (y) |U | 2 dy dx = ˆ|V (x)| 2 ˆB2ℓ (x) |U | 2 dx ˆ ˆB2ℓ * (x) |U | 2 B * (x)
|V | 2 dx.

We now turn to the proof of (2.32). We distinguish the generic case r * (0) ≤ ℓ from the non-generic case r * (0) > ℓ, and we start with the latter. By the 1 8 -Lipschitz continuity of r * and the assumption r * (0) > ℓ ≥ 1, we have |x| ≤ ℓ =⇒

|V | 2 ≤ r * (0) -d ˆB ℓ 2 (0) ˆB * (x) |U | 2 1 2 ˆB * (x) |V | 2 1 2 2 ˆBℓ * (0) B * (x) |U | 2 1 2 B * (x) |V | 2 1 2 2 .
Combined with (2.31), this yields the conclusion (2.32) for r * (0) > ℓ. We turn to the generic case r * (0) ≤ ℓ. We split the integral over R d into the far-field contribution |x| ≥ 4ℓ and the near-field contribution |x| < 4ℓ. For the former, we proceed as above,

ℓ -d ˆ|x|≥4ℓ ˆBℓ (x) |U V | 2 dx ≤ ˆ|x|≥4ℓ B ℓ (x) |V (y)| 2 ˆB2ℓ (y) |U | 2 dy dx ≤ ˆ|x|≥3ℓ |V (x)| 2 ˆB2ℓ (x) |U | 2 dx ˆ B * (x) |V (y)| 2 ˆB2ℓ (y) |U | 2 1 |y|≥3ℓ dy dx,
where the last bound follows from (3.4). By the 1 8 -Lipschitz continuity of r * and the assumption r * (0) ≤ ℓ, we infer that the condition |x| < ℓ implies r * (x) ≤ r * (0) + 1 8 ℓ < 2ℓ, hence B * (x) ⊂ B 3ℓ (0). The above inequality then reduces to

ℓ -d ˆ|x|≥4ℓ ˆBℓ (x) |U V | 2 dx ˆ|x|≥ℓ B * (x) |V (y)| 2 ˆB2ℓ (y) |U | 2 dy dx ≤ ˆ|x|≥ℓ ˆB2ℓ * (x) |U | 2 B * (x) |V | 2 dx. (3.5)
We turn to the near-field contribution |x| < 4ℓ. We start with the trivial estimate

ℓ -d ˆ|x|<4ℓ ˆBℓ (x) |U V | 2 dx ˆB5ℓ (0) |U V | 2 ,
and we use (3.4) in form of

ˆB5ℓ (0) |U V | ˆ B * (x) |U V |1 |y|<5ℓ dy dx.
By the 1 8 -Lipschitz continuity of r * and the assumption r * (0) ≤ ℓ, we infer that the

condition |x| > 7ℓ implies B * (x) ∩ B 5ℓ (0) = ∅, hence ˆB5ℓ (0) |U V | ˆ|x|<7ℓ B * (x) |U V | dx.
The Cauchy-Schwarz' inequality then leads to

ℓ -d ˆ|x|<4ℓ ˆBℓ (x) |U V | 2 dx ˆ|x|<7ℓ B * (x) |U | 2 1 2 B * (x) |V | 2 1 2 dx 2 ,
and (2.32) follows in combination with (3.5) in the generic case r * (0) ≤ ℓ.

3.3.

Proof of Proposition 2.5: Main estimates. We split the proof into two main steps, first addressing the case of the standard LSI (β > d), and then turning to the general multiscale case (β ≤ d). Let R ≥ 1 be arbitrary.

Step 1. Proof of (2.20) for standard LSI (β > d).

Since for standard LSI (β > d) we have

|||∂ fct J 0 (F )||| β ∂ fct J 0 (F ) 1 (cf. (2.14))
, it suffices to prove the following estimate: for all ℓ ≥ 1, α > d, and p > 1 with α(p -1) < d(2p -1),

∂ fct J 0 (F ) 2 ℓ α,p ℓ d r * (0) α p-1 p ˆrd p p-1 * w -α R p-1 p ˆwα(p-1) R |F | 2p 1 p . ( 3
∂ fct J 0 (F ) 2 ℓ ˆ ℓ + r * (x) d B * (x) |F | 2 + |∇S| 2 dx ℓ d ˆr * (x) d B * (x) |F | 2 + |∇S| 2 dx. (3.7)
This yields the conclusion (3.6) in combination with the following estimate applied for

s = q = 1 and v = S (cf. (2.17)): If v is the Lax-Milgram solution of -∇ • a∇v = ∇ • h with h ∈ C ∞ c (R d ), then for all s ≥ 0, q ≥ 1, α > d, and p > 1 with α(p -1) < d(2pq -1), ˆr * (x) ds B * (x)
|h| 2 + |∇v| 2 q dx α,p,q,s r * (0)

α p-1 p ˆrds p p-1 * w -α R p-1 p ˆwα(p-1) R [h] 2pq 2 1 p , (3.8)
where for s = 0 we may even choose p = 1, in which case (3.8) is replaced by

ˆ B * (x) |h| 2 + |∇v| 2 q dx q ˆ[h] 2q 2 , (3.9)
which we state and prove here for future reference only.

Here comes the argument for (3.8). For all α > d and p > 1, we smuggle in the weight w R * (x) := w R (|x| + r * (0)) to the power α p-1 p , and use Hölder's inequality with exponents

( p p-1 , p), so that ˆr * (x) ds B * (x) |h| 2 + |∇v| 2 q dx s ˆrds p p-1 * w -α R * p-1 p × ˆwR * (x) α(p-1) B * (x) |h| 2 + |∇v| 2 pq dx 1 p
, where the first RHS sum is bounded by

´rds p p-1 * w -α R since w R ≤ w R * .
Provided that α(p -1) < d(2pq -1), we may apply the large-scale weighted Calderón-Zygmund estimate of Lemma 2.7(c) to the equation for v, to the effect of ˆr * (x) ds B * (x)

|h| 2 + |∇v| 2 q dx α,p,q,s ˆrds p p-1 * w -α R p-1 p × ˆwR * (x) α(p-1) B * (x) |h| 2 pq dx 1 p
.

The claim (3.8) then follows from the bound w R * (x) ≤ r * (0) inf B * (x) w R , Jensen's inequality, and (3.4). For s = 0, we appeal to the large-scale (not weighted) Calderón-Zygmund estimate of Lemma 2.7(b), which amounts to choosing p = 1 in the above.

Step 2. Proof of (2.21) in the general multiscale case (β ≤ d).

The combination of (2.12) with (3.6) is not enough to prove (2.21), and we have to refine (3.6) in the regime ℓ ≥ R. By (2.32) in Lemma 2.9 and the mean-value property (2.26),

∂ fct J 0 (F ) 2 ℓ ˆ|x|≥ℓ ℓ + r * (x) d B * (x) |F | 2 + |∇S| 2 dx + ˆB7ℓ * (0) B * (x) |F | 2 + |∇S| 2 1 2 2 . (3.10)
Let now v be the Lax-Milgram solution of

-∇ • a∇v = ∇ • h with h ∈ C ∞ c (R d ).
In the following two substeps we estimate the far-field and near-field contributions separately.

Substep 2.1. Far-field estimate: For all s ≥ 0, q ≥ 1, 0 ≤ γ ≤ ds, α > d, and p > 1 with (dsγ)p + α(p -1) < d(2pq -1),

ˆ|x|≥ℓ ℓ + r * (x) ds B * (x) |h| 2 + |∇v| 2 q dx γ,α,p,q,s ℓ γ R ds-γ r * (0) ds-γ+α p-1 p × ˆrγ p p-1 * w -α R p-1 p ˆw(ds-γ)p+α(p-1) R [h] 2pq 2 1 p . (3.11)
We smuggle in the weight w R * to the power α p-1 p and the weight w 1 * to the power dsγ, and use Hölder's inequality with exponents ( p p-1 , p), to the effect of

ˆ|x|≥ℓ ℓ + r * (x) ds B * (x) |h| 2 + |∇v| 2 q dx s ˆ|x|≥ℓ w -α R * w -(ds-γ) p p-1 1 * (ℓ + r * ) ds p p-1 p-1 p × ˆwα(p-1) R * w (ds-γ)p 1 * B * (x) |h| 2 + |∇v| 2 pq dx 1 p .
In the first RHS factor, we use the bound w 1 * (x) ℓ + r * (x) for |x| ≥ ℓ, while in the second RHS factor we use w 1 * ≤ Rw R * . The above then leads to

ˆ|x|≥ℓ ℓ + r * (x) ds B * (x) |h| 2 + |∇v| 2 q dx γ,s R ds-γ ˆ|x|≥ℓ w -α R * (ℓ + r * ) γ p p-1 p-1 p × ˆw(ds-γ)p+α(p-1) R * B * (x) |h| 2 + |∇v| 2 pq dx 1 p
.

Provided (ds-γ)p+α(p-1) < d(2pq -1), we may apply the large-scale weighted Calderón-Zygmund estimate of Lemma 2.

|h| 2 + |∇v| 2 1 2 p ℓ d p-1 p r * (0) d p-1 p ˆrd(1-p 2 ) + * [h] p 2 1 p . (3.12)
Indeed, by Hölder's inequality with exponents

( p p-1 , p), ˆB7ℓ * (0) B * (x) |h| 2 + |∇v| 2 1 2 ℓ d p-1 p r * (0) d p-1 p ˆB7ℓ * (0) B * (x) |h| 2 + |∇v| 2 p 2 1 p ℓ d p-1 p r * (0) d p-1 p ˆ B * (x) |h| 2 + |∇v| 2
∂ fct J 0 (F ) 2 ℓ γ,α,p ℓ 2d p-1 p ˆrd(1-p 2 ) * [F ] p 2 2 p + ℓ γ R d-γ r * (0) d-γ+α p-1 p ˆrγ p p-1 * w -α R p-1 p ˆw(d-γ)p+α(p-1) R |F | 2p 1 p . (3.13)
We appeal to (2.12), which we combine wtih (3.6) for ℓ ≤ R and with (3.13) for ℓ > R.

Provided that 0 ≤ γ < β, we compute

ˆR 1 ℓ d-1-β dℓ R d-β : β < d log R : β = d ≃ R d π * (R) -1 , ˆ∞ R ℓ -1-β dℓ R -β , R d-γ ˆ∞ R ℓ -1-β+γ dℓ R d-γ-(β-γ) = R d-β ,
and the conclusion (2.21) follows.

∂ fct E(f, g) = G 1 + G 2 + G 3 with G 1 := g j (∇φ * j + e j ) ⊗ (∇w f + φ i ∇∇ i ū), G 2 := (φ * j ∇g j + ∇r) ⊗ ∇u, G 3 := -φ * j ∇(g j ∇ i ū) + ∇R i ⊗ (∇φ i + e i ),
so that it suffices to estimate the norms of each of the G i 's separately. We split the proof into two main steps: we first address the case of the standard LSI (β > d), and then turn to the general multiscale case (β ≤ d). Let R ≥ 1 be arbitrary.

Step 1. Proof of (2.22) for standard LSI (β > d).

Since for standard LSI (β > d) we have |||∂ fct J 0 (F )||| β ∂ fct J 0 (F ) 1 , it suffices to establish the following estimates: for all ℓ ≥ 1, 0 < αd ≪ 1, and

0 < p -1 ≪ α 1, G 1 2 ℓ α,p ℓ d ˆr4d * [g] 4 ∞ 1 2 ˆC4 µ 4 * [∇ 2 ū] 4 ∞ 1 2 , (3.14) G 2 2 ℓ α,p ℓ d r * (0) α p-1 2p ˆrd 2p p-1 * w -α R p-1 2p × ˆC4 µ 4 * [∇g] 4 ∞ 1 2 ˆwα(p-1) R |f | 4p 1 2p , (3.15)
(g∇ū)] ∞ [∇g] ∞ [∇ū] ∞ +[g] ∞ [∇ 2
ū] ∞ , and using Hölder's inequality, the estimate (3.16) easily leads to

G 3 2 ℓ α,p ℓ d r * (0) α p-1 2p ˆrd 2p p-1 * w -α R p-1 2p × ˆC4 µ 4 * [∇ 2 ū] 4 ∞ 1 2 ˆwα(p-1) R [g] 4p ∞ 1 2p + ˆC4 µ 4 * [∇g] 4 ∞ 1 2 ˆwα(p-1) R [∇ū] 4p ∞ 1 2p
, so that (2.22) follows in combination with (3.14) and (3.15). We address the estimates (3.14)-(3.16) separately, and split the proof into three substeps.

Substep 1.1. Proof of (3.14). By (2.31) in Lemma 2.9,

G 1 2 ℓ ˆ ˆB2ℓ * (x) |g| 2 |∇φ + Id | 2 B * (x) (|∇w f | + |φ||∇ 2 ū|) 2 dx
which by Cauchy-Schwarz' inequality turns into

G 1 2 ℓ ˆ ˆB2ℓ * (x) |g| 2 |∇φ + Id | 2 2 dx 1 2 × ˆ B * (x) (|∇w f | + |φ||∇ 2 ū|) 2 2 dx 1 2
. (3.17)

We start by treating the first RHS factor. Taking the local supremum of g, using the mean-value property (2.26), the Lipschitz continuity of r * , Jensen's inequality, and (3.4), we obtain

ˆ ˆB2ℓ * (x) |g| 2 |∇φ + Id | 2 2 dx ˆ ˆB2ℓ * (x) r d * [g] 2 ∞ 2 dx ˆ B 2ℓ * (x) r d * (ℓ + r * ) d [g] 2 ∞ 2 dx ≤ ˆr2d * (ℓ + r * ) 2d [g] 4 ∞ ℓ 2d ˆr4d * [g] 4 ∞ . (3.18)
We turn to the second RHS factor in (3.17). Note that the two-scale expansion error w f satisfies the following equation (cf. [12, proof of Theorem 3]),

-∇ • a∇w f = ∇ • (aφ j + σ j )∇∇ j ū . (3.19)
By (3.9) with q = 2 applied to w f , we obtain after taking local suprema of ∇ 2 ū, and controlling correctors by Lemma 2.8,

ˆ B * (x) (|∇w f | + |φ||∇ 2 ū|) 2 2 ˆ[(|φ| + |σ|)∇ 2 ū] 4 2 ˆC4 µ 4 * [∇ 2 ū] 4 ∞ . (3.20)
Combined with (3.17) and (3.18), this yields the conclusion (3.14).

Substep 1.2. Proof of (3.15). By Lemma 2.9 in form of (2.31) and Cauchy-Schwarz' inequality,

G 2 2 ℓ ˆ B * (x) (|φ||∇g| + |∇r|) 2 2 1 2 ˆ ˆB2ℓ * (x) |∇u| 2 2 1 2 . (3.21)
We start with the first RHS factor. By (3.9) with q = 2 applied to the solution r of (2.18), we obtain after taking local suprema of ∇g and controlling correctors by Lemma 2.8,

ˆ B * (x) (|φ||∇g| + |∇r|) 2 2 ˆ[(|φ| + |σ|)∇g] 4 2 ˆC4 µ 4 * [∇g] 4 ∞ . (3.22)
We turn to the second RHS factor in (3.21). By the Lipschitz continuity of r * , Jensen's inequality, and (3.4),

ˆ ˆB2ℓ * (x) |∇u| 2 2 ˆ B 2ℓ * (x) (ℓ + r * ) d |∇u| 2 2 ˆ(ℓ + r * ) 2d B * (x)
|∇u| 2 2 .

By (3.8) with s = q = 2 applied to the solution u of (1.3), we deduce for all α > d and p > 1 with α(p -1) < d(4p -1),

ˆ(ℓ + r * ) 2d B * (x) |∇u| 2 2
α,p ℓ 2d r * (0)

α p-1 p ˆrd 2p p-1 * w -α R p-1 p ˆwα(p-1) R |f | 4p 1 p . ( 3
G 3 2 ℓ ℓ d r * (0) α p-1 p ˆrd p p-1 * w -α R p-1 p ˆwα(p-1) R [φ * ∇(g∇ū)] 2p 2 1 p .
Taking local suprema of ∇(g∇ū) and using Lemma 2.8 to control correctors, (3.16) follows.

Step 2. Proof of (2.23) in the general multiscale case (β ≤ d).

As in Step 2 of the proof of Proposition 2.5, we need to refine (3.14)- (3.16) in the range ℓ > R. More precisely, we shall establish that for all ℓ ≥ 1, 0 ≤ γ ≤ d, 0 < αd ≪ 1, and

0 < p -1 ≪ α 1, G 1 2 ℓ γ,α,p ˆC2 µ 2 * [∇ 2 ū] 2 ∞ ˆrd * [g] 2 ∞ +ℓ γ R d-γ r * (0) d-γ+α p-1 2p ˆrγ 2p p-1 * w -α R p-1 2p × ˆr2pd * w 2p(d-γ)+α(p-1) R [g] 4p ∞ 1 2p ˆC4 µ 4 * [∇ 2 ū] 4 ∞ 1 2 , (3.24) G 2 2 ℓ γ,α,p ˆC2 µ 2 * [∇g] 2 ∞ ˆ|f | 2 +ℓ γ R d-γ r * (0) d-γ+α p-1 2p ˆrγ 2p p-1 * w -α R p-1 2p × ˆC4 µ 4 * [∇g] 4 ∞ 1 2 ˆw2p(d-γ)+α(p-1) R |f | 4p 1 2p , (3.25) G 3 2 ℓ γ,α,p ℓ 2d p-1 p r * (0) 2d p-1 p ˆCp µ p * [∇(g∇ū)] p ∞ 2 p +ℓ γ R d-γ r * (0) d-γ+α p-1 p ˆrγ p p-1 * w -α R p-1 p × ˆwp(d-γ)+α(p-1) R C 2p µ 2p * [∇(g∇ū)] 2p ∞ 1 p . (3.26) Replacing p by 2p p+1 , estimating [∇(g∇ū)] ∞ [∇g] ∞ [∇ū] ∞ + [g] ∞ [∇ 2 ū
] ∞ , and using Hölder's inequality, the estimate (3.26) easily leads to

G 3 2 ℓ γ,α,p ℓ d p-1 p r * (0) d p-1 p ˆC2 µ 2 * [∇g] 2 ∞ ˆ[∇ū] 2p ∞ 1 p + ˆC2 µ 2 * [∇ 2 ū] 2 ∞ ˆ[g] 2p ∞ 1 p + ℓ γ R d-γ r * (0) d-γ+α p-1 2p ˆrγ 2p p-1 * w -α R p-1 2p × ˆC4 µ 4 * [∇g] 4 ∞ 1 2 ˆw2p(d-γ)+α(p-1) R [∇ū] 4p ∞ 1 2p + ˆC4 µ 4 * [∇ 2 ū] 4 ∞ 1 2 ˆw2p(d-γ)+α(p-1) R [g] 4p ∞ 1 2p
.

Starting from (2.12) and appealing to (3.

|g| 2 |∇φ + Id | 2 2 dx 1 2 × ˆ B 2ℓ * (x) (|∇w f | + |φ||∇ 2 ū|) 2 2 dx 1 2 + ˆB7ℓ * (0) B * (x) |g| 2 |∇φ + Id | 2 dx ˆB7ℓ * (0) B * (x) (|∇w f | + |φ||∇ 2 ū|) 2 dx .
First, we take local suprema of g, apply Lemma 2.8 to control correctors, and use (3.4), to the effect of ˆB7ℓ * (0) B * (x)

|g| 2 |∇φ + Id | 2 dx ˆrd * [g] 2 ∞ .
Second, using (3.4)

(|∇w f | + |φ||∇ 2 ū|) 2 dx ˆ(|∇w f | + |φ||∇ 2 ū|) 2 ˆ(|φ| + |σ|) 2 |∇ 2 ū| 2 ˆC2 µ 2 * [∇ 2 ū] 2 ∞ .
Third, appealing to (3.11) with s = q = 2 and |h| = |g||∇φ + Id |, we obtain for all 0 ≤ γ ≤ d, α > d, and p > 1 with 2p(dγ) + α(p -1) < d(4p -1),

ˆ|x|≥ℓ ℓ + r * (x) 2d ˆB * (x) |g| 2 |∇φ + Id | 2 2 dx γ,α,p ℓ 2γ R 2(d-γ) r * (0) 2(d-γ)+α p-1 p × ˆrγ 2p p-1 * w -α R p-1 p ˆw2p(d-γ)+α(p-1) R [g(∇φ + Id)] 4p 2
|∇u| 2 2 dx 1 2 × ˆ B 2ℓ * (x) (|φ||∇g| + |∇r|) 2 2 dx 1 2 + ˆB7ℓ * (0) B * (x) |∇u| 2 dx ˆB7ℓ * (0) B * (x) (|φ||∇g| + |∇r|) 2 dx .
First, using (3.4), the energy estimate for (2.18), taking local suprema of g, and applying Lemma 2.8 to control correctors, ˆB7ℓ * (0) B * (x)

(|φ||∇g| + |∇r|) 2 dx ˆ(|φ||∇g| + |∇r|) 2 ˆ(|φ| + |σ|) 2 |∇g| 2 ˆC2 µ 2 * [∇g] 2 ∞ .
Second, the energy estimate for (1.3) yields

ˆB7ℓ * (0) B * (x) |∇u| 2 dx ˆ|∇u| 2 ˆ|f | 2 .
Third, appealing to (3.11) with s = q = 2 and h = u, we obtain for all 0 ≤ γ ≤ d, α > d,

and p > 1 with 2p(d -γ) + α(p -1) < d(4p -1), ˆ|x|≥ℓ ℓ + r * (x) 2d B * (x) |∇u| 2 2 dx γ,α,p ℓ 2γ R 2(d-γ) r * (0) 2(d-γ)+α p-1 p × ˆrγ 2p p-1 * w -α R p-1 p ˆw2p(d-γ)+α(p-1) R |f | 4p 1 p .
Finally, applying (3.9) with q = 2 (with r * replaced by 2ℓ + r * ) to the solution r of (2.18), taking local suprema of g, and applying Lemma 2.8 to control correctors,

ˆ B 2ℓ * (x) (|φ||∇g| + |∇r|) 2 2 dx ˆ[(|φ| + |σ|)∇g] 4 2 ˆC4 µ 4 * [∇g] 4 ∞ .
The combination of these four estimates yields the conclusion (3.25).

Proof of the main results

We mainly focus on the proof of the statements for the standard LSI (β > d), and quickly argue how to adapt the argument to general multiscale LSI (β ≤ d) in the last step.

Step 1. Proof of Proposition 1 for standard LSI

(β > d). Let F ∈ C ∞ c (R d ) d×d . Starting point is (2.20) in Proposition 2.5
. By Hölder's inequality, the triangle inequality in probability, and the stationarity of r * , we obtain for all R ≥ 1, 0

< α -d ≪ 1, 0 < p -1 ≪ α 1, and q ≫ 1 p-1 , E |||∂ fct J 0 (F )||| 2q 1 q α,p E r d+α p-1 p * q 1 q R d p-1 p ˆwα(p-1) R |F | 2p 1 p . Replacing F by ε d 2 F (ε•) and choosing R = 1 ε , this yields E |||∂ fct J ε 0 (F )||| 2q 1 q α,p E r d+α p-1 p * q 1 q ˆwα(p-1) 1 |F | 2p 1 p . ( 4.1)
We now recall the following implication (which follows from multiscale LSI in form of the moment bounds in [7, Proposition 3.1(i)]; see also [12, Step 1 of the proof of Theorem 1]): for all random variables Y 1 , Y 2 , given q 0 ≥ 1 and κ > 0,

E |||∂ fct Y 1 ||| 2q β 1 q ≤ E [Y q 2 ]
1 q for all q ≥ q 0 , and

E [exp(Y κ 2 )] ≤ 2 =⇒ ∃ C ≃ q 0 ,κ 1 : E exp 1 C Y 2 κ 1+κ 1 ≤ 2. (4.2)
Using this property and the moment bound of Lemma 2.7 for r * , the estimate (4.1) leads to the conclusion (2.3).

Step

all R ≥ 1, 0 < α -d ≪ 1, and 0 < p -1 ≪ α 1, |||∂ fct E(f, g)||| 2 α,p r * (0) α p-1 2p ˆr2d 2p p-1 * w -α R p-1 2p × ˆC4 µ 4 * (|∇f | + |∇ 2 ū|) 4 1 2 ˆwα(p-1) R |g| 4p 1 2p
+ ˆC4 µ 4 * |∇g| 4 . (4.4)

It remains to estimate the first two RHS terms of (4.4), which we will prove to be small not because the two-scale expansion is accurate, but because f 1f and g 1g are small themselves after rescaling. Arguing as in the proof of (2.16), we have the alternative formula

∂ fct E(f, g) = g ⊗ ∇ug ⊗ ∇ū -g∇ i ū ⊗ (∇φ i + e i ) + ∇t ⊗ ∇u -∇T i ⊗ (∇φ i + e i ), Using this decomposition and arguing as in the proof of Proposition 2.5, we obtain for all R ≥ 1, 0 < αd ≪ 1, and 0 < p -1 ≪ α 1,

|||∂ fct E(f, g)||| 2 r * (0) + ˆ|∇g| .

Combining this with (4.4) leads to the conclusion (4.3).

Substep 2.2. Conclusion. By Hölder's inequality, the triangle inequality in probability, and the stationarity of r * , the estimate (4.3) leads to the following: for all R ≥ 1, 0 < αd ≪ 1, 0 < p -1 ≪ α 1, and q ≫ 1 p-1 , .

E |||∂ fct E(f, g)||| 2q 1 q α,p E r 2d+α p-1 2p * C 2 q 1 q × R d 2 (1-1 p ) ˆµ4 * (|∇f | + |∇ 2 ū|)
We then apply the standard weighted Calderón-Zygmund theory to the constant-coefficient equation (1.4) for ū, and replace f and g by ε Step 3. General multiscale LSI (β ≤ d).

E |||∂ fct E ε (f, g)||| 2q 1 q α,p E r 2d+α p-1 2p * C 2 q 1 q × ε 2 µ * ( 1 ε ) 2 ˆµ4 * |∇f |
We start with Proposition 1. By Hölder's inequality, the triangle inequality in probability, and the stationarity of r * , the estimate (2.21) in Proposition 2.5 leads to the following: for all R ≥ 1, 0 < γ < β, 0 < αd ≪ 1, 0 < p -1 ≪ γ,α 1, and q ≫ 1 p-1 ,

E |||∂ fct J 0 (F )||| 2q β 1 q
γ,α,p E r .

d+α p-1 p * q 1 q R 2d π * (R) -1 × R -d p ˆw(d-γ)p+α(p-1) R |F | 2p 1 p + R -2d p ˆ[F ] p
The combination of this estimate with property (4.2) and with the moment bound of Lemma 2.7 for r * implies the desired estimate (2.3).

We finally turn to Theorem 1. Arguing as in Substep 2.1 above, we may get rid of local suprema in the estimate (2.23) in Proposition 2.6. Using then Hölder's inequality, the triangle inequality in probability, and the stationarity of r * , we obtain the following: for all R ≥ 1, 0 ≤ γ < β, 0 < αd ≪ 1, 0 < p -1 ≪ α 1, and q ≫ 1 p-1 ,

E |||∂ fct E(f, g)||| 2q 1 q γ,α,p E r 2d+α p-1 2p * C 2 q 1 q × R d+ d 2 (1-1 p ) π * (R) -1 ˆµ4 * (|∇f | + |∇ 2 ū|)

  )ˆRd g • (a ε ∇u ε -ā∇ū) = ˆRd ( P * L g) • (a ε ∇u ε -ā∇u ε ),in terms of the Helmholtz and Leray projections inL 2 (R d ) d , PH := ∇(∇ • ā∇) -1 ∇•, PL := Id -PH ā, P * H := ∇(∇ • ā * ∇) -1 ∇•, P * L := Id -PH ā * , (1.11)

( 2 . 5 )

 25 Set µ * (z) := µ * (|z|), recall the notation w 1 (z) := 1 + |z|, and consider the rescaled error functional E

  Let us further reformulate the RHS. On the one hand, since -∇ • ā∇ū = ∇ • f , the simplifying assumptions yield the pointwise bounds |∇ū| |f | and |∇ 2 ū| |∇f |. On the other hand, the function w f satisfies the equation -∇ • a∇w f = ∇ • ((aφσ)∇ 2 ū), (cf. [12, Remark 3]) so that our simplifying assumptions yield this time |∇w f | µ * |∇ 2 ū| µ * |∇f |. This leads to the bound

Proposition 2 . 5 ( 1 2

 251 Main estimates). Let the coefficient field a be Gaussian with parameter β > 0. Let π * and µ * be defined by (2.2) and (2.5), respectively, and let the random field r * be the minimal radius of Lemma 2.7 below. For F ∈ C ∞ c (R d ) we denote by [F ] 2 (x) := ( ffl B(x) |F | 2 ) the moving local quadratic average, and for R ≥ 1 we set w R (x) := |x| R + 1. Then the following hold: (i) If β > d, we have for all R ≥ 1, 0 < αd ≪ 1, and 0 < p -1 ≪ α 1, |||∂ fct J 0 (F )||| 2 α,p r * (0)

1 .

 1 ) d we denote by [F ] ∞ (x) := sup B(x) |F | the moving local supremum, and we recall the notation w R (x) = |x| R + Then the following hold: (i) If β > d, we have for all R ≥ 1, 0 < αd ≪ 1, and 0

  .26) (b) Large-scale Calderón-Zygmund estimates: Set B * (x) := B r * (x) (x), and more generally B ℓ * (x) := B ℓ+r * (x) (x). For all 1 < p < ∞, for all (sufficiently fast) decaying scalar fields u and vector fields g related in R d by -∇ • a∇u = ∇ • g, Large-scale weighted Calderón-Zygmund estimates: For all 2 ≤ p < ∞, 0 ≤ γ < d(p -1), and for all non-decreasing radial weights w ≥ 1 satisfying

2 , 3 . 3 . 1 .

 2331 (2.32) where we recall B * (x) = B r * (x) (x), B 2ℓ * (x) = B 2ℓ+r * (x) (x), and B 7ℓ * (0) = B 7ℓ+r * (0) (0), and where we have set B * (x) := B 5r * (x) (x).♦ Proof of the representation formulas and of the main estimates Proof of Lemma 2.4: Representation formulas.

7 8 r 4 , 2 B

 842 * (0) ≤ r * (0) -1 8 ℓ ≤ r * (x) ≤ r * (0) + 1 8 ℓ ≤9 8 r * (0), and similarly, |x| ≤ ℓ 2 =⇒ 15 16 r * (0) ≤ r * (x) ≤ 17 16 r * (0) =⇒ B 4r * (0) (0) ⊂ B * (x), where we recall the definition B * (x) = B 5r * (x) (x). Hence, ˆ|x|≤ℓ ˆB2ℓ * (x)|U |

  7(c) to the equation for v. Using the bound w R * (x) ≤ r * (0) inf B * (x) w R , Jensen's inequality, and (3.4), the conclusion (3.11) follows. Substep 2.2. Near-field estimate: For all ℓ ≥ 1 and p > 1, ˆB7ℓ * (0) B * (x)

,

  where we pass from B * (x) = B 5r * (x) (x) to B * (x) = B r * (x) (x). The claim (3.12) follows from the large-scale Calderón-Zygmund estimate of Lemma 2.7(b) applied to v. Substep 2.3. Conclusion. For all ℓ ≥ 1, 0 ≤ γ ≤ d, α > d, and p > 1 with (dγ)p + α(p -1) < d(2p -1) and p ≤ 2, the combination of Substeps 2.1-2.2 with (3.10) yields the following improvement of (3.6),

  p by 2p p+1 , estimating [∇

3 )

 3 We first apply(2.22) to the averaged functions f 1 and g 1 defined by f 1 (x) := ffl B(x) f andg 1 (x) := ffl B(x) g. Noting that [f 1 ] ∞ ffl B 2 (x) |f | and that the solution ū1 of the homogenized equation (1.4) with averaged RHS f 1 is given by ū1 (x) = ffl B(x) ū, and using the Lipschitz continuity of C, we obtain for all 0 < αd ≪ 1 and 0 < p -1 ≪ α 1,|||∂ fct E(f, g)||| 2 α,p |||∂ fct E(f 1f, g 1 )||| 2 + |||∂ fct E(f, g 1g)||| 2+ r * (0) α p-1

  where the auxiliary fields t and T are the Lax-Milgram solutions in R d of-∇ • a * ∇t = ∇ • ((a * -ā * )g), -∇ • a * ∇T i = ∇ • ((a * -ā * )g∇ i ū).

.

  Using in addition that|ff 1 | ≤ ´1 0 ffl tB |∇f (• + y)|dydt, this turns into |||∂ fct E(f 1f, g 1 )||| + |||∂ fct E(f, g 1g)||| r * (0)

4

 4

4

 4

d 4 f

 4 (ε•) and ε d 4 g(ε•). For the choice R = 1 ε , and by the bound µ * ( • ε ) µ * ( 1 ε )µ * (•), this implies

4

 4

Replacing F by ε d π * ( 1 ε ) 1 2F

 1 (ε•) and choosing R = 1 ε , this yields E |||∂ fct J ε 0 (F )||| 2q β

4

 4

1 p.Since in dimension d ≥ 2 the weights µ 2 * and µ 4 *

 124 +R -β ˆµ2 * |∇g| 2 ˆ(|f | + |∇ū|) 2 + R d-d p ˆ(|f | + |∇ū|) 2p1 p always belong to the Muckenhoupt classes A 2 and A 4 , respectively, we may apply the standard weighted Calderón-Zygmund theory to the constant-coefficient equation (1.4) for ū in order to simplify the above RHS. Replacing then f and g by π * ( 1 ε ) ε•), choosing R = 1 ε , and using the boundµ * ( • ε ) µ * ( 1 ε )µ * (•), the conclusion (2.6) follows as in Substep 2.2.

  Structure of the proof. We describe the string of arguments that leads to Proposition 1 and Theorem 1. Next to the corrector φ, we first need to recall the notion of flux corrector σ, which was recently introduced in the stochastic setting in [11,Lemma 1] and allows to put the equation for the two-scale homogenization error in divergence form (cf. (3.19) below). The extended corrector (φ, σ) is only defined up to an additive constant, and we henceforth choose the anchoring ffl

even when P * H F and P * L F do not have integrable decay. ♦ 2.2. B (φ, σ) = 0 on the centered unit ball B. Lemma 2.2 ([

  Before we turn to the (technical) estimates of |||∂ fct J 0 (F )||| β and |||∂ fct E(f, g)||| β , let us give an informal discussion of the scalings of the terms appearing in(2.15) and (2.16). To keep this discussion short, assume that ∇φ, ∇σ are bounded (which only holds after taking stochastic moments), that |φ(x)| + |σ(x)| µ * (|x|) (which again only holds after taking stochastic moments), and that the Helmholtz projections associated with -∇ • a * ∇ (and used to define S i , r, and R i via (2.17)-(2.19)) enjoy perfectly local bounds in the sense that

) and a * denotes the pointwise transpose coefficient field of a, and (φ * , σ * ) denotes the corresponding extended corrector (recall that a * = ā * ). ♦

  whereas for the last two RHS terms we use the definitions (2.18) and (2.19) of the auxiliary fields r and R, combined with equations (3.1) and (3.2), so that

  3.4. Proof of Proposition 2.6: Main estimates (cont'd). By (2.16) in Lemma 2.4, we have

  By (3.8) with s = q = 1 applied to the solution R of (2.19), we deduce for all α > d and p > 1 with α(p -1) < d(2p -1),

							.23)
	Combined with (3.21) and (3.22), this yields (3.15).
	Substep 1.3. Proof of (3.16).
	By (2.31) in Lemma 2.9 and the mean-value property (2.26), we find
		ˆ
	G 3	2 ℓ	ℓ + r * (x)	d	B * (x)	|φ * ||∇(g∇ū)| + |∇R|	2 dx.

  14)-(3.16) for ℓ ≤ R and to (3.24)-(3.26) for ℓ > R, we obtain the desired estimate (2.23) after arguing as in Substep 2.3 of the proof of Proposition 2.5. The rest of this step is split into three parts and is dedicated to the proof of (3.24)-(3.26).

	Substep 2.1. Proof of (3.24).
	By (2.32) in Lemma 2.9 and Cauchy-Schwarz' inequality,
		ˆ|x|≥ℓ
	G 1	2 ℓ	ℓ + r * (x)	2d	B * (x)

  .24) then follows from the combination of the above estimates with (3.20) (with r * replaced by 2ℓ + r * ).

						1
						p ,
	while the mean-value property (2.26) yields [g(∇φ + Id)] 4p 2 sion (3Substep 2.2. Proof of (3.25).	r 2pd * [g] 4p ∞ . The conclu-
	By (2.32) in Lemma 2.9 and Cauchy-Schwarz' inequality,
		ˆ|x|≥ℓ
	G 2	2 ℓ	ℓ + r * (x)	2d	B * (x)

  2. Proof of Theorem 1 for standard LSI (β > d). Let f, g ∈ C ∞ c (R d ).We split the proof into two substeps: we first improve (2.22) to avoid local suprema in the estimate, and then turn to (2.6) itself.

	Substep 2.1. Improvement of (2.22): for

  We now recall the following version of Hölder's inequality: for all random variables Y 1 , Y 2 , given κ 1 , κ 2 > 0, Using this property, the moment bounds of Lemmas 2.7 and 2.8 for r * and C yield, for all η > 0, E exp(1Cη (r 2d * C 2 )1 4 -η ) ≤ 2 for some C η ≃ η 1.Combining this with (4.5), property (4.2) yields the conclusion (2.6).

	E exp Y κ 1 1	≤ 2 and E exp Y κ 2 2 =⇒ ∃C ≃ κ 1 ,κ 2 1 : E exp ≤ 2	1 C	(Y 1 Y 2 )	κ 1 κ 2 κ 1 +κ 2	< ∞. (4.6)
				1
				2p
		+ ˆµ4 * |∇g| 4	1 2	ˆwα(p-1) 1	|f | 4p	1 2p	. (4.5)

In dimension d = 1, the homogenization commutator indeed simply takes the form Ξ(x) = ā(1 -ā a(x) ), which is exactly local wrt a.

Acknowledgements

The work of MD is supported by F.R.S.-FNRS through a Research Fellowship and by the CNRS-Momentum program. MD and AG acknowledge financial support from the European Research Council under the European Community's Seventh Framework Programme (FP7/2014-2019 Grant Agreement QUANTHOM 335410).

[20] S. Torquato. Random heterogeneous materials, volume [START_REF] Kozlov | The averaging of random operators[END_REF]


Muhammad Hamidullah

Nassim Rezzag

Céline Elie-Caille

Céline Élie-Caille

Thérèse Leblois

Lamb waves sensor in

Keywords: Lamb waves, GaAs, quasi-longitudinal modes, higher-order, liquid media I

INTRODUCTION

Plate acoustic waves (PAW) with two free surfaces have an advantage over surface acoustic waves (SAW) with only one free surface for sensor application in liquid media [START_REF] Caliendo | Guided acoustic wave sensors for liquid environments[END_REF]. In PAW, the sensing surface can be separated from the opposite surface with interdigital transducer (IDT), thus PAW-based sensors do not require an extra protection layer for the IDT to prevent contact with liquid. Several types of in-plane polarised PAW for sensors in liquid media have been reported, such as shear-horizontal acoustic plate modes (SH-APM) [START_REF] Déjous | A shear-horizontal acoustic plate mode (SH-APM) sensor for biological media[END_REF] and Lamb waves (LW) [START_REF] Zhou | A multi-parameter decoupling method with a Lamb wave sensor for improving the selectivity of label-free liquid detection[END_REF].

Fundamental antisymmetric (A0) and symmetric (S0) modes of LW are commonly used for sensors in liquid media without significant radiation of acoustic energy into the liquid under specific conditions. For the A0 modes, the phase velocity (vph) must be lower than the sound velocity of the liquid so that the acoustic energy is not converted into longitudinal pressure waves in the liquid. In the case of S0 modes, the particle displacement components must be a dominantly longitudinal component (u1) with close to zero shear vertical (u3) component. Both conditions are achieved at a low plate thickness-to-wavelength ((h/) ratio, typically below 0.1). Therefore, for A0 and S0 modes sensor in liquid media, the reduction in the plate thickness is necessary which causing the device to be fragile, or by using a larger wavelength with a trade-off in operating frequency and sensitivity reduction.

The limitations of low h/ ratio for A0 and S0 modes were solved by using higher-order symmetric modes with quasilongitudinal polarisation (QL-Sn). Anisimkin et al [START_REF] Anisimkin | A new type of acoustic modes of vibration of thin piezoelectric plates: Quasi-longitudinal normal modes[END_REF] reported QL-Sn modes in ST-cut quartz for a sensor in liquid media utilizing higher-order symmetry mode at h/ ratio of 1.485. Verona et al [START_REF] Verona | Quasi longitudinal Lamb acoustic modes along ZnO/Si/ZnO structures[END_REF] reported the occurrence of QL-Sn modes in layered thin film ZnO/Si-substrate/ZnO up the 4 th higherorder symmetric modes (S4) with h/ ratio of 2.6 [START_REF] Verona | Quasi longitudinal Lamb acoustic modes along ZnO/Si/ZnO structures[END_REF]. Further reduction in the wavelength and the device size is an advantage of using symmetric modes with a higher order number, however, the electromechanical coupling coefficient (K 2 ) reduces as the order number increases. For two-port delay line device implementation, the reduction of K 2 will affect the signal strength received in receiver IDT and reduces the signal to noise ratio. In this paper, we perform finite element method (FEM) simulation to compare the performances of the first 6 higher-order symmetric QL-LW on GaAs substrate and we verify experimentally the feasibility of using S5 and S6 modes for sensor application in liquid media.

II. FINITE ELEMENT SIMULATION

The occurrence of higher-order QL-LW modes at (100)cut GaAs plate propagating along the <110> direction is previously reported for S1 and S2 modes at h/ ratio of 0.6 and 1.2, respectively [START_REF] Hamidullah | Theoretical study and finite element simulation of ZnO/GaAs higher-order Lamb waves for microsensor application in liquid media[END_REF], where the vph of higher-order symmetric modes (Sn) are close to the vph of longitudinal bulk acoustic wave (LBAW). Due to the periodicity on the dispersion curves for higher-order modes, the h/ ratio of quasilongitudinal symmetric modes at order number n (QL-Sn) can be approximately obtained by multiplying the order number n with the factor of 0.6. Thus h/ ratio for the first 6 higher-order QL modes are 0.6, 1.2. 1.8, 2.4, 3 and 3.6. Figure 1 shows the total displacement profile of the first 6 QL-Sn modes on GaAs plate obtained by COMSOL Multiphysics eigenfrequency simulation with the wavelength of 168 µm and the plate thickness of 100 µm, 200 µm, 300 µm, 400 µm, 500 µm and 600 µm for QL-S1, QL-S2, QL-S3, QL-S4, QL-S5 and QL-S6 modes respectively. As shown in figure 1, the total particle displacement profiles at the surface for all six modes are dominantly longitudinal. In PAW with two free surfaces, the electrical boundary condition (BC) of the surfaces affects the value of K 2 . The electrically shorted BC is achieved by the deposition of a metal layer on the opposite side of the IDT. Metal-substrate-IDT (M-S-IDT) and substrate-IDT (S-IDT) are electrical coupling configurations with and without additional metal layers respectively. The K 2 of the first six QL-modes are shown in figure 2 for two different coupling configurations. As shown in figure 2, the K 2 decreases as the order number increases. The K 2 of S6 mode, for instance, is one order lower than the K 2 of S1 mode which will be equal to 10 dB reduction in the signal strength. Furthermore, the M-S-IDT configuration slightly improves the K 2 to around two times higher than the K 2 value of the S-IDT configuration. Figure 3a shows the particle displacement components: longitudinal (u1) and shear vertical (u3) for S5 and S6 modes at h/ of 3 and 3.6 respectively. The shear-vertical (u2) displacement component is zero across the plate. As shown in figure 3a, the longitudinal displacement component is dominant across the plate for both modes. At the surface of the plate, the u3 is around 20% of the maximum u1 inside the plate. Figure 3b shows the normalised u3 at the surface at h/ ratio of 2.8 to 3.2 for S5 mode and 3.3 to 3.9 for S6 mode. As shown in figure 3b, the lowest u3 component is at h/ of 3.1 for S5 and 3.7 at S6 with normalised u3 of 6.5% and 5.6% respectively. At these ratios, the lowest insertion loss is expected when the surface of the plate is in contact with liquid. Figure 4a shows the insertion loss simulation result when one surface of the plate is in contact with DI water for h/ ratio of 2.8 to 3.2 for S5 mode and 3.3 to 3.9 for S6 mode. Figure 4b shows the 3D total displacement profile of the S5 and S6 modes when the plate is in contact with the liquid. As expected, the lowest insertion loss is achieved at h/ of 3.1 for S5 mode and at 3.7 for S6 mode. For delay line with the centre to centre distance between transmitter and receiver IDT of 250, the minimum theoretical insertion loss due to the liquid damping is 1.34 dB for S5 mode and 1.04 dB for S6 mode. Furthermore, there is a margin of +/-0.1 in h/ ratio with relatively low insertion loss.

III. EXPERIMENTAL RESULTS

Two-port delay line devices were fabricated on (100)-cut GaAs wafer with the thickness of 500 µm 25 and 625 µm and thickness variation of +/-25 µm. For both wafers, the design of the IDTs is identical and the wavelength of the IDTs are 168 µm. Considering the thickness variation of the wafer, the h/ ratios are ranging from 2.82 to 3.12 for S5 mode and 3.57 to 3.86 for S6 mode. The waves propagate at <110> direction parallel to the wafer flat. The IDTs have 50 pairs of source and ground with an aperture of 50. The delay length is 200, thus the centre-to-centre distance from the IDT receiver to the IDT transmitter is 250 or equal to 42 mm. The metal electrode of 100 nm Al with 10nm Cr adhesion layer was deposited by a lift-off process. Gold metal layer was deposited at the back of the substrate to obtain the M-S-IDT coupling configuration. Furthermore, the metal layer at the sensing surface will neutralise the effect of liquid permittivity on the sensor responses. The GaAs wafers are attached to PCBs with a circular hole so that the delay line structures and the IDTs are suspended. 4194A Impedance/Gain-Phase Analyzer was used to obtain the frequency response and the power received in the IDT receiver normalised to the power sent from the IDT transmitter (S21 scattering parameter).

The S21 frequency responses of the sensor devices with the thickness of 500 µm and 625 µm are shown in figure 5a and 5b respectively. As shown in figure 5a, there are three resonant frequency peaks for the frequency range from 30 MHz to 33 MHz which correspond to A5, S5 and S6 modes when the device is in a dry environment. However, when the surface of the plate is in contact with DI water, the reduction in S21 signals are significant for A5 and A6 modes while the reduction in the S5 mode is only around 1 dB. Similarly, in figure 5b, there are three resonant frequency peaks in dry environment but only the S6 mode peak is observable when the plate is in contact with DI water with around 1.4 dB reduction in the S21 signal. The S21 signals of around -50 dB in contact with DI water for both S5 and S6 modes are distinguishable from the noise floor of -65 dB, thus showing the feasibility of those modes for sensor application in liquid media. 6, both S21 amplitudes and resonant frequencies reduce as the glycerol concentration increases. The results agree with the previously reported result of PAW viscosity sensors such as SH-APM [START_REF] Ricco | Acoustic wave viscosity sensor[END_REF]. The S21 amplitudes return to the baseline amplitude after the device is no longer in contact with liquid, however, the frequency is remaining in the lower value, especially after the DI water/glycerol mixture was in contact with the plate surface. One possible explanation is the trace of glycerol remains in the plate surface act as a mass loading which reduces the resonant frequency. Thus, the S21 amplitude is a more reliable signal than the frequency for viscosity sensor application.

IV. CONCLUSION

The feasibility of utilizing quasi-longitudinal modes at higher-order numbers (QL-S5 and QL-S6) for a sensor in liquid media is reported in this paper using (100)-cut GaAs substrate. The employment of modes with higher-order numbers allows the reduction of wavelength thus increases the resonant frequency without reducing the thickness of the plate and improves the robustness of the device. While the reduction in K 2 is inevitable, the S21 amplitude at the resonant frequency is distinguishable from the noise floor. The improvement in K 2 can be achieved by depositing highly piezoelectric thin film such as ZnO on GaAs [START_REF] Hamidullah | Theoretical study and finite element simulation of ZnO/GaAs higher-order Lamb waves for microsensor application in liquid media[END_REF] or by using plate substrate material with higher piezoelectricity constant.

Fig. 1 .

 1 Fig.1. Normalised total displacement profile of the first six higher-order quasi-longitudinal symmetric modes (QL-Sn) on (100)-cut GaAs propagating on <110> direction.

Fig. 2 .

 2 Fig. 2. K 2 of the first six QL-Sn modes with S-IDT and M-S-IDT coupling configuration

Fig. 3 .

 3 Fig. 3. (a) Particle displacement component profile of S5 and S6 modes and (b) Normalised shear vertical (u3) displacement at the surface of the plate

Fig. 4 .

 4 Fig. 4. (a) The insertion loss due to liquid damping for delay line with a centre to centre distance of 250 and at h/ ratio of 2.8 to 3.2 for S5 mode and 3.3 to 3.9 for S6 mode and (b) Displacement profile of S5 and S6 modes with h/ ratio of 3.1 and 3.7 respectively.

Fig. 6 .

 6 Fig. 6. Experimental results of QL-S5 mode as viscosity sensor with different concentrations of DI water/glycerol mixture

ACKNOWLEDGEMENT

  This conference proceeding is part of SmOoC project that has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 844135. The work was partly supported by the French Renatech network and its FEMTO-ST technological facility

Fig. 5 .

 5 Fig. 5. Experimental result of two-port delay line device utilising (a) S5 mode and (b) S6 mode in dry and liquid media


Dbibih Fatima-Ezzahraa

Virginie Blondeau-Pâtissier

Lyes Djoumi

Valérie Souman

Meddy Vanotti

Sensitivity Of Surface Acoustic Wave Sensors Based LiNbO3 128 ° Y-X and AT-quartz to PM10 and PM2.5

Introduction

Particulate matter (PM) was estimated to cause around 7 million premature deaths worldwide per year according to World Health organization [START_REF]Air pollution[END_REF]. These particles can cause diseases leading to premature death [START_REF] Lin | Particle size and chemical constituents of ambient particulate pollution associated with cardiovascular mortality in Guangzhou, China[END_REF]. Over the last decade, the monitoring of PM became an important issue on a worldwide scale.The miniaturization of the existing bulky systems has also become necessary to reduce their cost. To address this need, a new miniaturized system for PM monitoring was previously developed in our team [START_REF] Djoumi | Real Time Cascade Impactor Based On Surface Acoustic Wave Delay Lines for PM10 and PM2.5 Mass Concentration Measurement[END_REF].It takes advantage of both a 3 Lpm cascade impactor and surface acoustic waves (SAW) sensors to respectively separate and measure particles. The present study aims to compare the sensitivity of two SAW devices. The first exploits Love waves on an ATcut Quartz substrate and the second is based on Rayleigh waves from a LiNbO3 Y-X 128° substrates. The latter presents a high electromechanical coupling factor that could be exploited for the self-cleaning of the device after the measurement. This could allow overcoming the fouling issue faced with impactors that wouldn't be solved with quartz based sensors which suffer from an insufficient electromechanical coupling.

Method

In this study we used a dedicated 3 Lpm cascade impactor equipped with SAW delay lines as collecting plate. The Rayleigh wave sensor and the Love wave sensor were designed and fabricated respectively on LiNbO3 Y-X 128° cut and AT-Quartz cut substrates using a conventional photolithography process. The interdigital transducers (IDT) consist of double finger pairs made of 200 nm thick aluminum with a periodicity p=10 µm and a wavelength λ=40 µm. Hence, the working frequency is 125 MHz for the Love mode and 98 MHz for the Rayleigh mode.. The propagation of Love wave needs a guiding layer. For that purpose, a silica layer was deposited using sputtering technique with shear velocity lower than the piezoelectric substrates. The region between the two IDTs constitutes the sensing area.

Preliminary tests have been performed using candle smoke. As the obtained PM concentrations were unstable and difficult to reproduce, we have developed an experimental set-up bench consisting of a 1m 3 chamber, a particle's generator AGK 2000 and an optical system FIDAS ® 100. Thus, sodium chloride (NaCl) was used to produce PM2.5 while PM10 were generated from silicon carbide (SiC) in the 0 to 200 μg/m 3 concentration range.When the particles impact on the sensor surface, the velocity of the acoustic waves decreases due to the sensors' high gravimetric sensitivity. Accordingly, the wave phase velocity shift is measured by monitoring the phase at constant frequency with an open loop interrogation [START_REF] Rabus | High-sensitivity open-loop electronics for gravimetric acoustic-wave-based sensors[END_REF]. It's then correlated to the particles concentration.

Results and Conclusions

Figure 1 and 2 shows plots of phase derivative dϕ/dt of sensors-based AT-Quartz (black) and LiNbO3 (red) with the concentration of PM10 and PM2.5 respectively measured with the optical system FIDAS®. As expected, the sensitivity of the Rayleigh wave sensor based on LiNbO3 is lower than the Love wave sensor based on quartz for both PM10 and PM2.5. The sensor's sensitivity is estimated by applying a linear fit of the plotted data. The sensitivity of sensors based on quartz is 3.10-4 °/s.μg/m3 for PM 2.5 and 5.10-5 °/s.μg/m3for PM10. For sensors based on LiNbO3, the sensitivity is 1.10-4 °/s.μg/m3 for PM2.5 and 1.10-5 °/s.μg/m3 for PM10. The feature that the sensors' response is greater with PM2.5 than with PM10 can be explained by the fact that most adhesion forces are linearly dependent on particle diameter [START_REF] Kulkarni | Introduction to Aerosol Characterization[END_REF] which is often described as the rebound effect. As a result, the smaller particles settle more on the surface of the sensor resulting in a more significant slowdown of the wave. To ensure that each stage is measuring the targeted size range, we took images of the sensors surface at the end of the experiment using a microscope. Figures 3,4 show the surface of the PM2.5 stage and figures 5,6 the one of the PM10 stage. One can observe that, as expected, the PM2.5 contain more particles than the PM10 and that the particle sizes correspond well to the ranges [0.3 μm, 2.5 μm] and [2.5 μm, 10 μm]. It has therefore been shown that LiNbO3 sensors, although less sensitive, can also be used for fine particles detection. They still need work to improve their performance and limit the rebound effect, but they remain very promising for the development of selfcleaning systems.

Figure 1 :

 1 Figure 1 : Phase derivative of SAW sensors based LiNbO3 Y-X 128 ° (red) and AT-Quartz (black) as a function of PM 2.5 concentration.

Figure 2 :

 2 Figure 2 : Phase derivative of SAW sensors based LiNbO3 Y-X 128 ° (red) and AT-Quartz (black) as a function of PM 10 concentration.

Figure 3 :

 3 Figure 3: Image of the LiNbO3 Y-X 128 ° sensor's surface after exposition to PM 2.5, zoom x1500.Figure4: Image of the AT-quartz sensor's surface after exposition to PM 2.5, zoom x1500.

Figure 4 :

 4 Figure 3: Image of the LiNbO3 Y-X 128 ° sensor's surface after exposition to PM 2.5, zoom x1500.Figure4: Image of the AT-quartz sensor's surface after exposition to PM 2.5, zoom x1500.

Figure 5 :

 5 Figure 5: Image of the LiNbO3 Y-X 128 ° sensor's surface after exposition to PM 10, zoom x1500.Figure6: Image of the AT-quartz sensor's surface after exposition to PM 10, zoom x1500.

Figure 6 :

 6 Figure 5: Image of the LiNbO3 Y-X 128 ° sensor's surface after exposition to PM 10, zoom x1500.Figure6: Image of the AT-quartz sensor's surface after exposition to PM 10, zoom x1500.