On generalized Gaussian free fields and stochastic homogenization

We study a generalization of the notion of Gaussian free field (GFF). Although the extension seems minor, we first show that a generalized GFF does not satisfy the spatial Markov property, unless it is a classical GFF. In stochastic homogenization, the scaling limit of the corrector is a possibly generalized GFF described in terms of an"effective fluctuation tensor"that we denote by $\mathsf{Q}$. We prove an expansion of $\mathsf{Q}$ in the regime of small ellipticity ratio. This expansion shows that the scaling limit of the corrector is not necessarily a classical GFF, and in particular does not necessarily satisfy the Markov property.

Under assumptions described more precisely below, the scaling limit of the corrector was identified in [19,18] as an element of a slightly larger class of random fields, which we will call generalized GFFs (see also [8]). Intuitively, we define the generalized GFF as the field Φ such that ∇Φ is "as close as possible" to a white noise vector field. More precisely, a generalized GFF is a random field Φ solving − ∇ · a∇Φ = ∇ · W, where a ∈ Sym + d and W is a vector-valued white noise field with covariance matrix Q ∈ Sym + d . For simplicity, we will only consider such equations in the full space R d with d 3, but our arguments also cover domains with suitable boundary condition, or the addition of a massive term, with only minor modifications. We recall that a classical GFF is a Gaussian field whose covariance function is the Green function of −∇ · b∇, for some b ∈ Sym + d . When a and Q are proportional, the field Φ defined by (0.1) is a classical GFF, and thus satisfies the spatial Markov property [25].
The goal of this paper is twofold. In Section 1, we show that the Markov property does not extend to the wider class of generalized GFFs. In fact, no generalized GFF satisfies the Markov property, unless it is a classical GFF. Therefore, our apparently mild extension of the notion of GFF in fact wipes out one of its fundamental properties. In Section 2, we turn to the context of homogenization, and study the effective parameters describing the generalized GFF arising as the scaling limit of the corrector. We prove an asymptotic expansion for these coefficients, in the regime of small ellipticity contrast. This expansion is of independent interest, and shows that the scaling limit of the corrector is generically not a classical GFF, thereby justifying the relevance of the notion of generalized GFF.
We close this introduction with two remarks. First, in order to substantiate that the definition (0.1) is consistent with the intuition that ∇Φ is chosen to be "as close as possible" to a white noise vector field, it is simpler to discuss the case where the which is the set of solenoidal (i.e. divergence-free) vector fields. This provides us with the Helmholtz-Hodge decomposition of L 2 (T d , R d ) into the orthogonal sum of L 2 pot (T d ) and L 2 sol (T d ). If W were smooth, this would allow us to interpret a∇Φ as the orthogonal projection of W onto L 2 pot (T d ). For non-smooth W , the intepretation remains valid by testing and duality.
Second, we point out that although the generalized GFF fails to satisfy the Markov property, it does satisfy a related domain decomposition property: there exists a family of random fields (Φ A ) on R d indexed by all borel subsets A of R d such that the following holds for every Borel set A ⊆ R d : • we have Φ = Φ A + Φ A c , (A c denotes the complement of A in R d ); • the random fields Φ A and Φ A c are independent; EJP 22 (2017), paper 28.
• the random field Φ A is an a-harmonic function in the interior of A c .
Indeed, these properties are easy to verify from the definition of Φ A as solving −∇ · a∇Φ A = ∇ · (W 1 A ).

Non-Markov property
In this section, we show that a generalized GFF satisfies the Markov property if and only if it is a classical GFF.
is a centered Gaussian with variance R d f · Qf . We denote by (Ω, F, P) the underlying probability space, and by E the associated expectation. Informally, where δ is the Dirac distribution. One can extend the set of admissible test functions for W to every element of L 2 (R d , R d ) by density. We define the solution Φ to (0.1) to be the random distribution such that for every f ∈ C ∞ c (R d ), where (−∇ · a∇) −1 f is the unique function u tending to zero at infinity and such that −∇ · a∇u = f . Formally integrating by parts shows the consistency between (0.1) and (1.1). The latter makes sense since for f ∈ C ∞ c (R d ), the function ∇(−∇ · a∇) −1 f is in L 2 (R d , R d ), as can be checked for instance using the Green representation formula.
For every open U ⊆ R d , we define We simply write F := F(R d ). For every closed A ⊆ R d , we define We enlarge the σ-algebras defined above (without changing the notation) so that they contain all P-negligible measurable sets. We say that the field Φ is Markovian with respect to the open set U if conditionally on F(∂U ), the σ-algebras F(U ) and F(U c ) are independent (where U denotes the closure of U , U c the complement of U , and ∂U = U ∩ U c ).  In particular, if a and Q are proportional, then the field Φ is a classical GFF, and we recall that in this case the Markov property is well-known. In order to prove the theorem, it thus suffices to show that if Φ is Markovian with respect to U , then a and Q are proportional.
To our knowledge, the rigorous study of the Markov property of random fields was initiated with [15], where Lévy's "Brownian motion" indexed by a multidimensional parameter [14] is shown to be Markovian if and only if the space dimension is odd.
For random fields on discrete graphs, the Markov property is equivalent to the locality of the "energy function" (in the Gaussian case, this is the Dirichlet form, and more generally, we mean the logarithm of the probability density, up to a constant). This equivalence can be checked by a direct computation for Gaussian fields, and we refer to [10, Theorems 4.1 and 4.2] for a more general statement. It is natural to expect a similar phenomenon in the continuum. However, a counter-example to this conjecture was given in [23]. We will recall this counter-example in subsection 1.2. In spite of this, relying on the fact that the field Φ is "at least as regular as white noise", we will be able to justify this conjecture in our context.
Prior to the counter-example of [23], incorrect proofs of the general conjecture were published. In [22,13], the arguments are laid down pretending that the field is defined pointwise, and therefore the difficulty caused by the possibly low regularity of the field is missed. The paper [9] proceeds more carefully, but is also flawed 1 .
The equivalence between locality of L and the Markov property was also investigated in the framework of Dirichlet form theory [4,24,11]. The arguments given there rely on potential theory, and only apply to operators satisfying the maximum principle. This is unfortunately not the case of the operator L we consider here. In order to see this, we can use the general result of [3], which identifies the class of integro-differential operators satisfying the maximum principle. In our case, the operator is moreover invariant under translation. In this setting, the results of [3] can be understood as follows: if the operator satisfies the maximum principle, then it is the generator of a Lévy process. Moreover, this Lévy process must have the same scale invariance as Brownian motion, in view of the scaling properties of L. By the Lévy-Khintchine formula, it follows that such a Lévy process must be a multiple of Brownian motion. The operator L must therefore be of the form −∇ · b∇ for some matrix b, but we show below that this can only happen when a and Q are proportional.

Proof of Theorem 1.1
To sum up: using some mild regularity property of the field Φ defined by (1.1), we show that the Markov property implies that L is a local operator. We then observe that L is not a local operator, unless a and Q are proportional. As announced in the theorem, we will be sufficiently careful to actually only require the Markov property with respect to one non-trivial open set.
We denote by H the Hilbert space obtained as the closure of {Φ(f ), f ∈ C ∞ c } in L 2 (Ω, F, P). We denote by K the Hilbert space obtained by completing {L −1 f, f ∈ C ∞ c } with respect to the scalar product f, g L := f Lg.
The space K is usually called the reproducing kernel Hilbert space. Note that f → is well-defined and extends to an isometry from H onto K.
The isometry of Lemma 1.2 induces an isometry between H(U ) and K(U ). If A ⊆ R d is a closed set, we define The following lemma is classical. (2) The map is the orthogonal projection onto H(U ).
We have shown Since both spaces are closed, this completes the proof of (1). Part (2) follows at once, Proof. We decompose the proof into three steps.
In particular, and this proves (1.4).
Step 2. We show that The equality is clear. The orthogonality of the sets on the left-hand side follows from the definition of H 0 and the previous step. In particular, the set H(U ) + H(U c ) is closed.
Step 3. Let f ∈ C ∞ c . In order to complete the proof, it suffices to see that This is the step of the proof where we need to use the fact that Φ is "at least as regular as white noise". In more precise words, we show that we can control the quadratic form g → gL −1 g over local functions g by the L 2 norm of g squared, and that this is sufficient to conclude. Without loss of generality, we assume that the support of f is contained in the unit ball B(0, 1). Let χ : R → [0, 1] be a C ∞ function such that χ = 1 on (−∞, 0] and χ = 0 on [1, +∞), and let where dist(x, U ) denotes the distance between x and the open set U . Note that and that Φ(f [1 − χ n ]) ∈ H(U c ). We now argue that Φ(f χ n ) converges to an element of H(U ) as n tends to infinity. This would imply that Φ(f [1 − χ n ]) converges to an element of H(U c ) and therefore complete the proof of (1.6). If the sequence (Φ(f χ n )) converges in H, then the limit is necessarily in H(U ); so what needs to be argued is simply the so we need to check that f χ n is a Cauchy sequence with respect to the seminorm Letting Λ be the largest eigenvalue of Q and h := (−∇ · a∇) −1 g, we have Since d 3, using the Green function representation, we obtain that there exists a constant C < ∞ (depending only on a) such that To sum up, we have shown that there exists a constant C < ∞ such that for every g ∈ C ∞ c with support in B(0, 1), Since f χ n → f in L 2 , this completes the proof.
Proof. The isometry of Lemma 1.2 transports the decomposition of H in Lemma 1.4 into a decomposition of K. In particular, Let g ∈ C ∞ c with support disjoint from S. We have By density, we deduce that f 1 ∈ K(U c ) ⊥ ⊆ K(U ). By symmetry, we also have f 2 ∈ K(U ) ⊥ , which completes the proof.
Proof. In the Fourier domain, f 1 , f 2 L can be written explicitly as We change ξ to Q − 1 2 ξ, and then go back to physical domain, to obtain that Since a and Q are not proportional, Q − 1 2 aQ − 1 2 is not a multiple of identity. By denoting we only need to show that there exists g 1 , g 2 whose supports are in Q 1 2 U and Q 1 2 U c respectively and such that For x away from the support of g 2 , we can write with G the Green function of −∆. Since g 1 , g 2 have disjoint supports, we can further integrate by parts to obtain . By an explicit calculation, we have, for x = 0, for some constant c = 0. We show in Appendix A that if A is not a multiple of the identity, then (x t Ax) 2 /|x| 2 cannot be a quadratic form. As a consequence, the function x → |x| d+6 K(x) is a non-zero polynomial over R d \ {0}. This implies that we can find g 1 , g 2 whose supports are in Q 1 2 U and Q 1 2 U c respectively and satisfy R 2d The proof is complete.
Proof of Theorem 1.1. The result follows from Lemmas 1.5 and 1.6.

A counter-example with less regularity
For the reader's convenience, we briefly recall the counter-example given in [23]. Let For A open or closed, denote by F H (A) the σ-algebra generated by the random variables in H(A). We claim that for every such A, we have F( is dense in W 1,2 (the closure of the union above is W 1,2 perp (A)). Hence, by the martingale convergence theorem, if Z is a bounded F-measurable random variable, then

Homogenization and expansion of effective fluctuation tensor
We now turn to stochastic homogenization. We focus on discrete elliptic equations in divergence form: d is a field of diagonal matrices. The entries a ii (x), x ∈ Z d , i = 1, . . . , d are i.i.d. random variables defined on the probability space (Ω, F, P). We assume that a ii (x) ∈ [1 − τ 0 , 1 + τ 0 ] for some fixed τ 0 ∈ (0, 1). Let  as the random conductance on the edge linking x to x + e i . The discrete gradient and divergence for f : Z d → R and F : Z d → R d are defined respectively by . For any e = (e,ē) ∈ B and f : Z d → R, we also write ∇f (e) = f (ē) − f (e). For any ξ ∈ R d and i = 1, . . . , d, we define ξ(e i ) = ξ i .
There exists a deterministic matrix a ∈ Sym + d such that if α > 0 and u (ε) solves then it was shown in [12,Theorem 4 where u ε is extended to R d as a piecewise constant function. In other words, on large scales, the random coefficients a(x) behave like the homogeneous, deterministic coefficients a, and the discrete heterogeneous operator ∇ * a(x)∇ is averaged as the continuous homogeneous operator −∇ · a∇.
As was pointed out in the introduction, the corrector plays an important role in proving the convergence of u ε → u h and calculating the homogenized matrix a. The equation of the corrector in the direction of ξ ∈ R d says ∇ * a(x)(∇φ ξ (x) + ξ) = 0, x ∈ Z d . (2.1) Since we assume d 3, there exists a stationary zero-mean random field solving (2.1), as was shown in [7]. From now on, we assume furthermore that a(e) = a(ζ e ), where a : R → R is a fixed twice differentiable function with bounded first and second derivatives (and taking values in [1 − τ 0 , 1 + τ 0 ]), and {ζ e : e ∈ B} is a sequence of i.i.d.
standard Gaussian random variables. Under this technically convenient condition, the large-scale behavior of the corrector is approximately a generalized GFF. More precisely, it was shown in [19,18] that for a certain white noise vector field W ξ . A heuristic derivation of (2.3) can be found in Here · denotes the expectation in Ω and {φ i = φ ei , i = 1, . . . , d} are the correctors in the canonical directions. For sufficiently smooth functions on Ω, the weak derivative with respect to ζ e is denoted by ∂ e , and its adjoint is ∂ * e = ζ e − ∂ e . In (2.4), we have set and a (e) := ∂ e a(e) = a (ζ e ).
Note that the map ξ → Q ξ is quadratic. This object should thus be viewed as a four-fold tensor, which we propose to call the effective fluctuation tensor.
The convergence of the random distribution in (2.1) to Φ ξ motivates our study of generalized GFFs. Since we assume the coefficients to be i.i.d., the homogenized matrix inherits the symmetries of the lattice, and a is thus a multiple of the identity.
The goal of this section is to obtain an expansion of Q ξ in the regime of small ellipticity contrast. This is interesting per se, e.g. as a means to compute the effective fluctuation tensor when we expect only a small amount of random fluctuation in the underlying medium, in which case Q ξ may be replaced by the series expansion up to certain order depending on the desired accuracy. It also allows us to give examples of environments such that Q ξ is not a multiple of the identity, which implies, in view of Theorem 1.1, that the limiting field Φ ξ does not satisfy the Markov property. Without loss of generality, we may assume that a takes the form where b is a field of diagonal matrices with b = 0 and b L ∞ 1, and τ ∈ (0, 1) is a free parameter that we will take sufficiently small. We let δ ij be the Kronecker symbol, that is, δ ij = 1 if i = j, and δ ij = 0 otherwise. Theorem 2.1. There exists τ 0 > 0 such that for every τ ∈ [0, τ 0 ), ξ ∈ R d and i, j = 1, . . . , d, we have the convergent expansion where c ξ,l,i,j can be computed explicitly as explained below, and c ξ,0,i,j = δ ij ξ 2 i b(e) 2 , In particular, if τ > 0 is sufficiently small and ξ = 0, then Q ξ is not a multiple of identity.
We first perform a formal expansion of Q ξ and observe that (2.6)-(2.8) should hold.
We then justify the full expansion rigorously.

A formal expansion
In view of (2.5), the corrector equation (2.1) in the direction η ∈ R d can be rewritten where ∆ := −∇ * ∇ is the discrete Laplacian. Formally, we can write If the operator −τ ∇(−∆) −1 ∇ * b is a contraction (note that τ 1), then (2.10) There are four factors of η + ∇φ η appearing in (2.4) with η = e i , e j , ξ, which we will replace by the expansion in (2.9). Since a (e) = τ b (e) for any e ∈ B, we obtain an expansion of [Q ξ ] ij in terms of τ written as: where we suppressed the dependence of c ξ,l on i, j in the notation. We emphasize that ∞ l=0 c ξ,l τ l is an expansion of (2.12) where the only τ −dependent terms are those factors of η + ∇φ η with η = e i , e j , ξ. The following are simple calculations using the i.i.d. structure of the random coefficients.
We introduce some notation. Let G(x, y) be the Green function of the discrete Laplacian −∆, and ∇∇G(x, y) be the Hessian matrix such that where ∇ i , ∇ j are with respect to the x, y variable respectively. For any edge e = (e,ē) ∈ B, we also write ∇∇G(e, y) = ∇G(ē, y) − ∇G(e, y).

Second order.
To get the second order term, we replace all factors in (2.4) of the form η + ∇φ η by X 0,η = η to obtain where we used [19,Proposition 3.1] in the last step.
Third order. From now on, we focus on the case i = j. To get the third order term, we replace one of the four factors of η + ∇φ η in (2.4) by X 1,η and all other three by X 0,η = η. We write which is a vector for any x ∈ Z d . We also write which is a scalar for any e ∈ B. We claim that we only need to consider e i + ∇φ i or e j + ∇φ j . Otherwise if we replace ξ + ∇φ ξ by X 1,ξ , i.e., in (2.4) e i (e k ) + ∇φ i (e k ) → e i (e k ), e j (e k ) + ∇φ j (e k ) → e j (e k ), we observe that there is a factor of e i (e k )e j (e k ) = δ ik δ jk = 0 since i = j. Therefore, we For the first term on the r.h.s. of the above expression, we can write Since i = j, it is clear that ∇∇G(e j , y) · b(y)e i and b (e j )(1 + L ) −1 b (e j ) are independent for all y ∈ Z d , so we have the expectation equals zero because b has mean zero. The same discussion applies to the second term, thus c ξ,1 = 0.
Fourth order. To get the fourth order term, we have multiple options. First we consider the case when we have one factor of X 2,η coming from η + ∇φ η . By the same reason as before, we can not choose η = ξ, otherwise we have a factor of δ ik δ jk = 0. If instead we consider η = e i (the same discussion applies to e j ) and write: By the same discussion as before, b(z)e i is independent of b (e j )(1 + L ) −1 b (e j ) for all z ∈ Z d because i = j. Since ∇∇G(e j , y) · b(y)∇∇G(y, z)b(z)e i is a linear combination of b kk (y) and b ii (z), the only nonzero contribution after taking the expectation is when y = z and k = i: EJP 22 (2017), paper 28.
Since ∇∇ i G(e j , y) is a gradient, it follows that y∈Z d ∇∇ i G(e j , y) = 0. Now we consider the case when we have two factors of X 1,η . Recall that By symmetry, we only need to consider the following cases: Note that we do not consider replacing both factors of ξ + ∇φ ξ since it leads to a factor of δ ik δ jk = 0. For (i), we write and in order to get a nonzero contribution in (i), we only need the terms with y = z and the factor ξ i , i.e., τ 2 In the last step, we used the fact that a (e j )(1 + L ) −1 a (e j ) is independent of b ii (y) for all y ∈ Z d . For (ii), we write and it is clear that for any k = 1, . . . , d, we have an independent factor of b(y)e i or b(z)e j since i = j, thus the above expression equals to zero. For (iii), we write To summarize, the above formal calculation shows that for i = j, c ξ,0 = c ξ,1 = 0, (2.13) and which is non-zero if ξ i ξ j = 0.

Proof of the expansion
Before proving the expansion rigorously, we introduce more notation. Let ζ = (ζ e ) e∈B ∈ Ω denote the sample point, and for x ∈ Z d , we define the shift operator τ x on Ω by (τ x ζ) e = ζ x+e , where x + e := (x + e, x +ē) is the edge obtained by shifting e by x.
Since a = I d + τ b, (2.16) can be written as The operator D(−∆) −1 D * is defined by the following lemma.
..,d = D(−∆) −1 D * F be a weak limit of Dφ λ in L 2 (Ω), then Ψ is the unique function in L 2 (Ω, R d ) that satisfies the following three properties: where the last step comes from Ψ · DG = F · DG for any test function G. This implies On the other hand, Thus Dφ λ · Dφ λ → Ψ · Ψ as λ → 0. The proof is complete.
To justify the expansion, we need L 4 rather than L 2 boundedness of the operator D(−∆) −1 D * . The following lemma, whose proof is postponed to Appendix B, is the key to justify the expansion. Lemma 2.3. For any p ∈ (1, ∞), there exists C p > 0 such that Proof of Theorem 2.1. We first rewrite (2.4) as By Lemma 2.3 and the fact that b ∈ L ∞ (Ω, R d×d ), for any p ∈ (1, ∞) and n ∈ N + we have P n η p C n p η ∞ EJP 22 (2017), paper 28.
To show that Q ξ may not be a multiple of identity, we only need to repeat the formal calculation in Section 2.1 verbatim with G replaced by G λ to derive that with c ξ,2 given by (2.14). The proof is complete. for all x ∈ R n , (x t Ax) 2 = |x| 2 x t Bx (ii) the matrix A is a multiple of identity.
Proof. The implication (ii) =⇒ (i) is obvious. We prove the converse implication by induction on the size n of the matrix. Letting A be an n × n matrix and assuming that the result holds for matrices of size n − 1, we can choose x = (x 1 , . . . , x n−1 , 0) to obtain A ij = cδ ij and B ij = c 2 δ ij for some constant c > 0 with i, j = 1, . . . , n − 1. Now, for any i < n, by considering the coefficients of x 3 i x n and x i x 3 n , we have The goal here is to prove the boundedness of D(λ − ∆) −1 D * from L p (Ω) to itself, and we will borrow a deterministic estimate from [2].
The same proof works in our case since the only ingredient we need to change in their proof is the following bound on the triple gradient of G λ , which was given by [2, Lemma
Using Lemma B.1 and the fact that D(λ − ∆) −1 D * is bounded from L 2 (Ω, R d ) to itself, we can apply the standard interpolation argument, e.g., [2,Theorem 4.4], to conclude that D(λ − ∆) −1 D * is bounded from L p (Ω, R d ) to itself for any p ∈ (1, ∞). For the case λ = 0, we only need to note that for any F ∈ L 2 (Ω, R d ), in L 2 (Ω, R d ) as λ → 0, so by applying Fatou's Lemma, we conclude D(−∆) −1 D * is bounded from L p (Ω, R d ) to itself. The proof of Lemma 2.3 is complete.