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Random walks and exclusion processes among random conductances on random inﬁnite clusters: homogenization and hydrodynamic limit

We consider a stationary and ergodic random ﬁeld { ω ( b ) : b ∈ E d } parameterized by the family of bonds in Z d , d > 2. The random variable ω ( b ) is thought of as the conductance of bond b and it ranges in a ﬁnite interval [ 0, c 0 ] . Assuming that the set of bonds with positive conductance has a unique inﬁnite cluster C ( ω ) , we prove homogenization results for the random walk among random conductances on C ( ω ) . As a byproduct, applying the general criterion of [ F ] leading to the hydrodynamic limit of exclusion processes with bond–dependent transition rates, for almost all realizations of the environment we prove the hydrodynamic limit of simple exclusion processes among random conductances on C ( ω ) . The hydrodynamic equation is given by a heat equation whose diffusion matrix does not depend on the environment. We do not require any ellipticity condition. As special case, C ( ω ) can be the inﬁnite cluster of supercritical Bernoulli bond percolation.


Introduction
We consider a stationary and ergodic random field ω = ω(b) : b ∈ E d , parameterized by the set E d of non-oriented bonds in Z d , d 2, such that ω(b) ∈ [0, c 0 ] for some fixed positive constant c 0 . We call ω the conductance field and we interpret ω(b) as the conductance at bond b. We assume that the network of bonds b with positive conductance has a.s. a unique infinite cluster C, and call E the associated bonds. Finally, we consider the exclusion process on the graph (C, E) with generator L defined on local functions f as where the configuration η b is obtained from η by exchanging the values of η x and η y , if b = {x, y}. If ω(b) : b ∈ E is a family of i.i.d. random variables such that ω(b) is positive with probability p larger than the critical threshold p c for Bernoulli bond percolation, then the above process is an exclusion process among positive random conductances on the supercritical percolation cluster. If p = 1, then the model reduces to the exclusion process among positive random conductances on Z d . Due to the disorder, the above model is an example of non-gradient exclusion process, in the sense that the transition rates cannot be written as gradient of some local function on {0, 1} C [KL] (with exception of the case of constant conductances). Despite this fact, the hydrodynamic limit of the exclusion process can be proven without using the very sophisticated techniques developed for non-gradient systems (cf. [KL] and references therein), which in addition would require non trivial spectral gap estimates that fail in the case of conductances non bounded from below by a positive constant (cf. Section 1.5 in [M]). The strong simplification comes from the fact that, since the transition rates depend only on the bonds but not on the particle configuration, the function Lη x , where η x is the occupancy number at site x ∈ C, is a linear combinations of occupancy numbers. Due to this degree conservation the analysis of the limiting behavior of the random empirical measure π(η) = x∈C η x δ x is strongly simplified w.r.t. disordered models with transition rates depending both on the disorder and on the particle configuration [Q1], [FM], [Q2]. Moreover, the function Lη x can be written as (Lη) x , where L is the generator of the random walk on C of a single particle among the random conductances ω(b) and η ∈ {0, 1} C is thought of as an observable on the state space C of the random walk. This observation allows to derive the hydrodynamic limit of the exclusion process on C from homogenization results for the random walk on C. This reduction has been performed in [N] for the exclusion process on Z with bond-dependent conductances, the method has been improved and extended to the d-dimensional case in [F]. The arguments followed in [F] are very general and can be applied also to exclusion processes with bond-dependent transition rates on general (non-oriented) graphs, even with non diffusive behavior (see [FJL] for an example of application). The reduction to a homogenization problem can be performed also by means of the method of corrected empirical measure, developed in [JL] and [J]. In [JL] the authors consider exclusion processes on Z with bond-dependent rates, while in [J] the author proves the hydrodynamic limit for exclusion processes with bond-dependent rates on triangulated domains and on the Sierpinski gasket. Moreover, in [J] the author reobtains the hydrodynamic limit of the exclusion process on Z d among conductances bounded from above and from below by positive constants. Note that this last result follows at once by applying the standard non-gradient methods (in this case, their application becomes trivial) or the discussion given in [F][Section 4]. Moreover, it is reobtained in the present paper by taking (ω b : b ∈ E d ) as independent strictly positive random variables.
By the above methods [N], [F], [JL], [J], the proof of the hydrodynamic limit of exclusion processes on graphs with bond-dependent rates reduces to a homogenization problem. In our contest, we solve this problem by means of the notion of two-scale convergence, which is particularly fruitful when dealing with homogenization problems on singular structures. The notion of two-scale convergence was introduced by G. Nguetseng [Nu] and developed by G. Allaire [A]. In particular, our proof is inspired by the method developed in [ZP] for differential operators on singular structures. Due to the ergodicity and the Z d -translation invariance of the conductance field, the arguments of [ZP] can be simplified: for example, as already noted in [MP], one can avoid the introduction of the Palm distribution. Moreover, despite [ZP] and previous results of homogenization of random walks in random environment (see [Ko], [Ku], [PR] for example), in the present setting we are able to avoid ellipticity assumptions.
We point out that recently the quenched central limit theorem for the random walk among constant conductances on the supercritical percolation cluster has been proven in [BB] and [MP], and previously for dimension d 4 in [SS]. Afterwards, this result has been extended to the case of i.i.d. positive bounded conductances on the supercritical percolation cluster in [BP] and [M]. Their proofs are very robust and use sophisticated techniques and estimates (as heat kernel estimates, isoperimetric estimates, non trivial percolation results, ...), previously obtained in other papers. In the case of i.i.d. positive bounded conductances on the supercritical percolation cluster, we did not try to derive our homogenization results from the above quenched CLTs (this route would anyway require some technical work). Usually, the proof of the quenched CLT is based on homogenization ideas and not viceversa. And in fact, our proof of homogenization is much simpler than the above proofs of the quenched CLT. Hence, the strategy we have followed has the advantage to be self-contained, simple and to give a genuine homogenization result without ellipticity assumptions. This would have not been achieved choosing the above alternative route. In addition, our results cover more general random conductance fields, possibly with correlations, for which a proof of the quenched CLT for the associated random walk is still lacking.
The paper is organized as follows: In Section 2 we give a more detailed description of the exclusion process and the random walk among random conductances on the infinite cluster C. In addition, we state our main results concerning the hydrodynamic behavior of the exclusion process (Theorem 2.2) and the homogenization of the random walk (Theorem 2.4 and Corollary 2.5). In Section 3 we show how to apply the results of [F] in order to derive Theorem 2.2 from Corollary 2.5, while the remaining sections are focused on the homogenization problem. In particular, the proof of Theorem 2.4 is given in Section 6, while the proof of Corollary 2.5 is given in Section 7. Finally, in the Appendix we prove Lemma 2.1, assuring that the class of random conductance fields satisfying our technical assumptions is large.

Models and results
2.1. The environment. The environment modeling the disordered medium is given by a stationary and ergodic random field Stationarity and ergodicity refer to the natural action of the group of Z d -translations. ω and ω(b) are thought of as the conductance field and the conductance at bond b, respectively. We call Q the law of the field ω and we assume that for some fixed positive constant c 0 . Hence, without loss of generality, we can suppose that Q is a probability measure on the product space Ω := [0, c 0 ] E d . Moreover, in order to simplify the notation, we write ω(x, y) for the conductance ω(b) if b = {x, y}. Note that ω(x, y) = ω(y, x).
Below, we denote by E(ω) the bonds in E(ω) connecting points of C(ω) and we will often understand the fact that ω ∈ Ω 0 .
Define B(Ω) as the family of bounded Borel functions on Ω and let D be the d × d symmetric matrix characterized by the variational formula (a, Da) = 1 m inf ψ∈B(Ω) e∈B * Ω ω(0, e)(a e + ψ(τ e ω) − ψ(ω)) 2 I 0,e∈C(ω) Q(dω) , ∀a ∈ R d , and the translated environment τ e ω is defined as τ e ω(x, y) = ω(x + e, y + e) for all bonds {x, y} in E d . In general, I A denotes the characteristic function of A. Our last assumption on Q is that the matrix D is strictly positive: In conclusion, our hypotheses on the random field ω are given by stationarity, ergodicity, (H1),(H2) and (H3). The lemma below shows that they are fulfilled by a large class of random fields. In order to state it, given c > 0 we define the random fieldω c = ω c For c = 0 we simply setω :=ω 0 . We postpone the proof of the above Lemma to the Appendix.
2.2. The exclusion process on the infinite cluster C(ω). Given a realization ω of the environment, we consider the exclusion process η(t) on the graph G(ω) = C(ω), E(ω) with exchange rate ω(b) at bond b. This is the Markov process with paths η(t) in the Skohorod space D [0, ∞), {0, 1} C(ω) (cf. [B]) whose Markov generator L ω acts on local functions as We recall that a function f is called local if f (η) depends only on η x for a finite number of sites x. By standard methods [L] one can prove that the above exclusion process η(t) is well defined.
Every configuration η in the state space {0, 1} C(ω) corresponds to a system of particles on C(ω) if one considers a site x occupied by a particle if η x = 1 and vacant if η x = 0. Then the exclusion process is given by a stochastic dynamics where particles can lie only on sites x ∈ C(ω) and can jump from the original site x to the vacant site y ∈ C(ω) only if the bond {x, y} has positive conductance, i.e. x and y are connected by a bond in G(ω). Roughly speaking, the dynamics can be described as follows: To each bond b = {x, y} ∈ E(ω) associate an exponential alarm clock with mean waiting time 1/ω(b). When the clock rings, the particle configurations at sites x and y are exchanged and the alarm clock restarts afresh. By Harris' percolation argument [D], this construction can be suitably formalized. Finally, we point out that the only interaction between particles is given by site exclusion.
We can finally describe the hydrodynamic limit of the above exclusion process among random conductances ω(b) on the infinite cluster C(ω). If the initial distribution is given by the probability measure µ on {0, 1} C(ω) , we denote by P ω,µ the law of the resulting exclusion process.
with boundary condition ρ 0 at t = 0 and where the symmetric matrix D is variationally characterized by (2.1).
If a density profile ρ 0 can be approximated by a family of probability measures µ ε on {0, 1} C(ω) (in the sense that (2.5) holds for each δ > 0 and ϕ ∈ C c (R d )), then it must be 0 ρ 0 m a.s. On the other hand, if ρ 0 : R d → [0, m] is a Riemann integrable function, then it is simple to exhibit for Q-a.a. ω a family of probability measures µ ε on {0, 1} C(ω) approximating ρ 0 . To this aim we observe that, due to the ergodicity of Q and by separability arguments, for Q a.a. ω it holds for each Riemann integrable function ϕ : R d → R with compact support. Fix such an environment ω. Then, it is enough to define µ ε as the unique product probability measure on {0, 1} C(ω) such that µ ε (η x = 1) = ρ 0 (εx)/m for each x ∈ C(ω). Since the random variable ε d x∈C(ω) ϕ(εx) η x is the sum of independent random variables, it is simple to verify that its mean equals ε d x∈C(ω) ϕ(εx)ρ 0 (εx)/m and its variance equals ε 2d x∈C(ω) ϕ 2 (εx)[ρ 0 (εx)/m][1 − ρ 0 (εx)/m]. The thesis then follows by means of (2.8) and the Chebyshev inequality.
The proof of Theorem 2.2 is given in Section 3. As already mentioned, it is based on the general criterion for the hydrodynamic limit of exclusion processes with bond-dependent transition rates, obtained in [F] by generalizing an argument of [N], and homogenization results for the random walk on C(ω) with jump rates ω(b), b ∈ E(ω), described below.
2.3. The random walk among random conductances on the infinite cluster C(ω).
Given ω ∈ Ω we denote by X ω (t|x) the continuous-time random walk on C(ω) starting at x ∈ C(ω), whose Markov generator L ω acts on bounded functions g : C(ω) → R as The dynamics can be described as follows. After arriving at site z ∈ C(ω), the particle waits an exponential time of parameter and then jumps to a site y ∈ C(ω), |z − y| = 1, with probability ω(z, y)/λ ω (z). Since the jump rates are symmetric, the counting measure on C(ω) is reversible for the random walk.
In what follows, given ε > 0 we will consider the rescaled random walk with starting point x ∈ εC(ω). We denote by µ ε ω the reversible rescaled counting measure and write L ε ω for the symmetric operator on L 2 (µ ε ω ) defined as Due to (2.8), for almost all ω ∈ Ω the measure µ ε ω converges vaguely to the measure m dx, where the positive constant m is defined in (2.2). In what follows, · µ ε ω and (·, ·) µ ε ω will denote the norm and the inner product in L 2 (µ ε ω ), respectively. We recall a standard definition in homogenization theory (cf. [Z], [ZP] and reference therein): for every family ϕ ε ∈ L 2 (µ ε ω ) weakly converging to ϕ ∈ L 2 (m dx). The strong convergence f ε ω → f admits the following characterization (cf. [Z][Proposition 1.1] and references therein): We need now to isolate a Borel subset Ω * ⊂ Ω of regular environments. To this aim we first define Ω 1 as the set of ω ∈ Ω 0 (recall the definition of Ω 0 given before (H2)) such that From the the ergodicity of Q and the separability of C c (R d ) and C(Ω), it is simple to derive that Q(Ω 1 ) = 1. The set of regular environments Ω * will be defined in Section 4, after Lemma 4.4, since its definition requires the concept of solenoidal forms. We only mention here that Ω * ⊂ Ω 1 and Q(Ω * ) = 1. We can finally state our main homogenization result, similar to [ZP][Theorem 6.1]: where the symmetric matrix D is variationally characterized in (2.1).
The proof of Theorem 2.4 will be given in Section 6. We state here an important corollary of the above result: Set P ε t,ω = e tL ε ω , P t = e t∇·(D∇·) . Note that P ε t,ω : t 0 is the L 2 (µ ε ω )-Markov semigroup associated to the random walk X ε,ω (t|x), i.e.
while P t is the L 2 (mdx)-Markov semigroup associated to the diffusion with generator ∇ · (D∇·). As proven in Section 7 it holds: In particular, for each ω ∈ Ω * , given any function 3. Proof of Theorem 2.2 As already mentioned, having the homogenization result given by Corollary 2.5, Theorem 2.2 follows easily from the criterion of [F] for the hydrodynamic limit of exclusion processes with bond-dependent rates. The method discussed in [F] is an improvement of the one developed in [N] for the analysis of bulk diffusion of 1d exclusion processes with bond-dependent rates. Although in [F] we have discussed the criterion with reference to exclusion processes on Z d , as the reader can check the method is very general and can be applied to exclusion processes on general graphs with bond-dependent rates, also under non diffusive space-time rescaling and also when the hydrodynamic behavior is not described by heat equations (cf. [FJL] for an example).
The following proposition is the main technical tool in order to reduce the proof of the hydrodynamic limit to a problem of homogenization for the random walk performed by a single particle (in absence of other particles). Recall the definition (2.23) of the semigroup P ε t,ω associated to the rescaled random walk X ε,ω defined in (2.10).
Proposition 3.1. For Q-a.a. ω the following holds. Fix δ, t > 0, ϕ ∈ C c (R d ) and let µ ε be a family of probability measures on {0, 1} C(ω) . Then Proof. One can prove the above proposition by the same arguments used in [F][Section 3] or one can directly invoke the discussion of [F][Section 4] referred to exclusion processes on Z d with non negative transition rates, bounded from above. In fact, to the probability measure µ ε on {0, 1} C(ω) one can associate the probability measure ν ε on {0, 1} Z d so characterized: ν ε is concentrated on the event Note that, if η(t) has law Pµ ε , then σ(t) is the exclusion process on Z d with initial distribution ν ε and generator . In particular, Proposition 3.1 coincides with the limit (B.2) in [F][Section 4].
We can now complete the proof of Theorem 2.2. First we observe that whenever (2.5) is satisfied for functions of compact support, then it is satisfied also for functions that vanish fast at infinity. Indeed, for our purposes it is enough to show that the limit For such a function f , given ℓ > 0 we can find (3.6) The above limit together with (3.4) and (3.5) implies (3.2) for all functions f ∈ C(R d ) satisfying (3.3). In particular, (3.3) is valid for f = P t ϕ, where P t = e t∇·(D∇·) and ϕ ∈ C c (R d ). Indeed, in this case f decays exponentially. Due to this observation, Proposition 3.1 and the fact that for any ϕ ∈ C c (R d ) and δ > 0. Since (3.7) follows from (2.25) of Corollary 2.5. This concludes the proof of Theorem 2.2.

Square integrable forms
We now focus our attention on the proof of homogenization for the random walk on the infinite cluster. To this aim, in this section we introduce the Hilbert space of square integrable forms and show how the variational formula (2.1) can be interpreted in terms of suitable orthogonal projections inside this Hilbert space.
Let M(R d ) be the family of Borel measures on R d . Given x ∈ Z d , y ∈ R d , ω ∈ Ω and ν ∈ M(R d ), τ x ω ∈ Ω and τ y ν ∈ M(R d ) are defined as Note that the family of random measures µ ε ω satisfies the identity Let µ be the measure on Ω absolutely continuous w.r.t. Q such that and define B as Given real functions u defined on Ω and v defined on Ω × B, we define the gradient ∇ (ω) u : Ω × B → R and the divergence ∇ (ω) * v : Ω → R, respectively, as follows: where v is any bounded Borel function on Ω × B. Note that if u ∈ L p (µ) and v ∈ L p (M ) then ∇ (ω) u ∈ L p (M ) and ∇ (ω) * v ∈ L p (µ). The space L 2 (M ) is called the space of square integrable forms. Note that M gives zero measure to the set (ω, e) ∈ Ω × B : {0, e} ∈ E(ω) .
The above identity and (4.6) allows to conclude the proof of (4.5), while the second part of the lemma follows easily from (4.5).
We point out another integration by parts formula. Then, Proof. By definition of v, we have (4.8) Moreover, since I z∈C(ω) ω(z, z + e) = I z+e∈C(ω) ω(z, z + e), we can write Identities (4.8) and (4.9) allow to conclude the proof of (4.7).
Finally, we point out the simple identities valid for all functions a, b : εC(ω) → R d . In what follows, (4.10) and (4.11) will be frequently used without explicit mention. Due the translation invariance of Q we conclude that Since C(ω) is connected, (4.15) is equivalent to say that for Q-a.a. ω there exists a constant a(ω) such that u(τ x ω) = a(ω) for all x ∈ C(ω). Trivially, the function a(ω) is translation invariant. Hence, due to the ergodicity of Q we can conclude that a(ω) is constant Q-a.s. Since a(ω) = u(ω) if 0 ∈ C(ω), we conclude that u(ω) is constant for µ-a.a. ω.
We have now all the tools in order to define the set of regular environments Ω * . To this aim we first observe that L 2 sol is separable since it is a subset of the separable metric space L 2 (M ). We fix once and for all a sequence {ψ j } j 1 dense in L 2 sol . Since elements of L 2 (M ) are equivalent if, as functions, they differ on a zero measure set, we fix a representative ψ j and from now on we think of ψ j as pointwise function ψ j : Ω × B → R. Since ψ j ∈ L 2 (M ) it must be Ψ j,e (ω) := ω(0, e)ψ j (·, e) ∈ L 2 (µ) . j,e ∈ C(Ω) such that f (k) j,e converges to Ψ j,e in L 2 (µ) as k → ∞. We first make a simple observation: Lemma 4.4. Given j 1, define the Borel set Ω * ,j as the set of configurations ω ∈ Ω 0 such that for each e ∈ B and k, n 1. Then Q(Ω * ,j ) = 1.
Proof. Let us define the set A x as Since Ω 0 \ A coincides with the set of ω ∈ Ω 0 satisfying (4.17), we only need to prove that (4.18) is satisfied Q-a.s. for each k, n 1 and e ∈ B. This is a direct consequence of the L 1 -ergodic theorem.
Recall the definition of Ω 1 ⊂ Ω 0 given before Theorem 2.4. We can finally define the set Ω * : Definition 3. We define the set Ω * of regular environments as We conclude this section by reformulating the variational characterization (2.1) of the diffusion matrix D in terms of square integrable forms. To this aim, given a vector ξ ∈ R B * , we write w ξ for the square integrable form w ξ (ω, ±e) := ±ξ e , e ∈ B * , ω ∈ Ω .  Moreover, due to the definition of orthogonal projection, we get (4.23) By definition, (4.24) We can rewrite the last term in a more useful form. In fact, due to the translation invariance of Q, we get (4.25) Due to (4.23), (4.24) and (4.25) we conclude that (4.26) In particular, the matrix D is related to the matrix D via the identity D = 2mD . In particular, the above identity would hold with ψ = πw ξ . Due to (4.20) and (4.21), this would imply that (ξ, Dξ) = 0, which is absurd due to hypothesis (H3).
Finally, we conclude with a simple but crucial observation. Given ξ ∈ R B * , there exists a unique form v ∈ L 2 pot such that w ξ + v ∈ L 2 sol . In fact, these requirements imply that w ξ + v = πw ξ .

Two-scale convergence
In this section we analyze the weak two-scale convergence for our disordered model. We recall that Ω * denotes the set of regular environments ω defined in the previous section, and we recall that (·, ·) µ ε ω and · µ ε ω denote respectively the inner product and the norm in L 2 (µ ε ω ). In our context the two-scale convergence [ZP][Section 5] can be defined as follows: ⇀ v) if the following two conditions are fulfilled: and ψ ∈ C(Ω). Let us first collect some technical results concerning the weak two-scale convergence. For the next lemma, recall the definition of the function Ψ j,e ∈ L 2 (µ) given in (4.16).
Lemma 5.1. Fix ω ∈ Ω * and suppose that L 2  j,e ∈ C(Ω) given in Section 4. Then, by Schwarz inequality, we get Since ω ∈ Ω * ⊂ Ω * ,j and since f (k) j,e → Ψ j,e in L 2 (µ), we conclude that the upper limit of the r.h.s. as ε ↓ 0 and then k ↑ ∞ is zero. On the other hand, since v ε 2 ⇀ v and f (k) j,e → Ψ j,e in L 2 (µ), we obtain that This allows to get (5.3).
By the same arguments leading to [ZP][Lemma 5.1] and [Z][Prop. 2.2] one can easily prove the following result: Lemma 5.2. Fix ω ∈ Ω * . Suppose that the family of functions v ε ∈ L 2 (µ ε ω ) satisfies (5.1). Then from each sequence ε k converging to zero, one can extract a subsequence ε kn such that v ε converges along ε kn to some v ∈ L 2 (R d × Ω, dx × µ) in the sense of weak two-scale convergence.
We give the proof for the reader's convenience: (note that the first estimate follows from Schwarz inequality, while the second one follows from (2.17) and (5.1)). Using a standard diagonal argument and the separability of the space of test functions ϕ, ψ, we can conclude that there exists a subsequence {ε kn } n 1 along which the limit in the l.h.s. of (5.2) exists and can be extended to a continuous linear functional on L 2 (R d × Ω, dx × µ). Therefore, this limit can be written as the inner product in In what follows, we will apply the concept of weak two-scale convergence to the solution u ε ω ∈ L 2 (µ ε ω ) of (2.18) and to its gradients, for a fixed sequence f ε ω ⇀ f . To this aim we start with some simple observations. We note that, given u, v ∈ L 2 (µ ε ω ), it holds (5.6) In particular, we can write Moreover, taking the inner product of (2.18) with u ε ω , we obtain Hence, since f ε ω ⇀ f , for any λ > 0 it holds that (5.8) Lemma 5.3. Fixω ∈ Ω * . The family u ε ω converges along a subsequence to a function u 0 ∈ L 2 (R d ×Ω, dx×µ) in the sense of weak two-scale convergence and u 0 does not depend on ω, i.e. u 0 ∈ L 2 (R d , dx).
Proof. Due to Lemma 5.2, the sequence u ε ω converges along a subsequence ε k ↓ 0 to a function u 0 ∈ L 2 (R d × Ω, dx × µ) in the sense of weak two-scale convergence. In order to simplify the notation, we suppose that this convergence holds for ε ↓ 0. We need to prove that u 0 does not depend on ω. To this aim, fix e ∈ B, ϕ ∈ C ∞ c (R d ) and a function ψ ∈ C(Ω). We define v(ω, e ′ ) = ψ(ω)δ e,e ′ . Due to the definition of weak two-scale convergence, it holds The r.h.s. in (5.10) is bounded by where By Schwarz inequality and (5.7) we can bound for a suitable positive constant c(ψ, ϕ) depending on ψ and ϕ. Due to (5.8), we obtain that I 1 c(ψ, ϕ,ω)ε. Moreover, by Schwarz inequality we have and again from (5.8) we deduce that I 2 c(ψ, ϕ,ω)ε. Hence the r.h.s. of (5.10) is bounded by cε and due to (5.9) we get that Since this holds for all ϕ ∈ C ∞ c (R d ) we get that for Lebesgue a.a. x ∈ R d . Due to separability, we conclude that for Lebesgue a.a. x ∈ R d the above identity is valid for all v of the form v(ω, e ′ ) = ψ(ω)δ e,e ′ , for some function ψ ∈ C(Ω) and some e ∈ B. By Lemma 4.3 we conclude that for these points x, the function u 0 (x, ·) is constant µ-almost everywhere. This concludes the proof.
In what follows, u 0 will be as in Lemma 5.3 for a fixedω ∈ Ω * . We will prove at the end that u 0 coincides with the solution of (2.19) and in particular that u 0 does not depend oñ ω.
Lemma 5.4. Fixω ∈ Ω * . The function u 0 belongs to the Sobolev space H 1 (R d , dx). Moreover, along a suitable subsequence and for all e ∈ B it holds where ∂ e u 0 (x) denotes a representative of the weak derivative in L 2 (dx) of u 0 , along the direction e. Then, for Lebesgue a.a. x ∈ R d , it holds (5.17) where π : L 2 (M ) → L 2 sol (M ) is the orthogonal projection onto L 2 sol (M ).
Let us now prove (5.17). Since u 0 ∈ H 1 (R d , dx), we are allowed to rewrite the first identity in (5.22) as Then, by means of the arbitrariness of ϕ and separability arguments, we get that for Lebesgue a.a.
6. Proof of Theorem 2.4 We start with a technical result, which could be proven in much more generality: (6.1) and suppose that Proof. Trivially, it is enough to prove the following limits Since h ∈ L 2 (µ ε ω ), the integrals in the l.h.s. of (6.4) and (6.5) are meaningful. Moreover, observe that for each ℓ > 0 one can find a function g ℓ ∈ C c (R d ) such that h(x) = g ℓ (x) for any x ∈ R d with |x| ℓ, and |g ℓ (x)| c/(1 + |x| d+1 ).
In order to prove (6.4) we observe that for a positive constant c(ℓ) going to 0 as ℓ ↑ ∞. The above estimates (6.7) and (6.8), and the limit (due to the definition of Ω * ) allow to derive (6.4) by taking the limit ℓ ↑ ∞.
In order to prove (6.5) we observe that for a positive constant c(ℓ) going to 0 as ℓ ↑ ∞. Since h ε ω → h and g ℓ ∈ C c (R d ) we can conclude that The above limit together with (6.9) and (6.10) implies (6.5).
We have now all the main tools in order to prove Theorem 2.4. We take ω ∈ Ω * , define u 0 , v 0 e as in Lemma 5.4 and assume that f ε ω ⇀ f ω . We want to prove that u 0 solves equation (2.19) and that (2.20) holds.
First we observe that the weak two-scale convergence (5.13) implies the weak convergence L 2 (µ ε ω ) ∋ u ε ω ⇀ u 0 ∈ L 2 (mdx) (6.11) as ε ↓ 0 along the subsequence of Lemma 5.4. Taking the inner product of (2.18) with a test function ϕ ∈ C ∞ c (R d ) and using (5.6), we get the identity . (6.12) By taking the limit ε ↓ 0 (along the subsequence of Lemma 5.4) and then dividing by m, from the trivial identity (5.21), the limit (5.14) in Lemma 5.4, the limit (6.11) and the hypothesis The second member in (6.13) can be rewritten as (6.14) Due to Lemma 5.4, u 0 ∈ H 1 (R d , dx). Given x ∈ R d we consider the gradients ζ(x) := ∇ϕ(x) = ∂ e ϕ(x) e∈B * , ξ(x) := ∇u 0 (x) = ∂ e u 0 (x) e∈B * (the definition is well posed for Lebesgue a.a. x ∈ R d , since u 0 ∈ H 1 (R d , dx)). Due to Lemma 5.4, we know that for Lebesgue a.a. x ∈ R d the form θ x defined in (5.15) coincides with the form πw ξ(x) (recall definition (4.19)). Therefore, due to (4.20), (4.21) and (4.27), we can rewrite (6.14) as (6.15) In conclusion, (6.13) reads ( 6.16) Hence, the function u 0 of Lemma 5.4 is the solution of equation (2.19), which is unique (in particular u 0 does not depend from ω ∈ Ω * ). Due to Lemma 5.2 it is simple to verify that for each sequence ε k ↓ 0 one can extract a sub-subsequence ε kn satisfying Lemma 5.4. Hence, by the previous results, we conclude that for each sequence ε k ↓ 0 one can extract a sub-subsequence ε kn such that thus implying that the functions u ε ω ∈ L 2 (µ ε ω ) weakly converge to u 0 ∈ L 2 (mdx). This concludes the proof of point (i).
In order to conclude we only need to derive (2.25) from (2.24). To this aim, let Λ ℓ := [−ℓ, ℓ] d , ℓ > 0. We claim that, for any ω ∈ Ω * , given any Without loss of generality we can assume that f 0. Since P X(tε −2 |x) = z = P X(tε −2 |z) = x ∀t 0, ∀x, z ∈ C(ω) , we can write The above limit and the identity R d P t f (z)dz = R d f (z)dz, following from the symmetry of P t , implies (7.2) with Λ c ℓ replaced by R d . Therefore, in order to prove (7.2) it is enough to show that lim To this aim we apply Schwarz inequality and obtain the bounds Since by (2.16) µ ε ω (Λ ℓ ) → m(2ℓ) d for each ω ∈ Ω * , the above upper bound and (2.24) imply that the first member in (7.5) goes to 0 as ε ↓ 0 for each ω ∈ Ω * . To conclude the proof of (7.4) it is enough to observe that for each ω ∈ Ω * the integral Λ ℓ P t f (x)µ ε ω (dx) converges to m Λ ℓ P t f (x)dx since P t f is a regular function fast decaying to infinity (the proof follows the same arguments used in order to check (6.4)). This concludes the proof of (7.2).
Let us come back to (2.25). For each ℓ > 0 we can bound (7.6) Due (2.24) and (7.2), by taking ε ↓ 0 we get that for each ω ∈ Ω * lim sup By the arbitrariness of ℓ in the above estimate one derives (2.25).
Ifω is a Bernoulli bond percolation with parameter p > p c , then for each c > 0 the random fieldω c is a Bernoulli bond percolation with parameter p(c) such that lim c↓0 p(c) = p. Hence, taking c > 0 small enough, we obtain that hypotheses (H2) and (H3) are satisfied.
The last statement regarding the cases of D diagonal or multiple of the identity can be proved by the same arguments used in the proof of Theorem 4.6 (iii) in [DFGW].