Berry–Esseen theorem and quantitative homogenization for the random conductance model with degenerate conductances

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Introduction
Stochastic homogenization of elliptic equations in divergence form with random coefficients started from the pioneering works of Kozlov [37] and Papanicolaou-Varadhan [49]. They established a qualitative homogenization result, which (adjusted to a discrete setting) can be rephrased as follows. The unique bounded solution u ε to the elliptic finite difference equation with ω describing stationary and ergodic, uniformly elliptic, random coefficients, and f an appropriate right-hand side, e.g. f ∈ C c (R d ) with zero mean, converges after a rescaling to the solution u 0 of the deterministic, elliptic equation where ω hom denotes a deterministic coefficient matrix, the so-called homogenized coefficients. Quantitative stochastic homogenization is concerned with finding the rate of convergence of u ε towards u 0 . The first result in this direction was obtained by Yurinskii [51] and relied on probabilistic arguments. Recently, Gloria and Otto in [29,30] and together with the second author in [28] obtained the optimal scaling of the error for the discrete, uniformly elliptic case by combining input from elliptic and parabolic regularity theory with input from statistical mechanics, in particular a spectral gap inequality used to quantify ergodicity. Thereafter, an increasing interest emerged in quantitative stochastic homogenization, e.g. see [6][7][8][9][10]12,14,15,22,25,31,32,42]. Closely related to the topic of homogenization in PDE theory is the problem of deriving invariance principles or functional central limit theorems for the so-called random conductance model in probability theory, which refers to the random walk X in random environment generated by the operator in (1.1). Roughly speaking, an invariance principle states that the scaling limit of X converges to a Brownian motion with a non-random covariance matrix 2 only depending on the law of the conductances, see Theorem 1.2 below. In particular, the covariance matrix of the limiting process and the homogenized coefficients are related by the identity 2 = 2ω hom . Such invariance principles are subject of very active research since more than a decade, see the surveys [16,39] and references therein.
The goal of this paper is to extend the quantitative theories of stochastic homogenization and invariance principles to the case of non-uniformly elliptic conductances. In particular, we are interested in moment bounds and decay estimates for the socalled corrector problem of homogenization and a Berry-Esseen theorem. Regarding the latter, a first inspiring result in this direction has been obtained by Mourrat [43] for uniformly elliptic i.i.d. conductances. In the uniformly elliptic case, we improve this result (in terms of the convergence rate) in dimension d = 2, 3, and we extend this result to degenerate and correlated environments in dimension d ≥ 3 with a weaker rate of convergence in dimension d = 3. Our analysis invokes input from the quantitative theory of stochastic homogenization as developed; in particular, we extend some key estimates obtained in [28] under the assumption of uniform ellipticity to degenerate elliptic operators under moment conditions.

The model
Let d ≥ 2. We study the nearest-neighbour random conductance model on the ddimensional Euclidean lattice (Z d , E d ), where E d := {e = {x, x ± e i } : x ∈ Z d , i = 1, . . . , d } denotes the set of non-oriented nearest neighbour edges and {e 1 , . . . , e d } the canonical basis in R d . We endow the graph with positive random weights, which we describe by a family ω = {ω(e), e ∈ E d } ∈ := (0, ∞) E d . We refer to ω(e) as the conductance of an edge e ∈ E d . To simplify notation, for any x, y ∈ Z d , we set Henceforth, we consider random conductances that are distributed according to a probability measure P on , equipped with the σ -algebra F := B((0, ∞)) ⊗E d . We write E for the expectation operator with respect to P. The measure space ( , F) is naturally equipped with a group of space shifts τ x : x ∈ Z d , which act on as where the shift by x of an edge e = {e, e} ∈ E d is defined as e + x := {e + x, e + x}. The random conductance model is defined as follows. For any fixed realisation ω it is a reversible continuous time Markov chain, X = {X t : t ≥ 0}, on Z d with generator L ω acting on bounded functions f : Z d → R as (

1.2)
With the help of the discrete gradient ∇ and its adjoint ∇ * (see Sect. 1.3 below), we can represent the generator in the compact form L ω = −∇ * ω∇, which highlights the fact that L ω is a (finite difference) second order operator in divergence form. We denote by P ω x the law of the process starting at the vertex x ∈ Z d and by E ω x the corresponding expectation . This random walk waits at x an exponential time with mean 1/μ ω (x) with μ ω (x) := y∈Z d ω(x, y) and chooses the vertex y for its next position with probability ω(x, y)/μ ω (x). Since the law of the waiting times does depend on the location, X is also called the variable speed random walk (VSRW). A general assumption required in the context of invariance principles is stationarity and ergodicity of the environment.

Definition 1.1 (Stationarity and ergodicity)
We say the measure P is stationary with respect to translations of Z d , if P • τ −1 x = P for all x ∈ Z d . We say P is ergodic, if P[A] ∈ {0, 1} for any A ∈ F such that τ x (A) = A for all x ∈ Z d .
First results in this context are annealed (or averaged) functional central limit theorems that yield the convergence of the rescaled random walk under the annealed measure P 0 defined by P 0 [·] := P ω 0 [·] d P(ω). This has been established in [19] (cf. also [36]) for general ergodic environments under the assumption of strict positivity and a first moment condition It is of particular interest to understand the finer question whether an invariance principle also holds for P-a.e. ω, that is in a quenched form. In [1] the quenched invariance principle has been shown for general i.i.d. conductances. However, in the case of a general ergodic environment, due to a trapping phenomenon, it is clear that some moment conditions (stronger than the one in (1.3)) are needed. Indeed, Barlow, Burdzy and Timar [11] give an example on Z 2 which satisfies a weak moment condition, but not a quenched invariance principle. To formulate the moment condition used in our paper, we set for any p, q ∈ [1, ∞], ( 1.4) Note that in the special case, when M( p, q) < ∞ for p = q = ∞, we obtain the uniformly elliptic case, i.e. P[1/c ≤ ω(e) ≤ c] = 1 for some c > 0 and the generator L ω = −∇ * ω∇ defines a uniformly elliptic discrete operator. Recently, the following quenched invariance principle has been proven for random walks under ergodic conductances. Theorem 1.2 [2] Suppose that P is stationary and ergodic, and that the moment condition M( p, q) < ∞ holds for exponents p, q ∈ (1, ∞] satisfying p −1 + q −1 < 2/d. For n ∈ N, define X (n) t := 1 n X n 2 t , t ≥ 0. Then, for P-a.e. ω, X (n) converges (under P ω 0 ) in law towards a Brownian motion on R d with a deterministic non-degenerate covariance matrix 2 .
For further invariance principles in the setting of random degenerate conductances we refer to [1,16,39] and references therein; for recent results on (qualitative) stochastic homogenization of elliptic operators in divergence form with degenerate coefficients, see [13,23,40,48].

Main results
In this paper our main concern is to establish a quantitative central limit theorem for the random walk X . For definiteness, let ξ ∈ R d be fixed and set where 2 still denotes the covariance matrix in Theorem 1.2. Then, the invariance principle of Theorem 1.2 yields for P-a.e. ω, (1.5) where (x) := (2π) −1/2 x −∞ e −u 2 /2 du denotes the distribution function of the standard normal distribution. Our goal is to quantify the speed of convergence in (1.5) for d ≥ 3 by means of a Berry-Esseen theorem. For a general ergodic environment the speed of convergence can be arbitrarily slow, since ergodic environments may have very weak mixing properties. Therefore it is necessary to quantify the assumption of ergodicity. For this purpose, following the approach of [28][29][30], we assume that P satisfies a spectral gap estimate with respect to a Glauber dynamics on the field of conductances.

Assumption 1.3 (Spectral gap)
Suppose P is stationary, and assume that there exists ρ > 0 such that for any u ∈ L 2 ( ). Here, the vertical derivative ∂ e u is defined as where δ e : E d → {0, 1} stands for the Dirac function satisfying δ e (e) = 1 and δ e (e ) = 0 if e = e.

Remark 1.4 (i)
There is a certain freedom in the choice of the derivative that appears on the right-hand side in (SG). In [28] the following vertical derivative is considered, which basically follows by applying (SG) to the function |u| p , see [28,Lemma 11].
In addition to Assumption 1.3 we need to assume stronger moment conditions than in Theorem 1.2. We do not keep track of the precise lower bounds for p and q in the moment condition M( p, q) < ∞ that we require in our analysis, since our approach is not optimal in that direction. Recall that in view of the counterexample in [11] a moment condition is necessary already for the (non-quantitative) invariance principle to hold.
Our first result is an annealed Berry-Esseen theorem in dimension d ≥ 3 with speed t − 1 5 +ε for d ≥ 4 and t − 1 10 +ε for d = 3, as well as a quenched Berry-Esseen theorem with the same speed but in an integrated form. Theorem 1.5 (Berry-Esseen theorem) Let d ≥ 3 and suppose that Assumption 1.3 holds. For any ε > 0 there exist exponents p, q ∈ (1, ∞) (only depending on d, ρ, and ε) such that under the moment condition M( p, q) < ∞ the following hold.
(ii) There exists a random variable X = X (d, ρ, ε, M( p, q)) ∈ L 1 (P) such that for P-a.e. ω, In the case of uniformly elliptic i.i.d. conductances an annealed Berry-Esseen theorem as in (i) has been proven in [43] for arbitrary dimension d ≥ 1 with rate t −1/10 (plus logarithmic corrections) for d = 2, and rate t −1/5 for d ≥ 3 (with some logarithmic corrections for d = 3). Theorem 1.5 extends this result to unbounded and correlated random conductances. In Sect. 7 below we discuss some relevant examples linked to Ginzburg-Landau interface models that naturally yield correlated conductances. To our knowledge (ii) is the first quenched Berry-Esseen-type result for the random conductance model.
Let us anticipate that the general strategy of our proof is the same as the one in [43]. However, there is a genuine difference between the uniformly elliptic case treated there and the degenerate elliptic case considered here. In the uniformly elliptic case in [43] the comparability (on the level resolvents) of the simple random walk and the random walk in the random environment is exploited (see [43, Proof of Theorem 5.1]), which comes in hand in studying the variance decay of the associated semigroup. In our case, no such principle is available. Instead, following ideas in [28], we first establish a semigroup estimate that invokes the gradient of the heat kernel associated with the degenerate elliptic operator ∇ * ω∇. Another difference to [43] is that we introduce a representation in divergence form for the carré du champ applied to harmonic coordinates. It invokes the so called extended corrector (φ i , σ i ), which has been recently introduced in the random case in [25]. Our refined argument allows to improve the rates obtained in [43] in the uniformly elliptic case in low dimensions d = 2, 3: The above result can be obtained by modifying the proof of Theorem 1.5 and is based on recent estimates that are known to hold in the uniformly elliptic case. We describe the modifications in Appendix A for the reader's convenience. The proof of Theorem 1.5 is given in Sect. 6 below. In the following we explain the general strategy. The most classical approach to show an invariance principle is to decompose the process X into a martingale part and a remainder (cf. e.g. [36]). It turns out that for the invariance principle the remainder is negligible, and thus the scaling limit of X is the same as the one for the martingale part. The latter can be analysed by martingale theory as for instance the well-established Lindeberg-Feller functional central limit theorem or Helland's martingale convergence theorem (see [34]). In the proof of Theorem 1.5 we follow the same strategy. Yet, since we seek for an estimate on the rate of convergence, at various places we need to replace qualitative arguments by estimates. A key result in this procedure is the following decay estimate for the semigroup (P t ) t≥0 defined by where p ω (t, x, y) := P ω x [X t = y] denotes the transition densities or heat kernel associated with L ω . Throughout the paper we will often write p(t, y) := p ω (t, 0, y) in short and use the fact that p τ z ω (t, x, y) = p ω (t, x +z, y+z) due to the definition of the space shifts. The semigroup (P t ) t≥0 can be interpreted as the transition semigroup of the process (τ X t ω) t≥0 , which is known as the process of the environment as seen from the particle. It is a contraction and generated by the (degenerate) elliptic operator −D * ω(0)D, where D and D * denote the horizontal derivative and its adjoint (see Sect. 1.3 for the precise definition). Our key estimate is the following.

Remark 1.8 (Comparison to the uniformly elliptic case)
For uniformly elliptic conductances, in [28], Gloria, Otto and the second author established the decay estimate and deduced various estimates in stochastic homogenization based on this decay estimate. The estimate (1.9) is optimal in terms of the scaling in t and in terms of the exponent of the norm on the right-hand side. In the degenerate elliptic case, we do not expect to get the same scaling. However, under sufficiently strong moment conditions, our estimate shows that we can get arbitrarily close to the scaling of the uniformly elliptic case. The argument in [28] crucially relies on a deterministic parabolic regularity estimate for ∇ p ω with optimal scaling in t. In the degenerate elliptic case this estimate is not valid. We replace it by a non-deterministic estimate on ∇ p ω with nearoptimal scaling, see Proposition 2.1 below, which we combine with an interpolation argument that exploits the contraction property of the semigroup. The latter is the reason for the exponent 8n in our estimate.

Remark 1.9
Recently, a similar estimate has been obtained in [24] for conductances uniformly bounded from above satisfying a moment condition on ω(e) −1 . The proofs of both Theorem 1.7 and the results in [24] follow the strategy in [28]. However, in [24] the required on-diagonal estimate on the heat kernel is derived from the anchored Nash inequality established in [45], which makes a uniform upper ellipticity necessary, while in our setting the analogue heat kernel bound in Lemma 2.5 can be deduced from the upper off-diagonal heat kernel estimates in [4,5]. For the opposite case, i.e. conductances bounded from below having quadratic moments, we refer to [18], where the simpler situation P t f (instead of P t D * F) is studied.
We will use Theorem 1.7 to quantify the convergence of the martingale part in the decomposition of the process X mentioned above, which relies on harmonic coordinates : × Z d → R d defined as where φ i denotes the corrector from stochastic homogenization defined in Proposition 1.10 below. Roughly speaking, φ i is a sublinearly growing solution to the equation ∇ * ω(∇φ i + e i ). The corrector φ i is a fundamental object in the qualitative and quantitative theory of stochastic homogenization, see e.g. the seminal work by Papanicolaou-Varadhan [49] or [28] for quantitative results. In particular, the covariance matrix 2 of the limiting process in Theorem 1.2 may be represented in terms of φ i or ψ i , respectively, as It is well-known that for P-a.e. ω, (i) the process M = (ω, X ) is a martingale (see Corollary 6.2 below) and thus features an invariance principle; (ii) the remainder X − M vanishes in the scaling limit due to the sublinear growth of the corrector.
For our purpose we need to quantify both the speed of convergence in (i) and the smallness of the remainder in (ii). For this reason we establish the existence of high moments of the corrector φ i (and an additional flux corrector σ i , which we explain below). In the following we say that a random variable u is stationary, if u(ω, x + y) = u(τ y ω, x) for all x, y ∈ Z d and P-a.e. ω ∈ .
and ω hom ∈ R d×d denotes the homogenized coefficient matrix characterised by 3) The gradient fields ∇φ i and ∇σ i are stationary, have finite 2nd moments, and vanishing expectation, i.e.
While the sublinearity of the corrector can be established for general ergodic environments satisfying the relatively weak assumptions in Theorem 1.2 (cf. [2]), the existence of high moments of φ i and ∇φ i as in Proposition 1.10 only holds true under sufficiently strong mixing assumptions. In dimension d = 2, even in the case of uniformly elliptic, i.i.d. conductances, the stationary version of the corrector does not exist. On the other hand, for d ≥ 3, in the uniformly elliptic case and under sufficiently strong mixing assumptions, the stationary representations of φ i and σ i exist and satisfy (high) moment bounds. This has been first achieved on the level of φ i in [29] (see also [28] where bounds are obtained via a decay estimate similar to Theorem 1.7, and see [15] for moment bounds on σ i and ∇σ i ). To our knowledge Proposition 1.10 is the first result on moment bounds in the degenerate elliptic setting. The dependence on the growth exponent on the dimension in statement (c) is non-optimal. In fact, we expect the statements to hold for d ≥ 3 (with θ = 0 and arbitrary p θ < ∞) as in the uniformly elliptic case; see Remark 2.4. Using the moments of φ i and ergodicity we obtain a sufficiently good control on the remainder, see Proposition 6.8 below. In order to quantify the convergence of the martingale part in (i), we use a Berry-Esseen estimate for martingales in [33] (see Theorem 6.4 below). It requires to quantify the speed of convergence of towards zero, where ξ · M denotes the quadratic variation process of the martingale ξ · M. For that purpose we establish the following result which also exploits the moment bounds on (φ i , σ i ) obtained in Proposition 1.10.

Proposition 1.11
Let d ≥ 3, ε > 0, and suppose Assumptions 1.3 is satisfied. Then there exist p, q ∈ (1, ∞) (only depending on d and ε) such that under the moment condition M( p, q) < ∞ the following holds. For any direction ξ ∈ R d with |ξ | = 1, we denote by φ ξ the corrector associated with ξ , i.e. φ ξ = d i=1 ξ i φ i with φ i as in Proposition 1.10, and by the associated harmonic coordinate. Consider where ω denotes the opérateur carré du champ associated with L ω , i.e.
Then there exists a constant c = c(d, ρ, θ, M( p, q)) such that In the proof of the proposition we make use of the field σ i defined in Proposition 1.10, which allows us to represent the quadratic variation of M in divergence form, see Lemma 5.1 below. Similarly as φ i , the field σ i is a classical object in periodic homogenization. Yet, in the stochastic case it has been utilised just recently, see e.g. [15,25].

Remark 1.12
The exponent 1/5 in Theorem 1.5 is non-optimal. However, the decay rates in Theorem 1.7 and Proposition 1.11 (for d ≥ 4) are optimal apart from the ε which appears due to the degeneracy of the environment. The exponent 1/5 then arises by the application of the general Berry-Esseen theorem for martingales (see Theorem 6.4 below with the choice n = 2). In d ≥ 3 it can be deduced from the moments bounds on the corrector in Proposition 1.10 that the jumps of the martingale part are in L n (P ω 0 ) for any n ∈ N (cf. Proposition 6.7 below). In such a situation the results in [44] show that the decay is optimal (for the choice n = 2 in Theorem 6.4). Thus, a possibility to improve the exponent 1/5 (within the above approach) is to apply the Berry-Esseen theorem for martingales in Theorem 6.4 for larger values of n which would require control on higher moments of ξ ·M t t − σ 2 ξ (cf. the discussion in [43, p. 6]). An alternative PDE-approach towards an optimal result would be to estimate the difference between the heterogeneous and homogenized parabolic Green's function. Very recently, Armstrong et al. [7,Theorem 9.11] obtained such an estimate for the equation on R d in the case of uniformly elliptic coefficients and under a finite range of dependence assumption. The estimate suggests that the exponent in Theorem 1.5 (in the uniformly elliptic, i.i.d. case) can be improved to 1/2 (up to a logarithmic correction for d = 2). The rate 1/2 is the best one can hope for, since it is the convergence rate for the simple random walk. The argument in [7] relies on a large scale regularity theory for elliptic operators, which is not established in the degenerate case yet. For general non-uniformly elliptic and (possibly strongly) correlated coefficients, we expect the optimal rate of convergence to depend on the parameters p and q as well as on the mixing behaviour of the environment. We remark that the qualitative statement in form of a quenched invariance principle holds for general i.i.d. conductances (cf. [1]) and is conjectured to hold for ergodic conductances under the moment condition M(1, 1) < ∞, which is known to be necessary in the case of general ergodic environments (cf. [11]).

Notation
We finally introduce some further notation used in the paper. We write c to denote a positive, finite constant which may change on each appearance. Random constants depending on ω ∈ will be denoted by calligraphic letters such as X , Y etc. Further, means ≤ up to a constant depending only on some quantities specified in the particular context.
The cardinality of a set A ⊂ Z d will be denoted by # A.
and note that for f , g : Z d → R, the discrete product rule takes the form (1.17) We define the discrete divergence of a function F : . Since ∇ * can be seen as the adjoint of ∇. Note that the generator L ω defined in (1.2) above is a finite-difference operator in divergence form as it can be rewritten as We tacitly identify scalar functions defined on E d with vector-valued functions on Z d . In particular, given F : ; and given f : and denote by ∇∇ Further, we will also use the abbreviation With any random variable ζ : → R we associate its P-stationary extension Its adjoint in L 2 ( ) denoted by D * :

Structure of the paper
In Sect. 2 we establish the estimates on the gradient of the heat kernel and the Green's function needed in the proofs. Then Sect. 3 is devoted to the proof of the semigroup decay stated in Theorem 1.7 and Sect. 4 to the construction and the moment bounds of the extended corrector. The variance decay for the semigroup applied to the carré du champ operator in Proposition 1.11 is shown in Sect. 5. Then, the Berry-Esseen estimates in Theorem 1.5 are proven in Sect. 6. Finally, some relevant examples satisfying our assumptions are discussed in Sect. 7.

Gradient heat kernel and annealed Green's function estimates
In this section we establish regularity estimates for averages of the gradient of the heat-kernel and the mixed second gradient of the elliptic Green's function, which we require for the proofs in Sect. 3. We consider both annealed estimates, where the average is taken w.r.t. the probability measure P, and spatially averaged estimates in weighted 2 -spaces with weight m 2α where In the case of uniformly elliptic and bounded conductances the upcoming estimates are well-known. The estimates that we obtain in the degenerate case are weaker in two ways.
• We only obtain near optimal estimates, in the sense that the decay rate deviates from the optimal decay rate in the uniformly elliptic case by a small parameter ε > 0. • The estimates are random, in the sense that they hold up to a random constant, whose integrability is monitored by an exponent n ≥ 1. • The parameters ε and n can be chosen arbitrarily close 0 and ∞, respectively, provided we impose sufficiently strong moment conditions on the conductances.
Throughout this section we assume d ≥ 2. We start with the following (spatially averaged) decay estimate for the gradient of the heat-kernel, which is a key ingredient for the proof of Theorem 1.7. Proposition 2.1 Suppose that Assumption 1.3 holds. For any ε ∈ (0, 1), n ≥ 1 and α ≥ 0 there exist p, q ∈ (1, ∞) (only depending on d, ε, n, α) such that under the moment condition M( p, q) < ∞ the following holds. There exists a family of random variables In the uniformly elliptic case, in particular in the special case of ∇ * ω∇ = ∇ * ∇, the estimate holds with ε = 0 (which corresponds to the optimal decay in time), and with sup t≥0 Z t bounded by a deterministic constant, see e.g. [28,Theorem 3]. In the present situation the degeneracy of the conductances leads to a loss in the decay. As we will explain in Sect. 2.1.1, our proof of the estimate relies on an on-diagonal upper heat kernel bound. The result is then obtained by parabolic regularity arguments following [28].
From Proposition 2.1 we deduce a couple of annealed estimates.
The proofs of Corollaries 2.2 and the 2.3 below are presented in Sect. 2.2. Next we establish an annealed estimate on the gradient and the mixed second derivative of the elliptic Green's function, which for P-a.e. ω and all x, y ∈ Z d can be defined by the integral The decay exponent in Corollary 2.3 is not optimal. For n = 1 we miss the optimal decay by ε, and for large n by the additional exponent n−1 n and 2 n−1 n , respectively. As shown recently in [41] in the uniformly elliptic case and under the assumption that P satisfies a logarithmic Sobolev inequality, the estimate holds for any n ∈ N with the optimal decay exponent d − 1 and d, respectively. The argument in [41] lifts the estimate for n = 1 to higher moments by using the logarithmic Sobolev inequality and a deterministic regularity estimate, which is not available in our degenerate setting. Our (simple, yet suboptimal) argument is as follows. For n = 1 the estimate follows from Proposition 2.1 exploiting stationarity. For n 1 we obtain it by interpolating with a suboptimal estimate on high moments of ∇ p, that follows from Proposition 2.1 as well.

Remark 2.4
The non-optimality of the above estimate is the limiting factor that hinders us to improve the decay exponent in our main result, Theorem 1.5, in dimension d = 3. This becomes visible in the sublinear growth estimate for σ , see (1.15). In the proof of this estimate we exploit that thanks to Corollary 2.3 we have With the optimal estimate for ∇∇G at hand, the above estimate would hold for any s > 1, and we would obtain (1.15) with θ = 0 for any d ≥ 3. Eventually, this would improve the decay rate in Theorem 1.5 for d = 3 to 1 5 − ε.

Gradient heat kernel estimate: proof of Proposition 2.1
In this section we prove Proposition 2.
For the proof see Sect. 2.1.1 below. Next, we introduce the stationary weights , (2.4) and write μ := μ ω (0) and ν := ν ω (0) for abbreviation. Henceforth the random varable Y in (2.3) is fixed as defined in the proof of Lemma 2.5 below. In addition to Y, further random variables will appear in the estimates below. In this subsection, to keep the presentation lean, we say X denotes a random variable of class X (c 1 , c 2 , . . . , c n ) (in short X = X (c 1 , c 2 , . . . , c n )), if it can be written in the form with exponents p 1 , p 2 ≥ 1 and a constant c 0 ≥ 1 that can be chosen only depending on the parameters c 1 , . . . , c n . Evidently, the class is stable under taking products, sums and powers of such random variables. Moreover, finite moments of such a random variable are bounded, provided E[μ p ] < ∞ for p sufficiently large. Following [28], we lift the on-diagonal estimate of Lemma 2.5 to a weighted 2estimate for p and ∇ p.
Then there exists a random variable X = X (d, α, ε) such that for all t ≥ 0 and T ≥ 1,

Remark 2.7
In the case of conductances that are bounded from above, we may choose ε = 0 and thus recover the optimal scaling in t.
Based on Lemma 2.6 we establish the following variant of Proposition 2.1.

On-diagonal heat kernel estimate: proof of Lemma 2.5
The statement is a rather direct consequence of an on-diagonal estimate (see Lemma 2.9 below), which can be obtained from [5], and an application of the spectral gap estimate of Assumption 1.3 used to control moments of the estimate's random constant, see Lemma 2.10. Assuming M( p, q) < ∞ for any p, q ∈ (1, ∞), we denote by R = R(ω, p, q) ≥ 1 the smallest integer such that for all R ≥ R, with μ ω and ν ω defined in (2.4). Then, P-a.s., R < ∞ by the ergodic theorem.

Lemma 2.9
Let p, q ∈ (1, ∞) satisfy 1 p + 1 q < 2 d . Suppose that P is stationary and ergodic, and that the moment condition M( p, q) < ∞ is satisfied. Then, there exists c = c(d, p, q, M( p, q)) such that for t ≥ R 2 , Proof This on-diagonal bound follows immediately from the upper heat kernel bounds in [5,Theorem 2.5], which is based on arguments in [4]. Indeed, by our assumptions R = R(ω, p, q) defined via (2.9) is P-a.s. finite for p, q ∈ (1, ∞) with 1 p + 1 q < 2 d . Therefore the assumptions of [5, Theorem 2.5] are satisfied for P-a.e. ω. Alternatively, the estimate can be deduced from the parabolic Harnack inequality established in [3], see Proposition 4.7 and Remark 1.5 therein. Proof We only present the argument for μ ω , since the argument for ν ω is the same. To that end consider the random variable which is well-defined since E[μ p ] > 0 by assumption. We claim that for any k ∈ N, Indeed, since E[ f R ] = 0, the spectral gap inequality in form of (1.7) yields Now, the combination of the previous two estimates gives (2.10). Next, by a slight abuse of notation let R(ω) ≥ 1 be the smallest integer such that Since μ is stationary and E[ f R ] = 0, Birkhoff's ergodic theorem shows that P-a.s. sup r ≥R f r → 0 as R ↑ ∞. In particular, R < ∞ P-a.s. and R satisfies the first property in (2.9). We finally estimate the moments of R by using (2.10). To that end, note that for all R ∈ N with R ≥ 2, Hence, We choose k > (n + 1)/d and p = 2( p − 1)k to get the claim. Lemma 2.5 is now a simple consequence of the previous two results.

Proof of Lemma 2.5
Since p(2t, 0) = y∈Z d p(t, y) 2 by the symmetry of the kernel, and the fact that y∈Z d p(t, y) = 1, we deduce from Lemma 2.9 that By Lemma 2.10 we can achieve E[Y n ] < ∞ if sufficiently high moments of ω and ω −1 exist.
Step 2. Next we show (2.6). The starting point of the argument is (2.11), which we recall in an integrated and rearranged form By Step 1 the first term on the right-hand side is bounded by YX (T + 1) − d 2 −1+ε+α . Furthermore, by Hölder's inequality and Step 1 we have for all t ∈ (T , 2T ), The combination of the previous estimates yields (2.6).

Proof of Lemma 2.8
Throughout the proof, X denotes a generic random variable (that might change from line to line) of class X (d, α, ε). Moreover, we write if ≤ holds up to a constant only depending on d, α, ε.
Step 1. First we show that there exists X = X (d, α, ε) such that for all t ≥ 0 and We start with (2.12). First assume that α ≥ α 0 := d 2 + 1. Since m(t, y) 2α which combined with Lemmas 2.6 and 2.5 yields the desired estimate. In the case 0 ≤ α ≤ α 0 we proceed by interpolating the estimate for α = 0 and α = α 0 . Indeed, Hölder's inequality yields The first term on the right-hand side can be estimated as above, and the second term on the right-hand side is estimated by Lemma 2.5. Next, we prove (2.13). First note that (2.14) Indeed, by integrating the identity 1 2 d dt 2T y∈Z d p(T , y) 2 , which in combination with Lemma 2.5 yields (2.14). Now, we argue as above to obtain (2.13) for α ≥ α 0 by using Lemma 2.6. Finally, the estimate for 0 ≤ α ≤ α 0 follows (as in the proof of (2.12)) by interpolation.
Step 2. In this step we show that for any α ≥ 0 and 0 < ε < 1 there exists X = X (d, α, ε) such that for all z ∈ Z d and t ≥ 0. By the triangle inequality for the weight in form of m(2t, z) α m(t, z − y) α m(t, y) α , the semigroup property, and the shift property Using symmetry in form of p τ z ω (t, y − z, 0) = p τ z ω (t, 0, y − z) and applying (2.12) yields Step 3. Now we show the statement. First note that it suffices to prove the claimed estimate for t ≥ 1, since for 0 ≤ t ≤ 1 the estimate follows from (2.12) and the fact that |∇ p(t, e)| ≤ p(t, e) + p(t, e). For t ≥ 1, by the semigroup property and Jensen's inequality we have for all e ∈ E d , and by the triangle inequality for the weight in form of m(s, e) 2α m(t − s, z) 2α m(s, e − z) 2α ,

Now, note that
Step 2 implies for all s ∈ ( t 3 , 2t 3 ), where we used in the last step that t 3 ≤ t − s ≤ 2t 3 . Further, the shift property ∇ p τ z ω (s, 0, e − z) = ∇ p ω (s, z, e) gives that We conclude that Finally, sup t≥1 E |X t | n < ∞ for every n ∈ N since m(t, x) 1 1 uniformly in t and arbitrarily high moments of Y and X exist if M( p, q) < ∞ holds for p and q sufficiently large.

Proof of Corollary 2.2
Step 1. Estimates for ∇ p and ∇∇ p for n = 1.
Since the argument for ∇ p and ∇∇ p are similar, we only discuss the estimate for ∇∇ p, which follows the discussion below [ We multiply this identity with m(t, e − e ) α and obtain by the triangle inequality for the weight, i.e. m(t, e − e ) α ≤ 2 α m(t, e − y) α m(t, e − y) α , and the Cauchy Schwarz inequality in y∈Z d , We take the expectation, apply Cauchy Schwarz w.r.t. P and exploit stationarity and symmetry in form of E |∇ p(t, e, y)| 2 = E |∇ p(t, e − y, 0)| 2 to obtain where the last inequality holds by Proposition 2.1. Since e, e ∈ E d are arbitrary, the claim follows.
Step 2. Estimate for ∇ p and n 1.
Step 3. Estimates for ∇ p and ∇∇ p for n > 1.
For n > 1 we obtain the claimed estimates by interpolation of the estimates in Step 1 and Step 2 via Hölder's inequality in form of Indeed, applied to u = ∇ p, we obtain E ∇ p(t, y) and the claimed estimate follows by choosing λ close to 1 n . For ∇∇ p first notice that by the triangle inequality we have Now, the estimate follows by interpolating the estimate for ∇∇ p for n = 1 with the estimate for ∇ p for n n.

Proof of Corollary 2.3
First note that for θ > 0 and 0 ≤ α < 2θ , we have for all x ∈ Z d the elementary estimate

Semigroup decay: proof of Theorem 1.7
In this section we prove Theorem 1.7, which yields a rate for the decay of the semigroup P t : L ∞ ( ) → L ∞ ( ) defined by P t u(ω) := y∈Z d p ω (t, 0, y) u(τ y ω). First we recall some basic facts on the semigroup (P t ) t≥0 . We refer to [28] for details.
• Since the heat kernel is normalised such that y∈Z d p ω (t, 0, y) = 1, the semigroup P t is a contraction on L ∞ ( ). • The semigroup is characterised by a discrete heat equation on Z d . The connection is based on the stationary extension that associates a random variable, say u(ω), with the random fieldū(ω, x) := u(τ x ω) called the stationary extension of u. Now, consider u ∈ L ∞ ( ) and v(t) := P t u. Then for P-a.e. ω, the function v(ω, ·, ·) : to the Cauchy problem which directly follows from the definition of the semigroup. • An alternative characterisation by a Cauchy problem in L ∞ ( ) is as follows.
The stationary extension (·), the discrete derivatives ∇ i , ∇ * i and the horizontal derivatives Therefore, (3.1) is equivalent to the Cauchy problem in L ∞ ( ) given by (3.2) • The family (P t ) t≥0 is the Markovian transition semigroup associated with thevalued process {τ X t ω} t≥0 , which is known as the process of the "environment as seen from the particle". Furthermore, if P is stationary and ergodic, and (1.3) holds, then the measure P is stationary, reversible and ergodic for the environment process {τ X t ω} t≥0 and its semigroup (P t ) t≥0 , respectively (see e.g. [2, Lemma 2.4]).

Proof of Theorem 1.7
We follow the argument of [28,Theorem 1], where the optimal estimate is obtained in the case of uniformly elliptic coefficients. In our setting the lack of uniform ellipticity leads to a loss of decay, since at various places we use Hölder's inequality and the moment conditions in order to move the conductances outside (or inside) of some integrals. A central element is the weighted 2 -regularity estimate for the gradient of the heat kernel obtained in Proposition 2.1, which we apply in form of the following two estimates, the proof of which are postponed to Sect. 3.2. ε ∈ (0, 1), n ≥ d 2ε and θ ∈ (1, 2). There exists p, q ∈ (1, ∞) (only depending on d, n, ε, θ) and a constant c = c(d, n, ε, θ, M( p, q)) such that if M( p, q) < ∞ the following holds. For all F ∈ L 2nθ ( , R d ) and t ≥ 0,

Remark 3.2
In the uniformly elliptic case Lemma 3.1 holds with ε = 0 and θ = 1, which can be shown along the lines of [28]. For our purpose it is important that θ and ε can be chosen arbitrarily close to 1 and 0, respectively, provided we suppose a sufficiently strong moment condition.
For the proof of Theorem 1.7 we further need a non-linear Caccioppoli inequality for the operator D * ω(0)D. The following result is an extension of [28,Lemma 5] to the degenerate setting.

Lemma 3.4 (Caccioppoli)
Let v(t) = P t u. Then for every n ∈ N and 1 < θ < 2 we have

n, θ).
A last ingredient is the decay estimate on ordinary differential inequalities.

Lemma 3.5 Assume that
This is a generalisation of [28,Lemma 15], obtained in [24, Lemma 3.1]. θ ∈ (1, 2) to be chosen later. In the following stands for ≤ up to a constant only depending on d, ρ, n, θ, ε and M( p, q). For abbreviation we set γ : y). By the homogeneity of the estimate it suffices to consider the case

Proof of Theorem 1.7 Let
We claim that for any θ ∈ (1, 2), = 0 for all t ≥ 0. Hence, the n-version of the spectral gap estimate in (1.7) gives (3.5) In order to identify ∂ e v, recall thatv(ω, ·, ·) solves Now we apply ∂ e to this equation to get a characterisation of ∂ ev . More precisely, since ∂ e (∇ * ω∇v) = ∇ * ω∇∂ ev (s, ·) + ∇ * δ e (·)∇v(s, e) (here δ e : E d → {0, 1} denotes the Dirac function with δ e (e) = 1 and δ e (e ) = 0 for any e = e), we have and thus by Duhamel's formula We combine this identity with (3.5) and apply the triangle inequality to obtain For term I note thatū(ω, x) = ∇ * F (ω, x) whereF denotes the stationary extension of F. By definition of ∂ e , ∇ * and the stationary extension, we have for any random variable f the general calculus rules where H (ω, e) := ∂ e F(ω). Hence, using an integration by parts, Lemma 3.3 and (3.3) we get For term I I we use Lemma 3.1 and get for any θ ∈ (1, 2), which completes the argument for (3.4).
Next we apply Lemma 3.4 to obtain where we have used that which is the claimed estimate. 3.1, 3.3 and 3

Proofs of Lemmas
We take the expectation of the n-th power to obtain . By Hölder's inequality with exponents (θ, θ θ−1 ) and by stationarity this can be further estimated from above by where we used that y∈Z d m(t, y) −2nα 1 2n (t + 1) d 4n thanks to 2nα > d. Since we may assume that the 2n θ θ−1 -moment of Z t is bounded by a constant independent of t, and because d 4n ≤ ε 2 , the proof is complete.

Proof of Lemma 3.3 First we compute
In order to estimate the expectation, we apply Lemma 3.1 to F(ω) = H (ω, e ) for any e ∈ E d and obtain  (3.10) where here and below we write if the relation holds up to a constant only depending on d, n and θ . Indeed, using the elementary estimate

Moment bounds for the extended corrector: proof of Proposition 1.10
In this section, unless stated otherwise, ξ denotes one of the coordinate vectors e 1 , . . . , e d , and we drop the index in the notation for φ, σ and q. We split the proof of Proposition 1.10 into three steps. In Step 1 we prove (a) and (b), i.e. the existence of φ and σ , and the moment bounds for φ, which is a rather direct consequence of Theorem 1.7. In Step 2 we establish a sensitivity estimate for the right-hand side of (1.12c). In Step 3 we establish the growth bound for σ .
Since we may assume that high moments of Dφ 0 and ω(0) exist, Corollary 2.3 yields for any p > p and any ε > 0, provided the conductances satisfy sufficiently strong moment conditions with integrability exponents that depend on p and ε. Note that for p ↓ p and ε ↓ 0, we have Thanks to (4.3) we have γ p s d > 1, and thus we obtain γ > d s by choosing p and ε sufficiently close to p and 0. This completes the argument for (4.4).
We basically follow arguments in [26], where a similar statement is obtained for uniformly elliptic, continuous systems, and [47], where the argument is carried out for the corrector φ in the uniformly elliptic, discrete setting in dimension d = 2. The argument is split into three substeps. Then for any exponents r and p satisfying (4.2) and (4.3), we have For the argument let f : Z d → R denote the unique decaying solution to By representing ∇ f with help of the discrete Green's function for ∇ * ∇ we find that Since σ (ω, ·) grows sublinearly P-a.s., we deduce that In particular, we find that E[v L ] = 0, since E[∇σ ] = 0. Thus, the p-version of the Spectral Gap estimate, (1.7), yields Hence, we can estimate the right-hand side of (4.9) by appealing to Step 2 and (4.7). This completes the argument. Then for any r and p satisfying (4.2) and (4.3), we have For the argument let f : Z d → R denote the unique decaying solution to By representing ∇ f with help of the discrete Green's function for ∇ * ∇ we find that and, moreover, Hence, arguing as in Substep 3.2 we see that the claim follows from Step 2. Substep 3.3. Let L := |x| + 1. Then we have where the last identity holds due to the identities of the previous steps, and by stationarity of ∇σ in combination with identity (4.8). Hence, the triangle inequality, |x| ≤ L, and the estimates of Substep 3.1 und Substep 3.2 yield for any exponents r and p satisfying (4.2) and (4.6) the estimate In

Variance decay for the carré du champ: proof of Proposition 1.11
A key ingredient in the proof of Proposition 1.11 is the following lemma, which for d ≥ 3 yields a representation of g ξ in divergence form. Then,

Proof
Define and note that , 0), thanks to stationarity of ∇φ (see Proposition 1.10 (a.3)). Hence, where the last identity holds, since H is stationary in the sense that H (τ x ω, y) = H (ω, y + x).
Now the claimed representation follows from (5.1), (5.2) and (5.3), and the fact that the mean of the right-hand side in (5.2) is ξ · ω hom ξ .
Proof of Proposition 1. 11 First we recall that which holds for any f : Z d → R as can be seen by a direct calculation. In order to recover g ξ (ω), we need to consider f (x) = ψ ξ (ω, x) and evaluate at x = 0. By the definition of ψ ξ we have ψ ξ (ω, 0) = 0, and thus Lemma 5.1 yields where H is defined as in Lemma 5.1. Note that H (y) := H (ω, y) = H (τ y ω, 0). Thus, by the definition of P t , an integration by parts, Hölder's inequality, and Proposition 2.1, we get for all α > d and 0 < ε < 1, From the definition of H , the moment bounds of Proposition 1.10, and the property that high moments of Z t are bounded uniformly in t (see Proposition 2.1), we deduce that for any θ > 1 2 in dimension d = 3 and any θ > 0 in dimension d ≥ 4, we have and thus , we conclude that Note that in dimension d ≥ 4 (resp. d = 3) we might choose θ arbitrarily close 0 (resp. 1 2 ), while we can choose ε > 0 as small as we wish. This completes the proof.

Berry-Esseen theorem
In this section we prove Theorem 1.5. As mentioned earlier we will show a Berry-Esseen theorem for the martingale part and afterwards derive that the corrector converges sufficiently fast to zero along the path of the random walk. We start by setting-up the decomposition of X . Recall the definition of the corrector in Proposition 1.10 and of the harmonic coordinates : × Z d → R d in (1.10) above. Let χ : × Z d → R d be defined by

Remark 6.1
The harmonic coordinates satisfy the cocycle property, that is for P-a.e. ω, and a similar relation holds for χ . Note that conventions about the sign of the corrector χ differ, compare for instance [2,16] and [50].

Corollary 6.2
For P-a.e. ω, the process is a P ω 0 -martingale and Moreover, for every ξ ∈ R d , ξ · M is a P ω 0 -martingale with quadratic variation process given by Proof Clearly, ∇ * ω∇ i = 0 for every i = 1, . . . d, so i is L ω -harmonic. In particular, M and hence also ξ · M are P ω 0 -martingales. The decomposition (6.1) follows directly from the definition of χ . To compute ξ · M , which is the unique predictable process such that (ξ · M) 2 − ξ · M is a martingale, recall that where ω still denotes the opérateur carré du champ associated with L ω given by and ψ ξ = ξ · as defined in (1.16). Finally, since ψ ξ (ω, 0) = 0 we observe that for any x ∈ Z d , which completes the proof.

Quantitative CLT for the martingale part
In this section we show the following Berry-Esseen theorem for the martingale part.
(ii) There exists a random X = X (d, ρ, ε, M( p, q)) in L 1 (P) such that for P-a.e. ω, To prove Proposition 6.3 we will apply the following general quantitative central limit theorem for martingales. Theorem 6.4 Let (N t ) t≥0 be a locally square-integrable martingale (w.r.t. some probability measure P) and denote by N t := N t − N t− its jump process and by N t its quadratic variation process. Then, for any n > 1, there exists a constant c > 0 depending only on n such that .
(ii) There exists a random X = X (d, ρ, ε, M( p, q)) in L 1 (P) such that for P-a.e. ω, Proof (i) Recall that g ξ = ω (ψ ξ (ω, ·))(0) and note that σ 2 ξ = E[g ξ ] since 2 = 2ω hom (cf. (1.11) and (1.13) above). Setting G ξ := g ξ − E[g ξ ], we have by Corollary 6.2 that Arguing as in [43,Sect. 6] we get where we used the stationarity of (τ X s ω) in the second step and a change of variable in the last step. Since (τ X s ω) is reversible w.r.t. P, the semigroup operator P s is symmetric in L 2 (P) and we obtain that Now we apply Proposition 1.11, which gives for any ε > 0, for some constant c = c(d, ρ, ε, M( p, q)) provided M( p, q) < ∞ for suitable p and q. In view of (6.3) this finishes the proof of (i). Further, to show (ii), for a given ε > 0 we use Fubini's theorem and apply (i) for any ε ∈ (0, ε) to obtain for d ≥ 4, which implies (ii). In d = 3 the result follows by a similar argument.

Remark 6.6
Using the ergodicity of the environment process in (6.3) one can apply the theory of 'fractional coboundaries' of Derriennic and Lin [21] and deduce from the statement in Proposition 6.5 (i), e.g. if d ≥ 4, that for any ε > 0 we have for P-a.e. ω, for some random X = X (d, n) ∈ L 1 (P).
Proof Recall that for any function f : we obtain by the cocycle property (cf. Remark 6.1) and the ergodic theorem that as t tends to infinity P-a.s. and in L 1 ( , P), provided sufficiently high moments of ψ ξ (or the gradient of φ ξ , respectively) are finite, which can be ensured by Proposition 1.10, (1.14). Further, an application of the maximal ergodic theorem yields which finishes the proof.
Proof of Proposition 6. 3 We shall apply the general result in Theorem 6.4 with the choice n = 2 on the martingale Since we get by Proposition 6.5 that for any ε > 0, Moreover, for P-a.e. ω, and we obtain from Proposition 6.7 that Now we apply Theorem 6.4, first under the annealed measure P 0 , which gives 1 5 , so that (i) follows from (6.5) and (6.8). On the other hand, for P-a.e. ω we apply Theorem 6.4 under P ω 0 and obtain which implies (ii) by (6.6) or (6.7), respectively, and (6.8).

Speed of convergence for the corrector
Proof We denote by d be the natural graph distance on Z d , i.e. d(x, y) is the minimal length of a path between x and y and with a slight abuse of notation we set B(r ) := {y ∈ Z d | d(0, y) ≤ r }. Fix exponents α ∈ (d, d + 2δ), k ∈ N such that d + 2d/k < α and ε ∈ (0, δ). Hence, with m denoting the weight of (2.1), we get by appealing to Lemma 2.6 the estimate where X denotes a random constant (with arbitrarily hight moments provided M( p, q) < ∞ for p and q sufficiently large). Consider R t := d(0, X t ) and note that Now, let (z 0 , . . . , z R t ) denote a nearest-neighbour path connecting z 0 = 0 and z R t = X t . Then, by the cocycle property (cf. Remark 6.1), Hence, with H (ω) := |e|=1 ξ · χ(ω, e) k and H * (ω) := sup R>0

# B(R) y∈B(R)
H (τ y ω) (the associated maximal function), we get Note that E[H

Proof of Theorem 1.5
Let δ ∈ (0, 1 10 ) be arbitrary and let p and q be such that the statements in Propositions 6.3 and 6.8 hold. Recall that X t = M t − χ(ω, X t ), t ≥ 0, for P-a.e. ω. Hence, for any x ∈ R, 10 .
Recall that denotes the distribution function of the standard normal distribution. Since is bounded by 1, we have and we get On the other hand, since we can derive a similar lower bound by using the same arguments. Thus, and the same estimate holds if we replace P ω 0 by P 0 . Hence, the claim follows from Proposition 6.3.

Examples
In this section we discuss a class of environments satisfying Assumption 1.3 which are related to the Ginzburg Landau ∇φ-interface model (see [20]). This is a well known model for an interface separating two pure thermodynamical phases. We first explain a slightly more general construction and then revisit that specific class of environments at the end of the discussion. Our starting point is a shift-invariant probability measurẽ μ on˜ := R Z d . We suppose that for anyũ ∈ C 1 (˜ , R) the measureμ satisfies the Brascamp-Lieb inequality where G denotes the Green's function associated with the discrete Laplacian on Z d , and ∂ũ denotes the 2 (Z d )-gradient ofũ, which for sufficiently smoothũ is characterised by∂ũ  Recall that G denotes the discrete Green function for ∇ * ∇. In particular, v solves ∇ * ∇v = ∇ * ∂ū, and a standard energy estimate yields e∈E d |∇v| 2 ≤ e∈E d |∂ū(e)| 2 . We conclude that and potential function V ∈ C 2 (R; R + ), which we suppose to be even and strictly convex with c − ≤ V ≤ c + for some 0 < c − ≤ c + < ∞. Note that in the special case V (x) = 1 2 x 2 the field φ = {φ(x); x ∈ Z d } becomes a discrete Gaussian free field. For more details on the rigorous definition, which is based on taking the thermodynamical limit of Gibbs measures on finite volume approximations of the infinite lattice Z d , see [20,Sect. 4.5]. Then, thanks to the strict convexity we have the Brascamp-Lieb inequality (see [17], cf. also [46]) In particular, (7.1) holds for μ and the above considerations show that an environment with random conductances of the form {ω(e) = λ(∇φ(e)), e ∈ E d } for any positive, even, globally Lipschitz function λ ∈ C 1 (R) satisfies Assumption 1.3. As a further consequence from the Brascamp-Lieb inequality it is known that exponential moments for ∇φ(e) exist (cf. [20,46]). Thus, the environment {ω(e) = λ(∇φ(e)), e ∈ E d } with λ as above also satisfies M( p, q) < ∞ for all p, q ∈ [1, ∞). Lemma A.5 Let d ≥ 2, suppose Assumption 1.3 and uniform ellipticity, i.e. M( p, q) < ∞ for p = q = ∞. Then there exists a random variable X such that for all t ≥ 0 and n ∈ N, where c = c(d, ρ, M(∞, ∞), n).

Proof
We only discuss the case d = 2 since the argument for d ≥ 3 is similar but simpler. W.l.o.g. let ξ = e i . By Lemma A.1 we have and thus we get by Lemma A.2 (with α = d 2 + 1) for any n ∈ N, With these estimates at hand, Theorem 1.6 follows by the same argument as in the proof of Theorem 1.5.