Abstract
The article begins with a quantitative version of the martingale central limit theorem, in terms of the Kantorovich distance. This result is then used in the study of the homogenization of discrete parabolic equations with random i.i.d. coefficients. For smooth initial condition, the rescaled solution of such an equation, once averaged over the randomness, is shown to converge polynomially fast to the solution of the homogenized equation, with an explicit exponent depending only on the dimension. Polynomial rate of homogenization for the averaged heat kernel, with an explicit exponent, is then derived. Similar results for elliptic equations are also presented.
1 Introduction
1.1 Main results
The main goal of this article is to give quantitative estimates in the homogenization of discrete divergence-form operators with random coefficients. Writing \(\mathbb B \) for the set of edges of \(\mathbb Z ^d\), we let \(\omega = (\omega _e)_{e \in \mathbb B }\) be a family of i.i.d. random variables, assumed to be uniformly bounded away from \(0\) and infinity, and whose joint distribution will be written \(\mathbb P \) (with associated expectation \(\mathbb E \)). The operator whose homogenization properties we wish to investigate is
where we write \(y \sim x\) when \(x,y \in \mathbb Z ^d\) are nearest neighbours. For a bounded continuous \(f : \mathbb R ^d \rightarrow \mathbb R \), we consider \(u^{(\varepsilon )}\) the solution of

and \(u_\varepsilon (t,x) = u^{(\varepsilon )}(\varepsilon ^{-2} t,\lfloor \varepsilon ^{-1} x \rfloor )\). There exists a symmetric positive-definite matrix \(\overline{A}\) (independent of \(f\)) such that the function \(u_\varepsilon \) converges, as \(\varepsilon \) tends to \(0\), to the function \(\overline{u}\) solution of

The notions of being a solution to (DPE\(^\omega _\varepsilon \)) or (CPE), and of the convergence of \(u_\varepsilon \) to \(\overline{u}\), will be made precise later on. For every \(\alpha = (\alpha _1, \ldots , \alpha _d) \in \mathbb N ^d\), we call
a weak derivative of \(f\) of order \(|\alpha |_1\), where the derivative is understood in the sense of distributions.
Here and below, we write \(\lfloor x \rfloor \) for the integer part of \(x, a \wedge b = \min (a,b), a \vee b = \max (a,b), \log _+(x) = \log (x) \vee 1\), and \(|\xi |\) for the \(L^2\) norm of \(\xi \in \mathbb R ^d\). The main purpose of this paper is to prove the following theorems.
Theorem 1.1
Let \(m = \lfloor d/2 \rfloor + 3\) and \(\delta > 0\). There exist constants \(C_\delta \) (which may depend on the dimension) and \(q\) such that, if the weak derivatives of \(f\) up to order \(m\) are in \(L^2(\mathbb R ^d)\), then for any \(\varepsilon > 0, t > 0\) and \(x \in \mathbb R ^d\), one has
where
Remark 1.2
The Sobolev embedding theorem ensures that under the assumptions of Theorem 1.1, the function \(f\) is continuously differentiable and the norms \(\Vert \partial _{x_j} f \Vert _\infty \) are finite (see for instance [1, Theorem 5.4]).
Theorem 1.3
Let \(p_t^\omega (x,y)\) be the heat kernel associated to \(L^\omega \), let
be the heat kernel associated to \(\frac{1}{2} \nabla \cdot \overline{A} \nabla \), and let \(\delta > 0\). There exist constants \(c > 0\) (independent of \(\delta \)), \(q, C_\delta , \varepsilon _\delta > 0\) such that for any \(\varepsilon > 0, t > 0\) satisfying \(\varepsilon /\sqrt{t}\leqslant \varepsilon _\delta \) and any \(x \in \mathbb R ^d\), one has
In particular, for any \(s > 0\), there exists \(C_{\delta ,s}\) such that for \(\varepsilon \) small enough,
Remark 1.4
For a given smooth function \(f\) and a fixed \(t > 0\), the right-hand side of (1.2) is of the order of
where \(\delta ' = 2 \delta > 0\) is arbitrary. Similarly, for fixed \(t\) and \(x\), the right-hand side of (1.4) is of the order of
where \(\delta '' = 2\delta /(d+3) > 0\) is arbitrary.
Remark 1.5
Similar results concerning elliptic equations are presented in Theorems 7.1 and 7.3 below.
1.2 Context
Homogenization problems have a very long story, going back at least to [34, 40]. Rigorous proofs of homogenization for periodic environments were obtained in the 1960s and 1970s (see [4] for references), and then for random environments with [31, 32, 37, 42]. Classical methods used to show homogenization typically rely on a compactness argument, or on the ergodic theorem, both approaches leaving the question of the rate of convergence untouched.
For continuous space and periodic coefficients, [28, Corollary 2.7] uses spectral methods to show that
For random coefficients, available results are much less precise. For continuous space, [43] gives an algebraic speed of convergence of \(u_\varepsilon \) to \(\overline{u}\) for the elliptic problem and \(d \geqslant 3\), without providing an explicit exponent. In [8], the much more general case of fully nonlinear elliptic equations is considered, and a speed of convergence of a logarithmic type is proved.
Here, we focus on the convergence of the average of \(u_\varepsilon \) to \(\overline{u}\). This approach has been considered in [14] for the elliptic problem. There, it is shown that the suitably rescaled Green function, once averaged over the randomness of the coefficients, differs from the Green function of the homogenized equation by no more than a negative power of \(\varepsilon \). The exponent obtained is implicit, and depends on the ellipticity condition assumed on the random coefficients. Similar results for parabolic equations have been derived in [11].
In contrast, Theorems 1.1 and 1.3 provide explicit exponents, that depend only on the dimension. I conjecture that the correct order of decay with \(\varepsilon \) in Theorem 1.1 should be
This differs notably from what is obtained in Theorem 1.1 only when \(d = 2\). On the other hand, it may well be that the assumption of high regularity on \(f\) is only an artefact of the methods employed.
The fact that
is known at least since [3], where the much more difficult case where the random coefficients are Bernoulli random variables is considered (in this context, the heat kernel should be considered only within the unique infinite percolation cluster). Yet, for strictly positive random coefficients, this convergence does not hold if the distribution of the random coefficients is allowed to have a fat tail close to \(0\) and when \(d \geqslant 4\) [5, 6]. Under the same circumstance and when \(p^\omega \) is replaced by its average in (1.7), the convergence fails to hold in any dimension [20] (see however [2, Proposition 7.2] for a nice way to get around this problem).
Under our present assumption of uniform ellipticity, regularity properties of the average of \(p^\omega \) were proved in [12, 17] (more on this will come below).
An evaluation of the gap between the average of \(u_\varepsilon \) and \(\overline{u}\) naturally calls for estimates on the size of the random fluctuations of \(u_\varepsilon \) around its average. In this direction and for the elliptic problem, [12] obtains algebraic decay of the variance of \(u_\varepsilon \) (integrated over space). The exponent obtained is implicit, and depends on the ellipticity conditions.Footnote 1
1.3 Our approach
In order to prove Theorem 1.1, we will use the representation of \(u_\varepsilon \) as the expected value over the paths of a random walk, that we write \((X_t)_{t \geqslant 0}\). This random walk has inhomogeneous jump rates given by the \((\omega _e)_{e \in \mathbb B }\), and \(L^\omega \) is its infinitesimal generator. For instance, one has
where we write \(\mathbf P ^\omega _0\) for the distribution of the random walk starting from \(0\), and \(\mathbf E ^\omega _0\) for its associated expectation. The (pointwise) convergence of \(u_\varepsilon \) to \(\overline{u}\) is equivalent to the claim that the random walk, after diffusive scaling, satisfies a central limit theorem. Quantitative estimates should thus follow if one can provide with rates of convergence in this central limit theorem.
In [35], it is shown that there exist constants \(C,q \geqslant 0\) such that for any \(\xi \) of unit norm,
where \(\Phi \) is the cumulative distribution function of the standard Gaussian random variable, and \(\sigma (\xi ) = \xi \cdot \overline{A}\xi \).
This result has two important weak points: (1) the rates are far from the usual \(t^{-1/2}\) one obtains for sums of i.i.d. random variables, and (2) the theorem only gives information about the projections of \(X_t\) onto a fixed vector. We shall find ways to overcome these two problems.
The classical approach for the proof of a central limit theorem for the random walk consists in decomposing it as the sum of a martingale plus a remainder term, and then show that the martingale converges (after scaling) to a Gaussian random variable, while the remainder term becomes negligible in the limit.
In view of this, what should be done is clear: we should first find a quantitative estimate on how small the remainder term is, and second, show that the martingale converges rapidly to a Gaussian. This is indeed the method used in [35]. The control of the remainder term given there is satisfactory, and the problem lies with the quantitative central limit theorem for the martingale part.
This quantitative central limit theorem relies on the fact that one can have a sharp control of the variance of the quadratic variation of the martingale. It is shown that, after scaling, this variance decays like \(t^{-1}\) when \(d \geqslant 4\), which is the best possible rate. However, given such a control, the quantitative CLT (due to [24, 26]) used there only yields a decay of \(t^{-1/5}\) in this case.
Surprisingly, this exponent \(1/5\) is best possible in general [36]. To overcome this obstruction, we derive new quantitative CLT’s for martingales, that will not yield a Berry-Esseen type of estimate, but rather measure
where \(\mathfrak L \) is a class of functions (this is reminiscent of Stein’s method, see for instance [10]). When \(\mathfrak L \) is the class of bounded \(1\)-Lipschitz functions, the supremum is often called the Kantorovich(-Rubinstein) distance. We also consider \(\mathfrak L \) to be the class of bounded \(\mathcal C ^2\) functions that have first derivative bounded by \(1\) and second derivative bounded by \(k\), and call it the \(k\) -Kantorovich distance. The martingale CLT’s obtained hold for general square-integrable martingales, and are of independent interest.
Once equipped with these quantitative martingale CLT’s, we apply them to the one-dimensional projections of the random walk \((X_t)\), and for \(d \geqslant 3\), we obtain rates approaching the i.i.d. rate of \(t^{-1/2}\). To do so, we use estimates derived in [35], most importantly on the variance of the quadratic variation of the martingale. These in turn are consequences of the \(L^p\) boundedness of the corrector (for \(d \geqslant 3\), and with logarithmic corrections for \(d = 2\)), and of a spatial decorrelation property of this corrector, proved in [22, Theorem 2.1 and Proposition 2.1].
In order to obtain Theorem 1.1, we need to carry the information obtained on the projections of \(X_t\) to \(X_t\) itself, in a kind of quantitative version of the Cramér-Wold theorem. This is achieved through Fourier analysis, at the price of requiring the existence of weak derivatives of higher order.
The key observation that enables to go from Theorem 1.1 to Theorem 1.3 is the high regularity of the averaged heat kernel. In contrast to the true heat kernel, the averaged one has a gradient which is bounded by a constant times the gradient of \(\overline{p}\), as is proved in [12, 17].
The estimates due to [22] are the only place where the assumptions of independence and uniform ellipticity of the coefficients come into play. In particular, if it is shown that these estimates are valid for certain correlated environments, then the present results automatically extend to this context. The present results should also extend to continuous space with only minor change, as long as the estimates of [22] remain true in this setting.Footnote 2
1.4 Organization of the paper
We introduce the (\(k\)-)Kantorovich and Kolmogorov distances in Sect. 2. In Sect. 3, we consider general square-integrable martingales, and derive quantitative CLT’s with respect to the (\(k\)-)Kantorovich distances. We then apply these results to projections of the random walk \(X_t\) in Sect. 4. The homogenization setting is taken up in Sect. 5, and Theorem 1.1 is proved. Theorem 1.3 is then derived in Sect. 6. Finally, similar results for the homogenization of elliptic equations are presented in Sect. 7.
2 Distances between probability measures
A function \(f : \mathbb R ^m \rightarrow \mathbb R ^n\) is said to be \(k\)-Lipschitz if for any \(x,y \in \mathbb R ^m\), one has \(|f(y)-f(x)| \leqslant k |y-x|\). Let \(\nu , \nu '\) be probability measures on \(\mathbb R \), and let \(F_\nu , F_{\nu '}\) be their respective cumulative distribution functions. We define the Kantorovich distance between \(\nu \) and \(\nu '\) as
and the Kolmogorov distance between \(\nu \) and \(\nu '\) as
The notation for the Kantorovich distance becomes more transparent once we recall that (see for instance [41, Theorem 1.14 and (2.48)])
As we will see below, bounds in the martingale CLT are improved when measured with the Kantorovich distance instead of the Kolmogorov distance. We now introduce weaker forms of the Kantorovich distance, for which the rates of convergence will be even better. For any \(k \in [0,+\infty ]\), we define the \(k\) -Kantorovich distance as
where \(\mathcal C _b^2(\mathbb R ,\mathbb R )\) is the set of bounded twice continuously differentiable functions from \(\mathbb R \) to \(\mathbb R \). For \(k \leqslant k'\), one has \(\mathsf d _{1,k}\leqslant \mathsf d _{1,k'} \leqslant \mathsf d _{1,\infty } = \mathsf d _1\). Note that if \(f \in \mathcal C ^2_b(\mathbb R ,\mathbb R )\), then
In the sequel, if \(X\) follows the distribution \(\nu \) and \(Y\) the distribution \(\nu '\), we may write \(\mathsf d _1(X,Y)\) to denote \(\mathsf d _1(\nu ,\nu ')\), or also \(\mathsf d _1(X,F_{\nu '})\) if convenient. If \(X\) and \(Y\) are defined on the same probability space with probability measure \(P\) and associated expectation \(E\), then for any \(1\)-Lipschitz function \(f\), we have
and hence
Similarly, if \(X\) follows the distribution \(\nu \) and \(Y\) the distribution \(\nu '\), we write \(\mathsf d _{1,k}(X,Y), \mathsf d _{1,k}(X,F_{\nu '})\) or \(\mathsf d _{1,k}(\nu ,\nu ')\) as convenient.
3 Martingale CLT
For a square-integrable cadlag martingale \((M_t)_{t \in [0,1]}\) defined with respect to the probability measure \(P\) and the right-continuous filtration \((\mathcal F _t)_{t \geqslant 0}\), we write \((\langle M \rangle _t)_{t \in [0,1]}\) for its predictable quadratic variation,
and
Recall that we denote by \(\Phi \) the cumulative distribution function of the standard Gaussian random variable. In [24], the following is proved.
Theorem 3.1
([24]) For any \(p > 1\), there exists \(\overline{C}_p\) (independent of \(M\)) such that
Our first result consists in showing that one can get sharper bounds if one replaces the Kolmogorov distance by the (\(k\)-)Kantorovich distance in (3.1).
Theorem 3.2
For any \(p > 1\), there exists \(C_p\) (independent of \(M\)) such that
and for any \(k \geqslant 0\),
Remark 3.3
Naturally, one has \(\Vert \langle M \rangle _1 - 1\Vert _1 \leqslant \Vert \langle M \rangle _1 - 1\Vert _p\), and the statements are only interesting when this quantity, and also \(L_{2p}\), are small, so Theorem 3.2 indeed provides better rates of convergence than Theorem 3.1. It is shown in [36] that it is not possible to change the exponent \(p/(2p+1)\) appearing on the term \(\Vert \langle M \rangle _1 - 1\Vert _p\) in the right-hand side of (3.1) by any higher exponent. It would be interesting to investigate how sharp (3.2) is in this respect. The term \(\Vert \langle M \rangle _1 - 1\Vert _1\) appearing on the right-hand side of (3.3) cannot be improved. Indeed, let \((B_s)_{s \geqslant 0}\) be a standard Brownian motion, and consider the martingale \(M_s = B_{(1+\varepsilon ) s}\). Since the martingale is continuous, \(L_{2p}\) vanishes, while one has \(\Vert \langle M \rangle _1 - 1\Vert _1 = \varepsilon \). On the other hand, the cosine function has first and second derivatives bounded by \(1\), and thus
thus justifying the optimality of the exponent on \(\Vert \langle M \rangle _1 - 1\Vert _1\).
Remark 3.4
A quantitative martingale CLT expressed in terms of the Kantorovich distance was already formulated in [38, Theorem 8.1.16]. The terms involved in the bound are however difficult to estimate in practical situations, in contrast to what is obtained in Theorem 3.2.
In order to prove Theorem 3.2, we will rely on the following non-uniform version of Theorem 3.1.
Theorem 3.5
([24, 25]) For any \(p > 1\), there exists \(\tilde{C}_p\) (independent of \(M\)) such that if \(L_{2p} + \Vert \langle M \rangle _1 - 1\Vert _p^{p} \leqslant 1\), then for any \(x \in \mathbb R \),
[25, Theorem 1] is the equivalent statement concerning discrete-time martingales. Theorem 3.5 can be derived from its discrete-time version by applying the approximation procedure explained in [24, Section 4] (in [24], locally square-integrable martingales are considered, while we stick here with plainly square-integrable martingales. There is no loss of generality however, since a locally square-integrable martingale is in fact a square-integrable one if \(\Vert \langle M \rangle _1 - 1\Vert _1\) is finite. One can thus skip the localization procedure at the end of [24, Section 4]).
Proof of Theorem 3.2
We start by proving that there exists \(C_p\) (independent of \(M\)) such that (3.2) holds. We decompose the proof of this into three steps.
Step I.1. We first prove the claim assuming that
and that \(L_{2p} \leqslant 1\). Under this condition, Theorem 3.5 ensures that
We thus have, after possibly enlarging \(\tilde{C}_p\),
which is the desired result.
Step I.2. We now no longer impose that condition (3.4) holds, but keep with the assumption that \(L_{2p} \leqslant 1\). Following an idea probably due to [18], we introduce
Note that \(\tau \) is a stopping time, since the filtration is right-continuous, and thus
We define
and
Note that \(\langle M \rangle _{\tau ^-} \leqslant 1\). Let \((B_s)_{s \geqslant 0}\) be a standard Brownian motion, independent of the martingale. We define
By construction, \(\tilde{M}\) is a martingale, and
hence \(\langle \tilde{M} \rangle _2 = 1\). Naturally, the fact that \(\tilde{M}\) is defined on \([0,2]\) instead of \([0,1]\) plays no role, and this martingale satisfies condition (3.4) (at time \(2\)). Writing
we clearly have \(\tilde{L}_{2p} \leqslant L_{2p} \leqslant 1\). We learn from the first step of the proof that
We now want to use the fact that
to estimate \(\mathsf d _1(M_1,\Phi )\). In view of (2.5), we have
Note that
and thus
Let us write \(a_1 + a_2 + a_3\) for the latter sum, with obvious identifications. We bound the contribution of each of these terms successively.
since \(\tau \leqslant 1\) is a stopping time. Now, either \(\tau = 1\), in which case \(\langle M \rangle _1 - \langle M\rangle _\tau = 0\), or \(\tau < 1\), in which case \(\langle M \rangle _\tau \geqslant 1\). In both cases, we have
and thus
As for \(a_2\), we have
For the third term, we have
where \(c = E[|B_1|] \leqslant 1\). We decompose the last expectation as
The first term is bounded by \(\Vert 1-\langle M \rangle _1\Vert _1\), while the second term is smaller than
(to see this, consult for instance the proof of [27, Theorem 4.2]). The latter is bounded by
To sum up, we have shown that
Since we assume that \(L_{2p} \leqslant 1\), we have \(L_{2p}^{1/(2p)} \leqslant L_{2p}^{1/(2p+1)}\), and equations (3.6), (3.5) and (3.8) give us that
which is what we wanted to prove.
Step I.3. It remains to consider the case when \(L_{2p} > 1\). It follows from (2.5) that
where \(c\) is the \(L_1\) norm of a standard Gaussian, \(c \leqslant 1\). Moreover,
As a consequence, it is always true that
The theorem is thus clearly true when \(L_{2p} > 1\) as soon as \(C_p \geqslant 2\), and this finishes the proof of (3.2).
We now proceed to show that there exists \(C_{p}\) (independent of \(M\) and \(k\)) such that (3.3) holds, and decompose the proof of this fact into two steps.
Step II.1. We assume first that \(L_{2p} \leqslant 1\), and consider again the martingale \(\tilde{M}\) as constructed in step I.2. Since \(\langle \tilde{M} \rangle _2 = 1\), we know from step I.1 that
Let
and observe that
We have
The first term on the right-hand side is smaller than \(E[|\Delta M (\tau )|]\) by (2.5), and we have seen in (3.7) that this is smaller than \(L_{2p}^{1/(2p)} \leqslant L_{2p}^{1/(2p+1)}\). Using also (3.9), we obtain
Let \(f \in \mathcal C _b^2 (\mathbb R ,\mathbb R )\) be such that \(\Vert f'\Vert _\infty \leqslant 1\) and \(\Vert f''\Vert _\infty \leqslant k\). We will show that
Indeed, since \(f \in \mathcal C _b^2 (\mathbb R ,\mathbb R )\) and \(\Vert f''\Vert _\infty \leqslant k\), we have
But
and \(E[B_{1-\langle M \rangle _{\tau ^-}} \ | \ \mathcal F _{\tau }] = 0\) since \(B\) and \(M\) are independent. On the other hand,
and we have seen in step I.2, while treating the term \(a_3\), that
As a consequence, (3.11) is proved, and thus
We now show that
using the same technique. We write
and observe that
since \(M\) is a martingale and \(\tau \) a stopping time. On the other hand, we have seen while treating the term \(a_1\) in step I.2 that
and thus (3.13) is proved. Combining (3.10), (3.12) and (3.13), we thus obtain
and this proves (3.3) for \(L_{2p} \leqslant 1\).
Step II.2. We now conclude by considering the case when \(L_{2p} > 1\). We learn from step I.3 that
Since for any \(x \geqslant 0\), we have \(\sqrt{x} \leqslant 1 + x/2\), we thus obtain
and thus relation (3.3) holds when \(L_{2p} > 1\), provided we choose \(C_{p} \geqslant 3\). \(\square \)
4 The random walk among random conductances
Let \(0 < \alpha \leqslant \beta < + \infty \), and \(\Omega = [\alpha ,\beta ]^\mathbb B \). For any family \(\omega = (\omega _e)_{e \in \mathbb B } \in \Omega \), we consider the Markov process \((X_t)_{t \geqslant 0}\) whose jump rate between \(x\) and a neighbour \(y\) is given by \(\omega _{x,y}\). We write \(\mathbf P ^\omega _x\) for the law of this process starting from \(x \in \mathbb Z ^d, \mathbf E ^\omega _x\) for its associated expectation. Its infinitesimal generator is \(L^\omega \) defined in (1.1). We assume that the \((\omega _e)_{e \in \mathbb B }\) are themselves i.i.d. random variables under the measure \(\mathbb P \) (with associated expectation \(\mathbb E \)). We write \(\overline{\mathbb{P }} = \mathbb P \mathbf P ^\omega _0\) for the annealed measure. It was shown in [30] that under \(\overline{\mathbb{P }}\) and as \(\varepsilon \) tends to \(0\), the process \(\sqrt{\varepsilon } X_{\varepsilon ^{-1} t}\) converges to a Brownian motion, whose covariance matrix we write \(\overline{A}\) (in [39], it is shown that under our present assumption of uniform ellipticity, the invariance principle holds under \(\mathbf P ^\omega _0\) for almost every \(\omega \)).
Let \(\xi \in \mathbb R ^d\) be a vector of unit \(L^2\) norm. The purpose of this section is to give sharp estimates on the \(k\)-Kantorovich distance between \(\xi \cdot X_t/\sqrt{t}\) and \(\Phi _{\sigma (\xi )}\), where we write \(\Phi _\sigma \) to denote the cumulative distribution function of a Gaussian random variable with variance \(\sigma ^2\), and \(\sigma (\xi ) = (\xi \cdot \overline{A} \xi )^{1/2}\).
Theorem 4.1
For any \(\delta > 0\), there exists a constant \(C\) (which may depend on the dimension) such that for any \(k \geqslant 0\) and any \(\xi \) of unit norm, one has
for some \(q \geqslant 0\), where in the left-hand side, \(\xi \cdot X_t/\sqrt{t}\) stands for the distribution of this random variable under the measure \(\overline{\mathbb{P }}\), and where \(\Psi _{q,\delta }\) was defined in (1.3).
Remark 4.2
When \(d \geqslant 3\), the exponent of decay in (4.1) can thus be made arbitrarily close to \(1/2\), and this is the exponent one gets when considering sums of i.i.d. random variables with finite third moment.
Remark 4.3
By the same reasoning, one can also prove that there exist constants \(C\) (which may depend on the dimension) and \(q\) such that, for any \(\xi \) of unit norm, one has
where again \(\xi \cdot X_t/\sqrt{t}\) stands for the distribution of this random variable under the measure \(\overline{\mathbb{P }}\).
The proof of Theorem 4.1 follows a line of reasoning similar to that of [35, Theorem 2.1]. From now on, we fix \(\xi \in \mathbb R ^d\) of unit norm. The starting point is to approximate \(\xi \cdot X_t\) by a martingale, whose construction we now recall. To begin with, let us write \((\theta _x)_{x \in \mathbb Z ^d}\) to denote the action of translation of \(\mathbb Z ^d\) on the space of environments \(\Omega \), so that for \(\omega \in \Omega \) and \(x,y,z \in \mathbb Z ^d, y\sim z\),
Let \(\mathcal L \) be the operator acting on \(L^2(\Omega , \mathbb P )\) by
This operator comes out naturally as the infinitesimal generator of the Markov process of the environment viewed by the particle (i.e. the process \(t \mapsto \theta _{X_t} \ \omega \)). One can check that \(-\mathcal L \) is a positive self-adjoint operator on \(L^2(\Omega ,\mathbb P )\). We let
be the local drift in the direction \(\xi \), and for every \(\mu > 0\), we define \(\phi _\mu \in L^2(\Omega ,\mathbb P )\) to be such that
The parameter \(\mu > 0\) should be thought to be small (ideally, one would like to take it to be zero, but this is not possible in dimension \(2\)). We decompose \(\xi \cdot X_t\) as the sum \(M_\mu (t) + R_\mu (t)\), where
and
The next proposition collects several results, mostly from [35], that will be useful for our purpose.
Theorem 4.4
The process \((M_\mu (t))_{t \geqslant 0}\) is a square-integrable martingale under \(\overline{\mathbb{P }}\) (with respect to the natural filtration associated to \((X_t)_{t \geqslant 0}\)). Let \(\sigma _\mu = \mathbb E [(M_\mu (1))^2]^{1/2}\). There exist constants \(C\) and \(q\) such that for any \(\mu > 0, t > 0\), the following three estimates hold:
Moreover, for every integer \(p \geqslant 1\), there exist constants \(C\) and \(q\) such that for any \(\mu > 0, t > 0\), one has
In these four estimates, the constants do not depend on the vector \(\xi \in \mathbb R ^d\) of unit norm.
Inequality (4.7) was proved in [22, Theorem 1] (see also [21, Theorem 3 with \(k = 1\)] for a slightly different point of view). Inequalities (4.5) and (4.6) correspond respectively to [35, (3.10) and Proposition 3.4]. The last inequality with \(p = 2\) corresponds [35, (3.11)]; the extension to arbitrary \(p\) is straightforward.
Proof of Theorem 4.1
We first treat the case \(d \geqslant 2\). We have
with the understanding that random variables stand in place of their respective distributions under the measure \(\overline{\mathbb{P }}\). Let us write the three terms in the right-hand side above as \(b_1 + b_2 + b_3\), and proceed to evaluate each of these terms for the specific choice \(\mu = 1/t\). Considering (2.5), we can bound the term \(b_1\) by
and inequality (4.6) gives us adequate control of this upper bound.
To handle the term \(b_3\), consider a standard Gaussian random variable \(\mathcal N \). Then \(\sigma \mathcal N \) has \(\Phi _{\sigma }\) as its cumulative distribution function, hence
Since \(E[|\mathcal N |] \leqslant 1\), the term \(b_3\) is bounded by \(|\sigma _{1/t} - \sigma (\xi )|\). We can thus use inequality (4.7) (with \(\mu = 1/t\)), which is much better than what we need for our purpose.
We now turn to the term \(b_2\). For any \(p > 1\), we introduce
Theorem 3.2 tells us that if \(L_{2p}(t) \leqslant 1\), then \(b_2\) is smaller than
Inequality (4.8) ensures that
for some constants \(C\) and \(q\) depending on \(p\). In particular, it is always true that \(L_{2p}(t)\) tends to \(0\) as \(t\) tends to infinity. We fix \(p\) large enough so that
With such a choice for \(p\), we have \((L_{2p}(t))^{1/(2p+1)} = o(t^{\delta -1/2})\).
Finally, inequality (4.5) gives us that
Since
this finishes the proof of Theorem 4.1 for \(d\geqslant 2\) and \(t\) large enough, and it is easy to see that the left-hand side of (4.1) is bounded for smaller \(t\). The one-dimensional case is obtained in a similar way, following [35, Section 9]. \(\square \)
5 Homogenization
We consider the discrete parabolic equation with random coefficients

where \(f: \mathbb Z ^d \rightarrow \mathbb R , L^\omega \) is the operator defined in (1.1), and by \(L^\omega u(t,x)\), we understand \(L^\omega u(t,\cdot ) (x)\). Note that \(L^\omega \) is the discrete analog of a divergence form operator.
For a fixed \(\omega \in \Omega \), we say that \(u\) is a solution of (DPE\(^\omega \)) if it is continuous on \([0,+\infty ) \times \mathbb Z ^d\), has continuous time derivative there (in other words, \(u(\cdot ,x)\) is in \(\mathcal C ^1(\mathbb R _+,\mathbb R )\) for every \(x \in \mathbb Z ^d\)), and satisfies the identities displayed in (DPE\(^\omega \)).
Proposition 5.1
For any \(\omega \in \Omega \) and any bounded initial condition \(f\), there exists a unique bounded solution \(u\) of (DPE\(^\omega \)), and it is given by
This is a very well known result. Checking that (5.1) is indeed a solution is a direct consequence of the definition of the Markov chain. To see uniqueness, take \(\tilde{u}\) a bounded solution of (DPE\(^\omega \)). Letting \(\tilde{M}_s = \tilde{u}(t-s,X_s)\), one can show that \((\tilde{M}_s)_{0 \leqslant s \leqslant t}\) is a martingale under \(\mathbf P ^\omega _x\) for any \(x \in \mathbb Z ^d\), and as a consequence,
which is the function defined in (5.1).
For a symmetric positive-definite matrix \(\overline{A}\), we consider the equation (CPE) given in the introduction. We say that \(\overline{u}\) is a solution of (CPE) if it is continuous on \(\mathbb R _+ \times \mathbb R ^d\), has a continuous first derivative in the time variable and continuous first and second derivatives in the space variable on \((0,+\infty ) \times \mathbb R ^d\), and satisfies the identities displayed in (CPE).
Proposition 5.2
For any bounded continuous initial condition \(f\), there exists a unique bounded solution \(\overline{u}\) of (CPE), and it is given by
where, under the measure \(\mathbf P _x, B_t\) is a Brownian motion with covariance matrix \(\overline{A}\) that starts at \(x\).
Again, this result is standard. It is proved in the same way as Proposition 5.1, with the help of Itô’s formula.
Remark 5.3
The boundedness assumption in Propositions 5.1 and 5.2 could be changed for being subexponential. More precisely, let \(f : \mathbb Z ^d \rightarrow \mathbb R \) be such that for any \(\alpha > 0, |f(x)| = O(e^{\alpha |x|})\). Then there exists a unique solution \(u\) of (DPE\(^\omega \)) such that, for any \(\alpha > 0\) and any \(t \geqslant 0, \sup _{s \leqslant t} |u(s,x)| = O(e^{\alpha |x|})\). The boundedness condition was merely chosen for convenience.
We now define rescaled solutions of the parabolic equation with random coefficients. For a bounded continuous function \(f:\mathbb R ^d \rightarrow \mathbb R \), we let \(u^{(\varepsilon )}\) be the bounded solution of (DPE\(^\omega \)) with initial condition given by the function \(x \mapsto f(\varepsilon x)\), and for any \(t\geqslant 0\) and \(x \in \mathbb R ^d\), we let
It is well understood (see for instance [4, Chapter 3]) that the probabilistic approach yields pointwise convergence of \(u_\varepsilon \) to the solution of the homogenized problem. The following result is folklore (see also [33] where the homogenization of random operators in continuous space is obtained using the probabilistic approach).
Theorem 5.4
There exists a symmetric positive-definite matrix \(\overline{A}\) (independent of \(f\)) such that for every \(t \geqslant 0\) and \(x \in \mathbb R ^d\), we have
where \(\overline{u}\) is the bounded solution of (CPE) with initial condition \(f\).
Proof
Recall that we write \((\theta _x)\) to denote the translations on \(\Omega \), see (4.2). The distribution of \(X\) under \(\mathbf P ^\omega _x\) is the same as the one of \(X+x\) under \(\mathbf P ^{\theta _x \omega }_0\) (both are Markov processes with the same initial condition and the same transition rates). Using this observation in (5.3), we obtain that
where \(x_\varepsilon = \varepsilon \lfloor \varepsilon ^{-1} x \rfloor \).
Since the measure \(\mathbb P \) is invariant under translations, \(u_\varepsilon (t,x)\) has the same distribution as
It is proved in [15, 30] that for some symmetric positive-definite \(\overline{A}\) (independent of \(f\)), the quantity in (5.5) converges in probability to \(\mathbf E _0[f(B_t +x)]\) as \(\varepsilon \) tends to \(0\), where \(B\) is a Brownian motion with covariance matrix \(\overline{A}\). \(\square \)
Remark 5.5
It would be interesting to replace the convergence in probability in (5.4) by an almost sure convergence. Note that almost sure convergence for \(x = 0\) is equivalent to an almost sure central limit theorem for the random walk, and this is proved in [39]. Theorem 5.4 contrasts with for instance [28, Theorem 7.4], where weak convergence of an analogue of \(u_\varepsilon \) is proved, but for almost every environment.
We start the proof of Theorem 1.1 with two lemmas with a Fourier-analytic flavour.
Lemma 5.6
Let \(Z\) be a random variable with distribution \(\nu , \mathcal N \) be a standard \(d\)-dimensional Gaussian random variable independent of \(Z\), and \(\sigma > 0\). If \(f\) is in \(L^2(\mathbb R ^d)\), then
where
and
Remark 5.7
The definition of the Fourier transform given in (5.6) only makes sense for \(f \in L^1(\mathbb R ^d)\), but as is well known, the Fourier transform can then be extended to functions in \(L^2(\mathbb R ^d)\) by continuity.
Proof
Recall that we always assume \(f\) to be bounded. In order to prove the proposition, it suffices to prove it for functions \(f \in L^1(\mathbb R ^d)\), since we can then conclude by a density argument.
Let us write
Note first that
The distribution of \(Z + \sigma \mathcal N \) has a density (with respect to Lebesgue measure) at point \(z\) which is given by
As a consequence (and using the fact that \(\hat{\nu }\) is bounded), we have
Since
this proves the lemma. \(\square \)
Lemma 5.8
For any integer \(m \geqslant 0\), there exists a constant \(C_m\) such that if the weak derivatives of \(f\) up to order \(m\) are in \(L^2(\mathbb R ^d)\), then
Proof
See [19, Theorem 8]. \(\square \)
Proof of Theorem 1.1
Let \(t > 0\). We saw in the proof of Theorem 5.4 that
where in the last line, we used the fact that the measure \(\mathbb P \) is translation invariant, and we recall that we write \(\overline{\mathbb{E }}\) for \(\mathbb E \mathbf E ^\omega _0\) and \(x_\varepsilon \) for \(\varepsilon \lfloor \varepsilon ^{-1} x \rfloor \). Note that
which is the first term in the right-hand side of (1.2) (a “lattice effect”). We now focus on studying the difference
where we recall that \(\mathbf E _0[f(B_t+x)] = \mathbf E _x[f(B_t)] = \overline{u}(t,x)\). Possibly replacing \(f\) by \(f(\ \cdot \ + x)\), we may as well suppose that \(x = 0\). Let \(\sigma > 0\) be a small parameter, \(\mathcal N \) be a standard \(d\)-dimensional Gaussian random variable, independent of everything else, and write \(f_t = f(\sqrt{t} \ \cdot )\). Since \(f_t\) is bounded and continuous, we have
Similarly,
where we slightly abuse notation by using the same \(\mathcal N \) to denote a standard Gaussian (independent of everything else) under both the measures \(\mathbf E _0\) and \(\overline{\mathbb{E }}\). The random variable \(\sigma \mathcal N \) is introduced for regularization purposes, and in particular will enable us to use Lemma 5.6.
Let us write \(\nu _{\varepsilon }\) for the distribution of
under the measure \(\overline{\mathbb{P }}\), and \(\nu _0\) for the distribution of \(B_1\) under \(\mathbf E _0\). Note that
The function \( x \mapsto e^{i |\xi | x}\) has first derivative bounded by \(|\xi |\) and second derivative bounded by \(|\xi |^2\). In view of (2.4), we obtain from Theorem 4.1 that
Using Lemma 5.6, we thus obtain that
where \(C\) does not depend on \(\sigma \). We can thus take the limit \(\sigma \rightarrow 0\) in this inequality and use (5.8) and (5.9) to obtain
Since \(\hat{f}_t(\xi ) = t^{-d/2} \hat{f}(\xi /\sqrt{t})\), we can perform a change of variables on the integral underbraced above:
Note that \(|\xi | (\sqrt{t}|\xi | \vee 1) \leqslant (\sqrt{t}+1)(|\xi |^2+1)\). Hence, the integral above is bounded by
Let \(m = \lfloor d/2 \rfloor +3\). By the Cauchy–Schwarz inequality, this integral is bounded by
Since \(2m - 4 > d\), the first term of this product is finite, while Lemma 5.8 gives us that the second term is bounded by
Recalling (5.10), we thus get that
and this finishes the proof. \(\square \)
6 Heat kernel estimates
The heat kernel \(p_t^\omega (x,y)\) is defined so that \((t,y) \mapsto p^\omega _t(x,y)\) is the unique bounded solution to (DPE\(^\omega \)) with initial condition \(f = \mathbf 1 _x\). The heat kernel is symmetric: \(p_t^\omega (x,y) = p_t^\omega (y,x)\), and by translation invariance of the random coefficients, \(\mathbb E [p_t^\omega (x,y)] = \mathbb E [p_t^\omega (0,y-x)]\).
The aim of this section is to prove Theorem 1.3. In order to do so, we will need a regularity result on the averaged heat kernel. For \(f : \mathbb Z ^d \rightarrow \mathbb R \) and \(1 \leqslant i \leqslant d\), we write
where \((\mathbf e _i)_{1 \leqslant i \leqslant d}\) is the canonical basis of \(\mathbb R ^d\). The following result was proved in [12, Theorem 1.4], and then elegantly rederived in [17, (1.4)].
Theorem 6.1
There exist \(C, c_1 > 0\) such that for any \(t > 0\) and any \(x \in \mathbb Z ^d\), one has
We also recall the following upper bound on the heat kernel, taken from [16, Proposition 3.4] (see also [9, Section 3] for earlier results in this context).
Theorem 6.2
([16])
-
(1)
There exist constants \(C, \overline{c}\) such that for any \(t \geqslant 0\) and any \(x \in \mathbb Z ^d\),
$$\begin{aligned} p_t^\omega (0,x) \leqslant \frac{C}{1 \vee t^{d/2}} \exp \left( -D_{\overline{c}t}(x)\right) \!, \end{aligned}$$where
$$\begin{aligned} D_{t}(x) = |x|{{\mathrm{arsinh}}}\left( \frac{|x|}{t} \right) + t \left( \sqrt{1+\frac{|x|^2}{t^2}} - 1\right) . \end{aligned}$$ -
(2)
In particular, there exists \(c_2 > 0\) such that for any \(x\in \mathbb Z ^d\),
$$\begin{aligned} p_t^\omega (0,x) \leqslant \frac{C}{1 \vee t^{d/2}} \exp \left( -c_2\left( \frac{|x|^2}{t} \wedge |x| \right) \right) . \end{aligned}$$
Proof of Theorem 1.3
We decompose the proof into three steps.
Step 1. Possibly lowering the value of \(c_2 > 0\), we have that for any \(x \in \mathbb R ^d\),
Equation (6.2) and part (2) of Theorem 6.2 thus ensure that (possibly enlarging \(C\)),
Moreover, Theorem 6.1 remains true if we lower the value of the constant \(c_1 > 0\) in such a way that \(c_2 \geqslant c_1/2\sqrt{d}\).
Step 2. We now show that there exist \(c > 0\) (independent of \(\delta \)), \(\varepsilon _\delta > 0\) and \(C_\delta \) such that, for any \(\varepsilon \leqslant \varepsilon _\delta \) and any \(x \in \mathbb R ^d\), one has
Let \(f\) be a positive smooth function on \(\mathbb R ^d\) with support in \([-1,1]^d\) and such that \(\int f = 1\). We define, for any \(r > 0\), the function \(f_r : x \mapsto r^{-d} f(r^{-1} x)\).
Let \(u^{(\varepsilon )}\) be the bounded solution of (DPE\(^\omega \)) with initial condition \(f_r(\varepsilon \ \cdot )\) (we keep the dependence of \(u^{(\varepsilon )}\) in \(r\) implicit in the notation). By linearity, we have
Letting \(u_\varepsilon (t,x) = u^{(\varepsilon )}(\varepsilon ^{-2} t, \lfloor \varepsilon ^{-1} x \rfloor )\), we obtain
Let \(\overline{u}\) be the bounded solution of (CPE) with initial condition \(f_r\). Observing the proof of Theorem 1.1, we get that for any \(\delta > 0\), there exists \(C\) such that
Scaling relations ensures that \(\Vert \partial _{x_j} f_r \Vert _\infty \) is bounded, up to a constant, by \(r^{-(d+1)}\), while \(\hat{f_r}(\xi ) = \hat{f}(r \xi )\). As a consequence,
and the integrals on the right-hand side are finite since \(f\) is smooth (see Lemma 5.8). To sum up, for some constant \(C\) and any \(r \leqslant 1\), we have
The solution \(\overline{u}\) can be represented in terms of the heat kernel as
where we used the fact that \(\int f_r = 1\). For \(z \in \mathbb R ^d\) such that \(\Vert z\Vert _\infty \leqslant r\leqslant 1\) and up to a constant, \(|\overline{p}_1(z,x) - \overline{p}_1(0,x)|\) is bounded by \(r e^{-c_2 |x|^2}\) by (6.3). Since \(f_r\) has support in \([-r,r]^d\), we arrive at
On the other hand, if \(z \in \mathbb Z ^d\) is such that \(\Vert z\Vert _\infty \leqslant \varepsilon ^{-1} r\), then
We now argue that there exists \(c_3 > 0\) (independent of \(\delta \)) such that, uniformly over \(r \leqslant 1\) and \(x \in \mathbb R ^d\), one has
Theorem 6.1 tells us indeed that the left-hand side of (6.10) is smaller than
For any \(r \leqslant 1\) and \(\Vert x\Vert _\infty \geqslant 2\), the infimum above is larger than
so (6.10) holds in this case, with \(c_3 = c_1/2\sqrt{d}\). To control smaller values of \(\Vert x\Vert _\infty \), it suffices to enlarge the constant \(C\) in (6.10). To sum up, we have shown that
In the sum on the right-hand side of (6.6), only \(C (\varepsilon ^{-1} r)^d\) terms are non-zero, and \(\Vert f\Vert _\infty \leqslant r^{-d}\), so
Observe also that
This is a Riemann approximation of \(\int f = 1\), hence
and we are thus led to
Combining (6.8), (6.9), (6.11) and the fact that \(c_2 \geqslant c_3 = c_1/2\sqrt{d}\), we obtain that up to a constant,
is bounded by
uniformly over \(r \leqslant 1\). Since for \(\varepsilon \) small enough, one has \(\varepsilon \leqslant \Psi _{q,\delta }\left( {\varepsilon ^2}\right) \), the above is bounded, up to a constant, by
uniformly over \(r \leqslant 1\). Choosing
where
is here to ensure that \(r \leqslant 1\), we obtain that the expression in (6.12) is smaller than
This proves (6.5) when \(|x| \wedge |\varepsilon ^{-1} x| \leqslant M_\varepsilon \). Otherwise, we use the bound (6.4), together with the fact that \(c_2 \geqslant c_3\), to get
Hence, (6.5) holds also in this case, and we can always choose \(c = c_3(1-1/(d+3))\).
Step 3. We now extend the result to any time \(t > 0\). The heat kernel of the continuous operator satisfies the scaling relation
while we can write
For \(\varepsilon _\delta \) and \(C_\delta \) given by step 2, as soon as \(\varepsilon / \sqrt{t} \leqslant \varepsilon _\delta \), one thus has
which is the claim of the theorem. \(\square \)
7 Homogenization of elliptic equations
In this last section, we state and prove the counterparts of Theorems 1.1 and 1.3 for the homogenization of elliptic equations. For \(f : \mathbb R ^d \rightarrow \mathbb R \) bounded continuous, we consider the unique bounded solution of

Using integration by parts, one can check that
where \(u^{(\varepsilon )}\) is solution of (DPE\(^\omega _\varepsilon \)). For \(x \in \mathbb R ^d\), we let \(v_\varepsilon (x) = v^{(\varepsilon )}(\lfloor \varepsilon ^{-1} x \rfloor )\), so that
The function \(v_\varepsilon \) converges pointwise, as \(\varepsilon \) tends to \(0\), to \(\overline{v}\) the bounded solution of
and one has
where \(\overline{u}\) is the solution of (CPE). Equipped with the representations (7.2)–(7.3), it is straightforward to derive the following result from Theorem 1.1.
Theorem 7.1
Let \(m = \lfloor d/2 \rfloor + 3\) and \(\delta > 0\). There exist constants \(C_\delta \) (which may depend on the dimension) and \(q\) such that, if the weak derivatives of order \(m\) of \(f\) are in \(L^2(\mathbb R ^d)\), then for any \(\varepsilon > 0\) and \(x \in \mathbb R ^d\), one has
Remark 7.2
Note that on the other hand, it does not look so simple to deduce Theorem 1.1 from Theorem 7.1. A possibility for doing so may be to try to devise a quantitative version of [29, Theorem IX.2.16].
One can also consider the Green function \(G_\varepsilon ^\omega (x,y)\), the unique bounded function such that
Letting \(\overline{G}(x,y)\) be the Green function associated to equation (CEE), we can write the counterpart of Theorem 1.3.
Theorem 7.3
Let \(d \geqslant 2\) and \(\delta > 0\). There exist constants \(c > 0\) (independent of \(\delta \)), \(q, C_\delta \) such that for any \(\varepsilon > 0\) and any \(x \in \varepsilon \mathbb Z ^d {\setminus } \{0\}\), one has
When \(d = 1\), there exist \(C, c> 0\) such that for any \(\varepsilon > 0\) and any \(x \in \varepsilon \mathbb Z \), one has
Remark 7.4
The orders of magnitude, as \(\varepsilon \) tends to \(0\), of the right-hand side of (7.1) and (7.4), are given respectively by (1.5) and (1.6).
Proof
Our starting point is the fact that
while
Recall first that Theorem 1.3 ensures that there exist \(c >0, C_\delta ,\varepsilon _\delta > 0\) such that whenever \(t\geqslant (\varepsilon /\varepsilon _\delta )^2\), one has
The difference of interest
is bounded by
Let \(\eta = (\varepsilon /\varepsilon _\delta )^2 \vee (\varepsilon |x|)\). If \(t \geqslant \eta \), then the integrand above is bounded, up to a constant, by
In order to control the integral in (7.6), it thus suffices to bound the following three quantities:
We start with the integral in (7.7), which is the only non-negligible one. To begin with, note that for any \(\gamma \), a change of variables gives us the identity
and moreover, provided \(\gamma > 1\),
for some large enough \(C\) (and \(c \leqslant 2\)). We have thus shown that, for \(\gamma > 1\),
When \(d \geqslant 3\), we have \(\Psi _{q,\delta }(u) = u^{1/2-\delta }\), so that the integral in (7.7) is bounded, up to a constant, by
When \(d = 2\), the argument requires some minor modifications, due to presence of a logarithmic factor in \(\Psi _{q,\delta }\). One should consider instead integrals of the form
for some \(q' \geqslant 0\) and \(\gamma > 1\) (in fact, \(\gamma = 1+1/20\)). This last integral is bounded by
For the first integral, (7.11) gives us an upper bound. Inequality (7.11) also enables us to bound the second integral, using the fact that
These observations thus guarantee that (7.7) is also bounded by (7.13) when \(d = 2\).
We now turn to the evaluation of the integral in (7.8). Since, for \(z \geqslant 0\), one has \({{\mathrm{arsinh}}}(z) = \log (z+\sqrt{1+z^2}) \geqslant \log (1+z)\), and using part (1) of Theorem 6.2, one can bound the integral in (7.8) (up to a constant) by
A change of variables shows that this is equal to
where
Since we consider only \(x \in \varepsilon \mathbb Z ^d {\setminus } \{0\}\), the parameter \(\eta '\) is uniformly bounded, independently of the value of \(x\) and \(\varepsilon \). The integral in (7.14) is thus bounded (up to a constant) by
This finishes the analysis of the integral in (7.8), and there remains only to consider the integral in (7.9). This integral is bounded by a constant times
for some small enough \(c> 0\). A change of variables enables one to rewrite this integral as
Moreover,
for some large enough \(C'\), uniformly over \(\varepsilon > 0\) and \(x \in \varepsilon \mathbb Z ^d {\setminus } \{0\}\). The right-hand side of (7.15) is thus bounded by
We thus obtained the required bound on (7.9), and this finishes the proof of Theorem 7.3 for \(d \geqslant 2\).
For the one-dimensional case, the analysis must be slightly adapted. We need to bound the integrals appearing in (7.7), (7.8) and (7.9). The analysis of the integrals in (7.8) and (7.9) can be kept without change, except that only the case \(x \in \varepsilon \mathbb Z {\setminus } \{0\}\) was considered above, while here we want to consider also \(x = 0\). But this is a very easy case, since the upper bound \(t^{-1/2}\) on the heat kernels is integrable close to \(0\). As for the integral in (7.7), it is equal to
where \(\gamma = 1/2+1/16 < 1\). The integral above is uniformly bounded over \(x\) such that \(|x| \leqslant 1\). Otherwise, as noted in (7.10), we have
and we can bound the last integral by
where in the second part, we used the fact that for \(|x| \geqslant 1\) and \(s \geqslant |x|^{-1}\), we have \(s|x|^2 \geqslant |x|/2 + s/2\). We have thus shown that the integral in (7.7) is bounded, up to a constant, by
uniformly over \(x \in \mathbb R \), and this finishes the proof for \(d = 1\). \(\square \)
Notes
A. Gloria has announced improved estimates on this problem.
A. Gloria, S. Neukamm and F. Otto have announced results in this direction.
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Mourrat, JC. Kantorovich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients. Probab. Theory Relat. Fields 160, 279–314 (2014). https://doi.org/10.1007/s00440-013-0529-5
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DOI: https://doi.org/10.1007/s00440-013-0529-5
Keywords
- Quantitative homogenization
- Martingale
- Central limit theorem
- Random walk in random environment
Mathematics Subject Classification (2010)
- 35B27
- 35K05
- 60G44
- 60F05
- 60K37