# A Liouville theorem for stationary and ergodic ensembles of parabolic systems

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## Abstract

A first-order Liouville theorem is obtained for random ensembles of uniformly parabolic systems under the mere qualitative assumptions of stationarity and ergodicity. Furthermore, the paper establishes, almost surely, an intrinsic large-scale \(\mathcal {C}^{1,\alpha }\)-regularity estimate for caloric functions.

## Keywords

Stochastic homogenization Liouville theorem Large-scale regularity Parabolic equation Parabolic system## Mathematics Subject Classification

35B27 35B53 35K10 35K40 60H25 60K37## 1 Introduction and main results

*a*is assumed to be

*stationary*with respect to space-time translations and

*ergodic*. Precisely, for a probability space of coefficient fields \((\varOmega ,{\mathcal {F}},\left\langle \cdot \right\rangle )\), where \(\left\langle \cdot \right\rangle \) is used simultaneously to denote the law and expectation of the ensemble, the stationarity asserts that the coefficients are statistically homogeneous in time and space in the sense that

*F*:

However, conditions (2) and (3) are merely qualitative and contain no quantitative information about the mixing properties of the ensemble. Therefore, while the results of this paper apply to a very general class of environments, the corresponding homogenization may occur at an arbitrarily slow rate. In order to obtain more quantitative statements, such as in the recent work Armstrong et al. [3], it would be necessary to quantify the ergodicity in the way, for example, of a spectral gap inequality or a finite-range of dependence.

*a*, the limiting behavior, as \(\epsilon \rightarrow 0\), of solutions to the rescaled equation

*R*centered at the origin and let \({\mathcal {C}}_R\) denote the parabolic cylinder

*a*, for each \(i\in \{1,\ldots ,d\}\),where here, and throughout the paper, the integration variables will be omitted unless there is a possibility of confusion. This sublinearity is essentially equivalent to homogenization, see (13) below, and is crucial for the arguments of this paper.

*a*and as \(\epsilon \rightarrow 0\), of the solutions to (6). Indeed, by obtaining an energy estimate for the error in the asymptotic expansion

*a*, for every \(u_0\in L^2({\mathbb {R}}^d)\) and \(T>0\), as \(\epsilon \rightarrow 0\),

Looking ahead, observe that the behavior of the solution \(u^\epsilon \) to (6) on a unit scale, for \(\epsilon >0\) small, corresponds to a characterization of the large-scale behavior of the solution *u* satisfying (1). Namely, the behavior of the solution \(u^\epsilon \) on a unit scale corresponds to the behavior of *u* on scale \(\epsilon ^{-1}\) in space and \(\epsilon ^{-2}\) in time. The purpose of this paper will be to characterize the extent to which solutions of (1) inherit, on large-scales and for \(\left\langle \cdot \right\rangle \)-a.e. *a*, the regularity of solutions to constant-coefficient parabolic equations.

A concise statement of this large-scale regularity is contained in the following first-order Liouville theorem, which is the main theorem of the paper.

## Theorem 1

*a*satisfies the following first-order Liouville property: if

*u*is an ancient whole-space

*a*-caloric function, that is if

*u*is a distributional solution of

*a*-caloric function from the span of the

*a*-caloric gradients \(\{\xi +\nabla \phi _\xi \}_{\xi \in {\mathbb {R}}^d}\). The excess of an

*a*-caloric function measures this deviation, and is defined, for each \(R>0\) and

*a*-caloric function

*u*on \({\mathcal {C}}_R\), byIn Proposition 2 below, for \(\left\langle \cdot \right\rangle \)-a.e.

*a*, the excess of an

*a*-caloric function will be shown to decay like a power law in the radius. However, before the statement, it is useful to observe some essential differences between the parabolic and elliptic settings. In what follows, the superscript “\(\text {ell}\)” will be used to differentiate elliptic objects from their parabolic counterparts.

*a*. These notions motivated definition (14), and measured the deviation of the gradient of an \(a^\text {ell}\)-harmonic function

*u*on \(B_R\), by which is meant a solution

*u*on \(B_R\),The decay of the excess was controlled in [18, Lemma 2] through the introduction of a flux correction \(\sigma ^\text {ell}=\{\sigma ^\text {ell}_i\}_{i\in \{1,\ldots ,d\}}\). Namely, the flux \(q^\text {ell}=\{q^\text {ell}_i\}_{i\in \{1,\ldots ,d\}}\) is defined, for each \(i\in \{1,\ldots ,d\}\), by

*u*in \(B_R\),

*a*, the large-scale \(L^2\)-averages are sublinear. But, by taking this fact as an input, it follows from a Campanato iteration that the excess of an arbitrary \(a^\text {ell}\)-harmonic function decays according to (18). In this paper, the analogous result will also be obtained for the parabolic excess, as shown in Proposition 2 below.

In comparison with the elliptic setting, where the decay of the excess was determined by the sublinearity of the large-scale \(L^2\)-averages of \((\phi ^\text {ell},\sigma ^\text {ell})\), the decay of the parabolic excess will be determined by the sublinearity of the large-scale \(L^2\)-averages of the corrector \((\phi ,\psi ,\sigma )\), measured with respect to the scaling in space, and the sublinearity of the large-scale \(L^2\)-averages of \(\zeta \), measured with respect to the scaling in time. The first lemma of the paper establishes the existence of the extended corrector \((\phi ,\psi ,\sigma ,\zeta )\).

## Lemma 1

Suppose that the ensemble \(\left\langle \cdot \right\rangle \) satisfies (2), (3), (4), and (5). There exist \(C=C(d,\lambda )>0\) and random fields \(\phi =\{ \phi _i \}_{i\in \{1,\ldots ,d\}}\), \(\psi = \{\psi _i\}_{i\in \{1,\ldots ,d\}}\), \(\sigma =\{\sigma _{ijk}\}_{i,j,k\in \{1,\ldots ,d\}}\) and \(\zeta =\{\zeta _{ij}\}_{i,j\in \{1,\ldots ,d\}}\) on \({\mathbb {R}}^{d+1}\) with the following properties:

*a*, the following equations are satisfied in the sense of distributions on \({\mathbb {R}}^{d+1}\). The field \(\phi \) satisfies (7): for each \(i\in \{1,\ldots ,d\}\),

The following two propositions effectively split the probabilistic and deterministic aspects of the paper. Proposition 1 contains the probabilistic parts, and uses the stationarity and ergodicity of the ensemble to prove that the large-scale \(L^2\)-averages of \((\phi ,\psi ,\sigma )\) are sublinear with respect to the spatial scaling and that those of \(\zeta \) are sublinear with respect to the time scaling. This fact is essentially classical for the case of the correctors \(\phi \) and \(\zeta \), although a new argument for the sublinearity of \(\phi \) is presented which may be of independent interest. A new argument is required to prove the sublinearity of \(\sigma \) and \(\psi \).

The difference is the following. The corrector \(\phi \) is, in general, not stationary in either space or time but Eq. (7) yields some control over both its spatial and temporal derivatives. Similarly, the corrector \(\zeta \) has an explicit, stationary time derivative but is itself not stationary. In the second case, since Eqs. (19) and (20) yield only the spatial regularity for \(\psi \) and \(\sigma \), it is necessary to use the fact that both fields are stationary in time in order to obtain the convergence.

In fact, the following proposition will prove the sublinearity of the normalized corrector where, in the case of \(\phi \), the components are normalized by their large-scale averages on a parabolic cylinder, in the case of \(\zeta \), using the fact that the Sobolev embedding implies that \(\zeta \) is continuous, the components are normalized by their values at time zero and, in the case of \(\psi \) and \(\sigma \), the functions are normalized, for each fixed time, by their large-scale averages on a ball. This is in fact equivalent to the sublinearity of the corrector \((\phi ,\psi ,\sigma ,\zeta )\) without a normalization, see for instance [12, Lemma 2], but since this observation is not necessary for the arguments of the paper it is omitted.

## Proposition 1

*a*, the corrector \((\phi ,\psi ,\sigma )\) is strictly sublinear with respect to the spatial scaling and the corrector \(\zeta \) is strictly sublinear with respect to the time scaling in the sense that, for each \(i,j,k\in \{1,\ldots ,d\}\),andFurthermore, for \(\left\langle \cdot \right\rangle \)-a.e.

*a*, for each \(i\in \{1,\ldots ,d\}\), the large-scale \(L^2\)-averages of the components of the flux satisfy

It is important to observe at this point that Eqs. (19) and (20) defining \(\psi \) and \(\sigma \) are invariant if either \(\psi \) or \(\sigma \) is altered by a time stationary constant. This explains why in (23), for each \(R>0\), it is possible and, for our arguments required, to allow for a time-dependent normalization. The Eqs. (7) and (22) defining \(\phi \) and \(\zeta \) are not likewise invariant, and therefore the corresponding normalizations appearing in (23) and (24) are necessarily achieved by subtracting a true constant.

*a*finite radius \(r_*(a)\) such that, whenever \(r_*<r<R<\infty \), for every

*a*-caloric function

*u*in \({\mathcal {C}}_R\), the parabolic excess satisfies, for \(C_1=C_1(\alpha ,d,\lambda )>0\),

## Proposition 2

Suppose that the ensemble \(\left\langle \cdot \right\rangle \) satisfies (2), (3), (4), and (5). Fix a Hölder exponent \(\alpha \in (0,1)\). Then, there exist constants \(C_0=C_0(\alpha , d,\lambda )>0\) and \(C_1(\alpha ,d,\lambda )>0\) with the following property:

*u*to the parabolic equation

*u*in \(B_R\), define its \(a_{\mathrm {hom}}^\text {ell}\)-harmonic extension

*v*into \(B_R\) to be the solution

In particular, by analyzing the right hand side of (26), the energy of \(w^\text {ell}\) can be controlled by the growth of the extended corrector \((\phi ^\text {ell},\sigma ^\text {ell})\), the choice of the cutoff function \(\eta \) and the interior and boundary regularity of the \(a_{\mathrm {hom}}^\text {ell}\)-harmonic function *v*. The argument is completed by observing that, owing to the regularity of \(a_{\mathrm {hom}}^\text {ell}\)-harmonic functions, the energy of the homogenization error is a good approximation for the excess.

The methods of this paper apply the same philosophy to the parabolic setting. However, similarly to what was done in the proof of [12, Theorem 2], it is furthermore necessary to introduce a spatial regularization of the *a*-caloric function *u*. The purpose of this is to quantify the regularity in time, since such functions are already sufficiently regular in space. Precisely, if *u* is an *a*-caloric function then, in general, its time derivative \(u_t\in H^{-1}\) and no better, where \(H^{-1}\) denotes the dual space of the Sobolev space \(H^1\). However, for every \(\epsilon >0\), if \(u^\epsilon \) denotes the spatial convolution of *u* on scale \(\epsilon >0\), then it is possible to show that \(u^\epsilon _t\in L^2\) with a precise quantification of the \(L^2\)-norm of \(u^\epsilon _t\) in terms of the energy of \(\nabla u\), see Sect. 5.1 below. This additional approximation is necessary in order to quantify the boundary estimate of Sect. 5.3, which is necessary for the Campanato iteration and the quantitative homogenization result contained in Proposition 3 below.

*a*-caloric function

*u*in \({\mathcal {C}}_{R+\epsilon }\), define the \(a_{\mathrm {hom}}\)-caloric extension \(v^\epsilon \) of \(u^\epsilon \) into \({\mathcal {C}}_R\) to be the solution

*w*will be defined as

*w*, for an appropriately chosen cutoff \(\eta \), will be the starting point of the Campanato iteration used to control the decay of the excess. In this case, there is a contribution from the boundary, which will be controlled first by fixing \(\epsilon >0\) small. From the right hand side of (28), the energy of the homogenization error will then be controlled by the growth of the extended parabolic corrector \((\phi ,\psi ,\sigma )\) and, after integrating in parts by time, the growth of \(\zeta \) and

*q*. It is furthermore necessary to make a good choice for the cutoff function \(\eta \) and to use the interior and boundary regularity of the \(a_{\mathrm {hom}}\)-caloric function \(v^\epsilon \). The argument is completed by observing that, owing to the interior regularity of \(a_{\mathrm {hom}}\)-caloric functions, the homogenization error

*w*provides a good approximation for the excess.

The following proposition makes the previous intuition rigorous, and its proof is contained in Sect. 5.4 of the proof of Proposition 2. In the following, the constant \(\epsilon >0\) quantifies the regularization on the boundary, and the constant \(\rho >0\) quantifies a boundary layer introduced by a cutoff function. This is done in the proof to exploit the interior regularity of \(a_{\mathrm {hom}}\)-caloric functions, and in order to impose useful boundary conditions for the homogenization error.

## Proposition 3

*a*-caloric function

*u*on \({\mathcal {C}}_R\) and \(\epsilon \in (0,\frac{R}{4})\), there exists \(R_\epsilon =R_\epsilon (u,\epsilon )\in \left( \frac{R}{2},\frac{3R}{4}\right) \) such that, for the \(a_{\mathrm {hom}}\)-caloric extension \(v^\epsilon \) of

*u*on \({\mathcal {C}}_{R_\epsilon }\) in the sense of (27), for the homogenization error

*a*, for each \(\rho \in (0,\frac{1}{8})\),for the extended corrector \((\phi ,\psi ,\sigma ,\zeta )\) from Lemma 1.

Finally, the following parabolic Caccioppoli inequality will be used in the proofs of Theorem 1 and Proposition 2. The proof is classical, and is included for the convenience of the reader.

## Lemma 2

*a*, for every \(R>0\) and distributional solution

*u*of the equation

In comparison with the elliptic setting, the qualitative homogenization theory of divergence-form operators with coefficients depending on time and space is relatively under studied. While the case of periodic coefficients has long been understood, and a full explanation can be found in the classic reference Bensoussan et al. [13, Chapter 3], the qualitative stochastic homogenization of stationary and ergodic ensembles like (1) was obtained only more recently by Rhodes [25, 26]. However, related problems were earlier handled, such as the case of a Brownian motion in the presence of a divergence-free drift, by Komorowski and Olla [19], Landim et al. [21] and Oelschläger [22]. In the discrete setting, related questions have been considered, for instance, by Andres [1], Bandyopadhyay and Zeitouni [10] and Rassoul-Agha and Seppäläinen [24] in the uniformly elliptic setting and, for degenerate environments, by Andres et al. [2].

The quantitative homogenization of such ensembles has only recently been considered, and the preprint [3] contains, to our knowledge, the first results in this direction. In particular, in [3, Theorem 1.2], a full hierarchy of Liouville theorems is obtained for ensembles satisfying a finite-range dependence in space and time. Their method is motivated by the work of Armstrong and Smart [6] from the elliptic setting, which adapted the approach of Avellaneda and Lin [7] from the context of periodic homogenization.

In [7], a full hierarchy of Liouville properties was established for uniformly elliptic and periodic coefficient fields based upon the previous works Avellaneda and Lin [8, 9], which developed a large-scale regularity theory in Hölder and \(L^p\)-spaces. In [6], the approach of [7] was adapted to stationary and ergodic ensembles satisfying a finite-range dependence. Their proof, which obtained a large-scale \(C^{0,1}\)-regularity theory, was based upon a variational approach and the quantification of the convergence of certain sub-additive and super-additive energies. Their work was later extended by Armstrong and Mourrat [5] to more general mixing conditions, and subsequently gave rise to a significant literature on the subject. The interested reader is pointed to the recent monograph Armstrong et al. [4], and the references therein.

The approach of this paper follows closely the work [18], which derived, for uniformly elliptic ensembles, a large-scale \(C^{1,\alpha }\)-regularity estimate and first-order Liouville property under the qualitative assumptions of stationarity and ergodicity. The method was based upon the introduction of an intrinsic notion of excess, as defined in (16), as well as the construction of the flux correction \(\sigma ^\text {ell}\) defined in (17). The introduction of \(\sigma ^\text {ell}\) was used to prove that the homogenization error solves the divergence-form Eq. (26), which provided the starting point for a Campanato iteration as explained above.

Subsequently, Fischer and Otto [16] obtained a full hierarchy of Liouville properties under a mild quantification of the ergodicity. In Fischer and Otto [17], the necessary quantification of ergodicity from [16] was shown to be satisfied by a general class of Gaussian environments. However, absent some mild quantification of ergodicity in the sense of either [5] or [16], the existence of higher order Liouville and large-scale regularity statements remains an open question.

Finally, motivated by the work of Chiarini and Deuschel [15], the Bella et al. [12] derived a large-scale \(C^{1,\alpha }\)-regularity theory and first-order Liouville theorem for degenerate elliptic equations, where the boundedness and uniform ellipticity (5) was replaced by certain moment conditions. It is expected that the results of this paper can be similarly extended to degenerate environments, and the setting of [2] will serve as the starting point for future work.

In principle, one could also hope to combine the methods of this paper with those of [18], in the presence of a logarithmic Sobolev inequality like that used in [18, Theorem 1], to obtain more quantitative information. For example, the minimal radius \(r_*(a)>0\) quantifying the first scale for which the assumptions of Proposition 2 are satisfied, and which effectively defines the initial scale on which the \(C^{1,\alpha }\)-regularity of Proposition 2 begins to take effect, is expected to have stretched exponential moments in the sense of [18, Theorem 1]. Furthermore, again assuming a logarithmic Sobolev inequality, it should be possible to obtain a quantitative two-scale expansion for *a*-caloric functions like [18, Corollary 3]. Lastly, following the methods of [16], it may be possible to prove higher order Liouville statements under a mild quantification of the ergodicity.

The paper is organized as follows. The proofs are presented in the order of their appearance: Theorem 1, Lemma 1, Propositions 1, 2 and Lemma 2. In order to simplify the notation, the statements and proofs are written for the non-symmetric scalar setting. However, at the cost of increasing some constants, all of the arguments carry through unchanged for non-symmetric systems. Throughout, the notation \(\lesssim \) is used to denote a constant whose dependencies are specified in every case by the statement of the respective theorem, proposition or lemma.

## 2 The proof of Theorem 1

*a*satisfying the conclusions of Lemma 1, Propositions 1 and 2, and suppose that

*u*is a distributional solution of

*u*is strictly subquadratic,This implies that, for every \(\rho >0\),

*z*is a distributional solution of

*z*is necessarily constant in time as well. Therefore, there exists \(c\in {\mathbb {R}}\) such that \(u=c+\xi \cdot x+\phi _\xi \), which completes the argument.

## 3 The proof of Lemma 1

The construction of the corrector \((\phi ,\psi ,\sigma ,\zeta )\) will be achieved by lifting the relevant Eqs. (7), (19), (20) and (22) to the probability space \(\varOmega \), and thereby identifying \(\phi \) by its stationary, finite energy gradient and time derivative, \(\psi \) and \(\sigma \) by their stationary, finite energy gradients, and \(\zeta \) by its stationary time derivative. For this, it is necessary to define the horizontal derivative of a random variable as induced by shifts of the coefficient field in space and time. Then, these will be used to define an analogue of the Sobolev space \(H^1\) on the probability space.

*f*, define, for each \(i\in \{1,\ldots ,d\}\), the horizontal derivative

The following general fact about potential vector fields will be used in the construction of \(\sigma \) and to prove the sublinearity for the corrector \((\phi , \psi , \sigma , \zeta )\). It will be shown that, with respect to the sub-sigma-algebra of subsets that are invariant with respect to spatial translations of the coefficient fields, the conditional expectation of a potential vector field vanishes as a random variable.

## Lemma 3

*g*is invariant with respect to spatial shifts of the coefficient field as an \({\mathcal {F}}_{{\mathbb {R}}^d}\)-measurable function. In combination, (36) and (37) imply (34). Since the \({\mathcal {F}}_{{\mathbb {R}}^d}\)-measurable \(g\in L^2(\varOmega )\) and \(F\in L^2_{\text {pot}}(\varOmega ;{\mathbb {R}}^d)\) were arbitrary, this completes the proof of (33).

### 3.1 The construction of \(\phi \)

*a*, by integration. Observe that in order to define \(\phi \) via integration it is necessary to choose a base point, and it is this choice that ruins the stationarity.

We remark that the following proof is essentially the Lax–Milgram argument, where the only small subtlety is that the operator defining Eq. (38) is not coercive for \({\mathcal {H}}^1\), since it is not coercive with respect to the horizontal derivative in time.

### 3.2 The construction of \(\psi \)

## Lemma 4

The existence of \(\psi \) follows from Lemma 4 in the following way. For each \(k\in \{1,\ldots ,d\}\), choose \(F=Q_k\) and define \(D\psi _k:=\varPsi \), which defines \(\psi _k\), for \(\left\langle \cdot \right\rangle \)-a.e. *a*, as a function on \({\mathbb {R}}^d\) via integration. In this case, it is the choice of spatial base point that destroys the spatial stationarity of \(\psi \). However, for \(\left\langle \cdot \right\rangle \)-a.e. *a*, for each \(k\in \{1,\ldots ,d\}\), the function \(\psi _k\) can then be extended to \({\mathbb {R}}^{d+1}\) as a stationary function in time.

### 3.3 The construction of \(\sigma \)

*a*, on \({\mathbb {R}}^d\) via integration in space. As in the case of \(\psi \), the choice of base point ruins the spatial stationarity of \(\sigma \). However, for each \(i,j,k\in \{1,\ldots ,d\}\), for \(\left\langle \cdot \right\rangle \)-a.e.

*a*, the corrector \(\sigma _{ijk}\) can then be extended to \({\mathbb {R}}^{d+1}\) as a stationary function in time.

### 3.4 The construction of \(\zeta \)

*a*, by fixing a base point and integrating in time, which destroys the stationarity, and then extended to \({\mathbb {R}}^{d+1}\) as a spatially constant function.

### 3.5 The boundedness and uniform ellipticity of \(a_{\mathrm {hom}}\)

*X*,

*v*), and therefore the map (58) is convex. Hence, for each \(\xi \in {\mathbb {R}}^d\), using the corrector Eq. (7) and Jensen’s inequality,

*a*, and where the final inequality is obtained using the boundedness of the ensemble (5) and the vanishing expectation of the gradient from Lemma 3. This completes the proof of (57).

## 4 The proof of Proposition 1

### 4.1 The sublinearity of \(\psi \) and \(\sigma \)

## Lemma 5

*a*, the normalized large-scale \(L^2\)-averages of \(\varphi \) are strictly sublinear in the sense that

*a*, for each \(\epsilon \in (0,1)\),The proof now follows from a straightforward application of Egorov’s theorem. Let \(\epsilon \in (0,1)\) be arbitrary but fixed. For each \(\eta \in (0,1)\), use Egorov’s theorem and (66) to find a measurable subset \(A_\eta \subset \varOmega \) and \(R_\eta >0\) such that, for every \(a\in A_\eta \) and \(R>R_\eta \),where \(\chi _{A_\eta }\in L^\infty (\varOmega )\) denotes the indicator function of \(A_\eta \). Returning to (64), form the decompositionFor the second term of (68), it follows from the stationarity of \(\nabla \varphi \), the stationarity of \(\chi _{A_\eta }\), and the ergodic theorem [11, Theorems 2, 3] thatIn combination (64), (67), (68), and (69) imply that, since \(\eta \in (0,1)\) was arbitrary,where the final equality follows from (65), (67), and the dominated convergence theorem. Hence, returning to (61), it is immediate from the stationarity of \(\nabla \varphi \), the ergodic theorem [11, Theorems 2, 3], (63), and (70) thatThis, since \(\epsilon \in (0,1)\) is arbitrary, completes the proof.

### 4.2 The sublinearity of \(\phi \)

The sublinearity of \(\phi \) will follow from the following general fact.

## Lemma 6

*a*, the field \(\varphi \) satisfies

*a*, the normalized large-scale \(L^2\)-averages of \(\varphi \) are strictly sublinear in the sense that

*a*,It remains to prove that, for \(\left\langle \cdot \right\rangle \)-a.e.

*a*,Let \(\rho \in \mathcal {C}^\infty _c({\mathbb {R}}^d)\) be a smooth, symmetric convolution kernel supported in \(B_1\) and, for each \(\epsilon >0\), define the rescaling \(\rho ^\epsilon (\cdot )=\epsilon ^{-d}\rho (\frac{\cdot }{\epsilon })\). Then, for each \(\epsilon >0\), define the spatial convolution, for each \(x\in {\mathbb {R}}^d\) and \(t\in {\mathbb {R}}\),

*a*,it follows from (81) that, for \(\left\langle \cdot \right\rangle \)-a.e.

*a*, for every \(\delta \in (0,1)\),Therefore, since \(\delta \in (0,1)\) is arbitrary,In combination, (74), (75) and (82) combine to prove (73), and thereby complete the proof of Lemma 6. The sublinearity of the corrector \(\phi \), for \(\left\langle \cdot \right\rangle \)-a.e.

*a*, is then immediate from Lemma 1.

### 4.3 The sublinearity of \(\zeta \)

*a*, for each \(i,j\in \{1,\ldots ,d\}\),is essentially classical. For each \(i\in \{1,\ldots ,d\}\), the Poincaré inequality and the ergodic theorem [11, Theorems 2, 3] together with the Rellich-Kondrachov embedding theorem imply that the familyis compact in \(L^2([0,1])\) and converges weakly to zero, as \(\epsilon \rightarrow 0\), in \(H^1([0,1])\). Therefore, for \(\left\langle \cdot \right\rangle \)-a.e.

*a*, for each \(i,j\in \{1,\ldots ,d\}\), as \(\epsilon \rightarrow 0\),Furthermore, now exploiting the fact that \(\zeta \) is a one-dimensional function, it follows from the Sobolev embedding theorem and the Arzelà-Ascoli theorem that, for \(\left\langle \cdot \right\rangle \)-a.e.

*a*, for each \(i,j\in \{1,\ldots ,d\}\), the familyis pre-compact in \(C^{0,\frac{1}{2}}([0,1])\) and, by repeating the argument of (61), converges weakly to zero, as \(\epsilon \rightarrow 0\), in \(H^1([0,1])\). Therefore, for \(\left\langle \cdot \right\rangle \)-a.e.

*a*, for each \(i,j\in \{1,\ldots ,d\}\), as \(\epsilon \rightarrow 0\),In particular, for each \(i,j\in \{1,\ldots ,d\}\), as \(\epsilon \rightarrow 0\),Hence, in combination, (83) and (84) prove after rescaling that, for \(\left\langle \cdot \right\rangle \)-a.e.

*a*, for each \(i,j\in \{1,\ldots ,d\}\),which completes the argument since \(\zeta \) is constant in space.

### 4.4 The large-scale averages of *q*

## 5 The proof of Proposition 2

The proof of Proposition 2 is split into five steps. The first defines the augmented homogenization error. The second proves that the augmented homogenization error satisfies a parabolic equation. The third recalls some classical estimates governing the interior and boundary regularity of \(a_{\mathrm {hom}}\)-caloric functions. The fourth uses the equation satisfied by the augmented homogenization error to derive an energy estimate. And, finally, the fifth uses the energy estimate to complete the proof of excess decay.

### 5.1 The augmented homogenization error

*u*is an

*a*-caloric function in \({\mathcal {C}}_1\). That is, in the sense of distributions, suppose that

*u*satisfies

*u*and its convolution.

*a*from (5) that

*a*imply that

*u*. Precisely, for each \(\epsilon \in (0,\frac{1}{4})\), it follows from Jensen’s inequality and the definition of the convolution kernel that

*w*according to the rule

### 5.2 The equation satisfied by the augmented homogenization error

*w*be defined by (94). Since the boundary condition is immediate from the definition, it remains only to compute the equation. First, using definition (94), the gradient is defined by

*u*satisfies (86),

*v*satisfies (91),

### 5.3 Interior and boundary estimates for \(a_{\mathrm {hom}}\)-caloric functions

In this subsection, three classical estimates are presented to control the interior and boundary regularity of \(a_{\mathrm {hom}}\)-caloric functions.

### 5.4 The energy estimate for the augmented homogenization error

*w*defined in (94). Precisely, it will be shown that

*w*. However, for this it is necessary to introduce a cutoff to ensure that

*w*vanishes along the upper boundary of the cylinder. For each \(\delta \in (0,1)\), define a smooth cutoff function \(\gamma _\delta :{\mathbb {R}}\rightarrow {\mathbb {R}}\) which is non-increasing and satisfies \(0\le \gamma _\delta \le 1\) with

*a*from (5) and \(a_{\mathrm {hom}}\) from Lemma 1 and Hölder’s inequality imply that, after bounding the time derivative of \(v^\epsilon \) by its Hessian matrix,

*a*, Hölder’s inequality and the estimate for the Dirichlet to Neumann map, see [14], that

*w*from (94) and the fact that \(\zeta \) vanishes at \(t=0\) owing to (85), it follows after integrating by parts variously in time and space that

*u*and

*v*respectively,

*a*and the uniform ellipticity of \(a_{\mathrm {hom}}\) from Lemma 1, after bounding the time derivative of

*v*by the norm of its Hessian matrix,

*v*by its Hessian matrix,

*a*from (5), the definition of \(\gamma _\delta \), the Poincaré inequality in space, Hölder’s inequality, and Young’s inequality that

*a*,

*u*is an

*a*-caloric function on \(B_R\). Then, for each \(\epsilon \in (0,\frac{R}{4})\), there exists a radius \(R_\epsilon \in (\frac{R}{2},\frac{3R}{4})\) and a cutoff function \(\eta ^{R_\epsilon }_\rho \) with \(0\le \eta ^{R_\epsilon }_\rho \le 1\) and such that

### 5.5 The proof of excess decay

*u*is an

*a*-caloric function \({\mathcal {C}}_R\). Then, for each \(\epsilon \in (0,\frac{R}{4})\) and \(\rho \in (0,\frac{1}{8})\), choose a radius \(R_\epsilon \in (\frac{R}{2},\frac{3R}{4})\) and a cutoff \(\eta ^{R_\epsilon }_\rho \) such that, for the \(a_{\mathrm {hom}}\)-caloric extension \(v^\epsilon \) of \(u^\epsilon \) into \({\mathcal {C}}_{R_\epsilon }\), the conclusion of (129) is satisfied for the augmented homogenization error

*w*defined by

**Step 1:**In the first step of the proof, it will be shown that, for any \(\delta >0\), there exists \(C_2=C_2(d,\lambda ,\delta )>0\) such that, whenever, for each \(i\in \{1,\ldots ,d\}\),

*a*imply thatTherefore, first choose \(\epsilon _0\in (0,\frac{1}{4})\) satisfying

**Step 2:**The second step will show that the left hand side of (135) is a good approximation for the excess by using the interior regularity of \(a_{\mathrm {hom}}\)-caloric functions and the Caccioppoli inequality. To simplify the notation in what follows, define

*v*by the norm of its Hessian matrix, and using the uniform ellipticity of

*a*and the choice \(R_{\epsilon _0}\in (\frac{R}{2}, \frac{3R}{4})\), for each \(r\in (0,\frac{R}{4}]\),

*a*and \(R_{\epsilon _0}\in (\frac{R}{2},\frac{3R}{4})\), it follows that, for each \(r\in (0,\frac{R}{4}]\),

*a*-caloric coordinate \((x_i+\phi _i)\) satisfies

*a*and the choice \(R_{\epsilon _0}\in (\frac{R}{2}, \frac{3R}{4})\), imply that, for each \(r\in (0,\frac{R}{4}]\),

**Step 3:**In the third step, inequality (141) will be combined with (133) to prove the excess decay along a subsequence. Namely, for every \(\alpha \in (0,1)\), it will be shown that there exists \(C_0=C_0(d,\lambda ,\alpha )>0\) and \(\theta _0=\theta _0(\alpha ,d,\lambda )\in (0,\frac{1}{4})\) such that, if \(r_1=\theta _0R\) and if, for each \(r\in [r_1,R]\),

*a*-caloric gradient \((\xi +\nabla \phi _\xi )\) in the sense that, with (149), for every \(\xi \in {\mathbb {R}}^d\),

**Step 4:**The final step completes the proof using (151) and an iteration argument. Fix \(r_1<R\) such that, for \(C_0>0\) defined following (147), both (142) and (143) are satisfied for the constant \(C_0\) for every \(r\in [r_1,R]\). It will be shown that, in this case,

*n*be the unique positive integer satisfying \(\theta _0^{n-1}R \le r < \theta _0^nR\). Proceeding inductively, and relying upon the fact that (151) obtains an exact inequality, for constants \(C=C(\theta _0)>0\) which can change between inequalities,

## 6 The proof of Lemma 2

*a*satisfying (5). Fix \(R>0\) and suppose that

*u*is a distributional solution of

*a*, it follows that

## Notes

### Acknowledgements

Open access funding provided by Max Planck Society. We would like to thank the referee for their careful reading of the manuscript and their many helpful suggestions.

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