Higher-order pathwise theory of fluctuations in stochastic homogenization

Abstract

We consider linear elliptic equations in divergence form with stationary random coefficients of integrable correlations. We characterize the fluctuations of a macroscopic observable of a solution to relative order $\frac{d}{2}$, where d is the spatial dimension; the fluctuations turn out to be Gaussian. As for previous work on the leading order, this higher-order characterization relies on a pathwise proximity of the macroscopic fluctuations of a general solution to those of the (higher-order) correctors, via a (higher-order) two-scale expansion injected into the “homogenization commutator”, thus confirming the scope of this notion. This higher-order generalization sheds a clearer light on the algebraic structure of the higher-order versions of correctors, flux correctors, two-scale expansions, and homogenization commutators. It reveals that in the same way as this algebra provides a higher-order theory for microscopic spatial oscillations, it also provides a higher-order theory for macroscopic random fluctuations, although both phenomena are not directly related. We focus on the model framework of an underlying Gaussian ensemble, which allows for an efficient use of (second-order) Malliavin calculus for stochastic estimates. On the technical side, we introduce annealed Calderón–Zygmund estimates for the elliptic operator with random coefficients, which conveniently upgrade the known quenched large-scale estimates.

Appendix A. More Malliavin calculus

In this appendix, we include for completeness a short, essentially self-contained proof of Proposition 4.1.

Proof of Proposition 4.1

We split the proof into four steps.

Step 1. Proof of (i).

In terms of the Ornstein–Uhlenbeck semigroup $e^{-t\mathcal {L}}$ (e.g. [44, Section 1.4]), we may write

$$\begin{aligned} {\mathrm{Var}}\left[ X\right] =-\int _0^\infty \partial _t\mathbb {E}\left[ (e^{-t\mathcal {L}}X)^2 \right] \,dt \end{aligned}$$

where Mehler’s formula (e.g. [44, (1.67)]) indeed ensures that $\mathbb {E}\left[ (e^{-t\mathcal {L}}X)^2 \right] \rightarrow \mathbb {E}\left[ X \right] ^2$ as $t\uparrow \infty$. Computing the derivative in t yields

$$\begin{aligned} {\mathrm{Var}}\left[ X\right] \,=\,2\int _0^\infty \mathbb {E}\left[ (e^{-t\mathcal {L}}X)\mathcal {L}(e^{-t\mathcal {L}}X) \right] \,dt\,=\,2\int _0^\infty \mathbb {E}\left[ \Vert De^{-t\mathcal {L}}X\Vert _{\mathfrak {H}}^2 \right] \,dt, \end{aligned}$$

hence, appealing to the commutation relation (4.5) in the form $De^{-t\mathcal {L}}=e^{-t}e^{-t\mathcal {L}}D$,

$$\begin{aligned} {\mathrm{Var}}\left[ X\right] \,=\,2\int _0^\infty e^{-2t}\,\mathbb {E}\left[ \Vert e^{-t\mathcal {L}}DX\Vert _{\mathfrak {H}}^2 \right] \,dt, \end{aligned}$$

and the positivity of $\mathcal {L}$ leads to the conclusion,

$$\begin{aligned} {\mathrm{Var}}\left[ X\right] \,\le \,2\int _0^\infty e^{-2t}\,\mathbb {E}\left[ \Vert DX\Vert _{\mathfrak {H}}^2 \right] \,dt\,=\,\mathbb {E}\left[ \Vert DX\Vert _{\mathfrak {H}}^2 \right] . \end{aligned}$$

Step 2. Proof of (ii).

Let $X,Y\in \mathbb {D}^{1,2}$ with $\mathbb {E}\left[ X \right] =\mathbb {E}\left[ Y \right] =0$. First note that the Poincaré inequality (i) implies that the restriction of $\mathcal {L}$ to ${\text {L}}^2(\Omega )/{\mathbb {C}}:=\{U\in {\text {L}}^2(\Omega ):\mathbb {E}\left[ U \right] =0\}$ is invertible. In particular, there exists $Z\in {\text {L}}^2(\Omega )$ such that $Y=\mathcal {L}Z$. By definition of the adjoint $D^*$ (cf. (4.4)), we then find

$$\begin{aligned} \mathbb {E}\left[ XY \right] =\mathbb {E}\left[ X\mathcal {L}Z \right] =\mathbb {E}\left[ XD^* D Z \right] =\mathbb {E}\left[ \langle DX,DZ\rangle _{{\mathfrak {H}}} \right] . \end{aligned}$$

(A.1)

Appealing to the commutator relation (4.5) in the form $DY=D\mathcal {L}Z=(1+\mathcal {L})DZ$, we conclude

$$\begin{aligned} \mathbb {E}\left[ XY \right] =\mathbb {E}\left[ \langle DX,(1+\mathcal {L})^{-1}DY\rangle _{{\mathfrak {H}}} \right] . \end{aligned}$$

Step 3. Proof of (iii).

Similarly as in Step 1, in terms of the Ornstein–Uhlenbeck semigroup $e^{-t\mathcal {L}}$, we may write

$$\begin{aligned} {\mathrm{Ent}}\!\left[ X^2\right]= & {} -\int _0^\infty \partial _t\,\mathbb {E}\left[ (e^{-t\mathcal {L}}X^2)\log (e^{-t\mathcal {L}}X^2) \right] \,dt\\= & {} \int _0^\infty \mathbb {E}\left[ (\mathcal {L}e^{-t\mathcal {L}}X^2)\log (e^{-t\mathcal {L}}X^2) \right] \,dt \,=\,\int _0^\infty \mathbb {E}\left[ \frac{\Vert De^{-t\mathcal {L}}X^2\Vert _{{\mathfrak {H}}}^2}{e^{-t\mathcal {L}}X^2} \right] \,dt, \end{aligned}$$

hence, by the commutation relation (4.5) in the form $De^{-t\mathcal {L}}=e^{-t}e^{-t\mathcal {L}}D$,

$$\begin{aligned} {\mathrm{Ent}}\!\left[ X^2\right]= & {} \int _0^\infty e^{-2t}\,\mathbb {E}\left[ \frac{\Vert e^{-t\mathcal {L}}DX^2\Vert _{{\mathfrak {H}}}^2}{e^{-t\mathcal {L}}X^2} \right] \,dt. \end{aligned}$$

Appealing to Mehler’s formula for the Ornstein–Uhlenbeck semigroup $e^{-t\mathcal {L}}$ (e.g. [44, (1.67)]), the Cauchy–Schwarz inequality leads to

$$\begin{aligned} \Vert e^{-t\mathcal {L}}DX^2\Vert _{{\mathfrak {H}}}^2\,=\,4\,\Vert e^{-t\mathcal {L}}(XDX)\Vert _{{\mathfrak {H}}}^2\,\le \, 4\,\big (e^{-t\mathcal {L}}X^2\big )\big (e^{-t\mathcal {L}}\Vert DX\Vert _{{\mathfrak {H}}}^2\big ), \end{aligned}$$

so that the above becomes

$$\begin{aligned} {\mathrm{Ent}}\!\left[ X^2\right] \,\le \,4\int _0^\infty e^{-2t}\,\mathbb {E}\left[ e^{-t\mathcal {L}}\Vert DX\Vert _{{\mathfrak {H}}}^2 \right] \,dt \,\le \,2\,\mathbb {E}\left[ \Vert DX\Vert _{\mathfrak {H}}^2 \right] , \end{aligned}$$

which proves the logarithmic Sobolev inequality. Next, higher integrability is deduced by integration as e.g. in [2, Theorem 3.4]: we write

$$\begin{aligned} \mathbb {E}\left[ |X|^{2p} \right] ^\frac{1}{p}-\mathbb {E}\left[ X^2 \right] =\int _1^p\frac{1}{q^2}\mathbb {E}\left[ |X|^{2q} \right] ^{\frac{1}{q}-1}{\mathrm{Ent}}\!\left[ |X|^{2q}\right] dq, \end{aligned}$$

so that the logarithmic Sobolev inequality implies

$$\begin{aligned} \mathbb {E}\left[ |X|^{2p} \right] ^\frac{1}{p}-\mathbb {E}\left[ X^2 \right]\le & {} 2\int _1^p\frac{1}{q^2}\mathbb {E}\left[ |X|^{2q} \right] ^{\frac{1}{q}-1}\mathbb {E}\left[ \Vert D|X|^q\Vert _{\mathfrak {H}}^2 \right] dq\\= & {} 2\int _1^p\mathbb {E}\left[ |X|^{2q} \right] ^{\frac{1}{q}-1}\mathbb {E}\left[ |X|^{2(q-1)}\Vert DX\Vert _{\mathfrak {H}}^2 \right] dq\\\le & {} 2\int _1^p\mathbb {E}\left[ \Vert DX\Vert _{\mathfrak {H}}^{2q} \right] ^\frac{1}{q}dq~\le ~2p\,\mathbb {E}\left[ \Vert DX\Vert _{\mathfrak {H}}^{2p} \right] ^\frac{1}{p}, \end{aligned}$$

and the conclusion follows from the Poincaré inequality (i).

Step 4. Proof of (iv).

Let $X\in {\text {L}}^2(\Omega )$ with $\mathbb {E}\left[ X \right] =0$ and ${\mathrm{Var}}\left[ X\right] =1$. For $h\in {\text {L}}^\infty ({\mathbb {R}})$, we define its Stein transform $S_h$ as the solution of Stein’s equation

$$\begin{aligned} S_h'(x)-xS_h(x)=h(x)-\mathbb {E}\left[ h({\mathcal {N}}) \right] . \end{aligned}$$

As in Step 2, there exists $Z\in {\text {L}}^2(\Omega )$ such that $Y=\mathcal {L}Z$. We then compute

$$\begin{aligned} \mathbb {E}\left[ h(X) \right] -\mathbb {E}\left[ h({\mathcal {N}}) \right] =\mathbb {E}\left[ S_h'(X)-XS_h(X) \right] =\mathbb {E}\left[ S_h'(X)-(D^*DZ)S_h(X) \right] , \end{aligned}$$

and hence, integrating by parts and using (A.1) in the form $\mathbb {E}\left[ \langle DZ,DX\rangle _{\mathfrak {H}} \right] ={\mathrm{Var}}\left[ X\right] =1$,

$$\begin{aligned} \big |\mathbb {E}\left[ h(X) \right] -\mathbb {E}\left[ h({\mathcal {N}}) \right] \!\big |= & {} \big |\mathbb {E}\left[ S_h'(X)\big (1-\langle DZ,DX\rangle _{\mathfrak {H}}\big ) \right] \!\big |\nonumber \\\le & {} \Vert S_h'\Vert _{{\text {L}}^\infty }{\mathrm{Var}}\left[ \langle DZ,DX\rangle _{\mathfrak {H}}\right] ^\frac{1}{2}. \end{aligned}$$

(A.2)

Noting that $\Vert S_h'\Vert _{{\text {L}}^\infty }\le 2\Vert h\Vert _{{\text {L}}^\infty }$ (e.g. [42, Theorem 3.3.1]) and taking the supremum over all $h\in {\text {L}}^\infty ({\mathbb {R}})$, we deduce

$$\begin{aligned} {\text {d}}_{{\text {TV}}}\left( {X};{{\mathcal {N}}}\right) \,\le \,2\,{\mathrm{Var}}\left[ \langle DZ,DX\rangle _{\mathfrak {H}}\right] ^\frac{1}{2}. \end{aligned}$$

Noting that for h Lipschitz-continuous there holds $\Vert S_h'\Vert _{{\text {L}}^\infty }\le \sqrt{\frac{2}{\pi }}\Vert h'\Vert _{{\text {L}}^\infty }$ (e.g. [42, Proposition 3.5.1]), we can deduce a similar bound on the 1-Wasserstein distance. The corresponding bound on the 2-Wasserstein distance takes the form

$$\begin{aligned} W_2(X;{\mathcal {N}})\le \mathbb {E}\left[ |\langle DZ,DX\rangle _{\mathfrak {H}}-1|^2 \right] ^\frac{1}{2}={\mathrm{Var}}\left[ \langle DZ,DX\rangle _{\mathfrak {H}}\right] ^\frac{1}{2}; \end{aligned}$$

its proof is of a different nature and is based on an optimal transport argument in density space (cf. [35, Proposition 3.1]).

It remains to estimate the variance ${\mathrm{Var}}\left[ \langle DZ,DX\rangle _{\mathfrak {H}}\right]$. For that purpose, we apply the first-order Poincaré inequality (i) in the form

$$\begin{aligned} {\mathrm{Var}}\left[ \langle DZ,DX\rangle _{\mathfrak {H}}\right]\le & {} \mathbb {E}\left[ \Vert \langle D^2Z,DX\rangle _{\mathfrak {H}}+\langle DZ,D^2X\rangle _{\mathfrak {H}}\Vert _{{\mathfrak {H}}}^2 \right] . \end{aligned}$$

Noting that the commutation relation (4.5) leads to $DX=(1+\mathcal {L})DZ$ and $D^2X=(2+\mathcal {L})D^2Z$, we deduce

$$\begin{aligned} {\mathrm{Var}}\left[ \langle DZ,DX\rangle _{\mathfrak {H}}\right]\le & {} \mathbb {E}\left[ \big \Vert \big \langle D^2X,\big ((1+\mathcal {L})^{-1}+(2+\mathcal {L})^{-1}\big )DX\big \rangle _{\mathfrak {H}}\big \Vert _{{\mathfrak {H}}}^2 \right] \\\le & {} \mathbb {E}\left[ \Vert D^2X\Vert _{{\text {op}}}^4 \right] ^\frac{1}{4}\mathbb {E}\left[ \big \Vert \big ((1+\mathcal {L})^{-1}+(2+\mathcal {L})^{-1}\big )DX\big \Vert _{\mathfrak {H}}^4 \right] ^\frac{1}{4}, \end{aligned}$$

Noting as in [38, Proposition 3.2] that $(1+\mathcal {L})^{-1}$ and $(2+\mathcal {L})^{-1}$ have operator norms on ${\text {L}}^4(\Omega )$ bounded by 1 and $\frac{1}{2}$, respectively, the conclusion follows. $\square$