Normal approximation for a random elliptic equation

Nolen, James

doi:10.1007/s00440-013-0517-9

Cited by 40 publications

(61 citation statements)

References 21 publications

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“…We also mention related work on the fluctuations ofā L (cf. (1.15)) by Nolen [47,48] (who was first to use Chatterjee's second-order Poincaré inequalities [7,8] in stochastic homogenization), the second author and Nolen [23], Rossignol [50], and Biskup, Salvi, and Wolff [6]. Note that the present contribution originates in the attempt to upgrade the results in [23] into a functional CLT for an energy density.…”

Section: Resultsmentioning

confidence: 91%

The Structure of Fluctuations in Stochastic Homogenization

Duerinckx

Gloria

Otto

2020

Commun. Math. Phys.

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Four quantities are fundamental in homogenization of elliptic systems in divergence form and in its applications: the field and the flux of the solution operator (applied to a general deterministic right-hand side), and the field and the flux of the corrector. Homogenization is the study of the large-scale properties of these objects. For random coefficients, these quantities fluctuate and their fluctuations are a priori unrelated. Depending on the law of the coefficient field, and in particular on the decay of its correlations on large scales, these fluctuations may display different scalings and different limiting laws (if any). In this contribution, we identify a fifth and crucial intrinsic quantity, a random 2-tensor field, which we refer to as the homogenization commutator. In the model framework of discrete linear elliptic equations in divergence form with independent and identically distributed coefficients, we show what we believe to be a general principle, namely that the homogenization commutator drives at leading order the fluctuations of each of the four other quantities in probability, which reveals the pathwise structure of fluctuations in stochastic homogenization. In addition, we show in this framework that the (rescaled) homogenization commutator converges in law to a (2-tensor) Gaussian white noise, the distribution of which is thus characterized by some 4-tensor, and we analyze to which precision this tensor can be extracted from the representative volume element method. All these results are optimally quantified and hold in any dimension. This constitutes the first complete theory of fluctuations in stochastic homogenization. Extensions to the (non-symmetric) continuum setting are discussed, while details are postponed to forthcoming work.

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Section: Resultsmentioning

confidence: 91%

The Structure of Fluctuations in Stochastic Homogenization

Duerinckx

Gloria

Otto

2020

Commun. Math. Phys.

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“…Versions of (11.3) have been proved in [34,37,8,35,18], with the quantity J replaced by spatial averages of the energy density of the correctors and approximations to the corrector. A version of (11.5) was proved in [31,30], while a version of (11.6) with u r (z) replaced by ∫ Φz,r (u − E[u]) was proved in [24].…”

Section: Informal Heuristics and Statement Of Main Resultsmentioning

confidence: 99%

“…In particular, they were the first to obtain estimates for the correctors at the critical scalings, albeit with suboptimal stochastic integrability (typically finite moment bounds) and with somewhat restrictive ergodic assumptions. Later, central limit theorems for the spatial averages of the gradients and the energy densities of the correctors were obtained using these techniques [34,31,30,18]. These important and influential results were the first to give a complete quantitative picture of the behavior of the first-order correctors on any stochastic model, and have inspired a huge amount of subsequent research.…”

Section: 2mentioning

confidence: 99%

The additive structure of elliptic homogenization

2016

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Abstract. One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in [2]: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

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“…This was extended by Gu and Mourrat, in a non-obvious way, to characterize the leading order of the fluctuations of the homogenization error and to show that they are Gaussian [36], still relying on [39,Theorem 1]. Incidentally, the Gaussianity of fluctuations in stochastic homogenization was first established on the level of the error in the representative volume element method: [18] in the smallcontrast case, [30] based on Nolen's [42], and [44]. A quite different approach to, among other things, Gaussianity of leading-order fluctuations of the corrector, motivated by the approach to quantitative stochastic homogenization in [7], was carried out by Armstrong, Kuusi and Mourrat [5] (after being announced in [36]).…”

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confidence: 99%

Effective multipoles in random media

Bella

Giunti

Otto

2020

Communications in Partial Differential Equations

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In a homogeneous medium, the far-field generated by a localized source can be expanded in terms of multipoles; the coefficients are determined by the moments of the localized charge distribution. We show that this structure survives to some extent for a random medium in the sense of quantitative stochastic homogenization: In three space dimensions, the effective dipole and quadrupole -but not the octupole -can be inferred without knowing the realization of the random medium far away from the (overall neutral) source and the point of interest.Mathematically, this is achieved by using the two-scale expansion to higher order to construct isomorphisms between the hetero-and homogeneous versions of spaces of harmonic functions that grow at a certain rate, or decay at a certain rate away from the singularity (near the origin); these isomorphisms crucially respect the natural pairing between growing and decaying harmonic functions given by the second Green's formula. This not only yields effective multipoles (the quotient of the spaces of decaying functions) but also intrinsic moments (taken with respect to the elements of the spaces of growing functions). The construction of these rigid isomorphisms relies on a good (and dimension-dependent) control on the higher-order correctors and their flux potentials.

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Normal approximation for a random elliptic equation

Cited by 40 publications

References 21 publications

The Structure of Fluctuations in Stochastic Homogenization

The Structure of Fluctuations in Stochastic Homogenization

The additive structure of elliptic homogenization

Effective multipoles in random media

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