Title :

The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations

Author(s) :

Gloria, Antoine [Auteur]
Quantitative methods for stochastic models in physics [MEPHYSTO]
Département de Mathématique [Bruxelles] [ULB]
Otto, Felix [Auteur]
Max Planck Institute for Mathematics in the Sciences [MPI-MiS]

English keyword(s) :

stochastic homogenization
corrector
quantitative estimate

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]
Mathématiques [math]/Probabilités [math.PR]

English abstract : [en]

We consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of a finite range of dependence. We show that the gradient and flux (∇ϕ,a(∇ϕ+e)) of the corrector ϕ, when ...
Show more >We consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of a finite range of dependence. We show that the gradient and flux (∇ϕ,a(∇ϕ+e)) of the corrector ϕ, when spatially averaged over a scale R≫1 decay like the CLT scaling R−d2. We establish this optimal rate on the level of sub-Gaussian bounds in terms of the stochastic integrability, and also establish a suboptimal rate on the level of optimal Gaussian bounds in terms of the stochastic integrability. The proof unravels and exploits the self-averaging property of the associated semi-group, which provides a natural and convenient disintegration of scales, and culminates in a propagator estimate with strong stochastic integrability. As an application, we characterize the fluctuations of the homogenization commutator, and prove sharp bounds on the spatial growth of the corrector, a quantitative two-scale expansion, and several other estimates of interest in homogenization.Show less >

Language :

Anglais

European Project :

Quantitative methods in stochastic homogenization

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL