Quantitative stochastic homogenization and large-scale regularity

Abstract

This is a preliminary version of a book which presents the quantitative homogenization and large-scale regularity theory for elliptic equations in divergence-form. The self-contained presentation gives new and simplified proofs of the core results proved in the last several years, including the algebraic convergence rate for the variational subadditive quantities, the large-scale Lipschitz and higher regularity estimates and Liouville-type results, optimal quantitative estimates on the first-order correctors and their scaling limit to a Gaussian free field. There are several chapters containing new results, such as: quantitative estimates for the Dirichlet problem, including optimal quantitative estimates of the homogenization error and the two-scale expansion; optimal estimates for the homogenization of the parabolic and elliptic Green functions; and $W^{1,p}$-type estimates for two-scale expansions.

504 pages, 14 figures

References in corpus (3)

• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• A new method of normal approximation
• Quantitative results on the corrector equation in stochastic homogenization

Cited by in corpus (45)

• Quantitative estimates in stochastic homogenization for correlated coefficient fields
• Higher-order pathwise theory of fluctuations in stochastic homogenization
• Quenched local limit theorem for random walks among time-dependent ergodic degenerate weights
• Quantitative homogenization of the parabolic and elliptic Green's functions on percolation clusters
• Extrema of the two-dimensional Discrete Gaussian Free Field
• Local Limit Theorems for the Random Conductance Model and Applications to the Ginzburg-Landau $\nabla \varphi$ Interface Model
• Robustness of the pathwise structure of fluctuations in stochastic homogenization
• Homogenization of Parabolic Equations with Non-self-similar Scales
• Higher-order linearization and regularity in nonlinear homogenization
• Lower Gaussian heat kernel bounds for the Random Conductance Model in a degenerate ergodic environment
• The random heat equation in dimensions three and higher: the homogenization viewpoint
• Optimal corrector estimates on percolation clusters
• An informal introduction to quantitative stochastic homogenization
• Quantitative linearization results for the Monge-Ampère equation
• A scalar version of the Caflisch-Luke paradox
• Ratio convergence rates for Euclidean first-passage percolation: Applications to the graph infinity Laplacian
• Quantitative nonlinear homogenization: control of oscillations
• Quantitative estimates for homogenization of nonlinear elliptic operators in perforated domains
• Massless Phases for the Villain model in $d\ge 3$
• Space-time homogenization for nonlinear diffusion
• Homogenization of locally periodic parabolic operators with non-self-similar scales
• Quantitative Estimates in Reiterated Homogenization
• 3D image based stochastic micro-structure modelling of foams for simulating elasticity
• Eigenvalue fluctuations for random elliptic operators in homogenization regime
• Optimal artificial boundary conditions based on second-order correctors for three dimensional random elliptic media
• Homogenization of symmetric stable-like processes in stationary ergodic medium
• Uniform Boundary Estimates in Homogenization of Higher Order Elliptic Systems
• Absence of weak disorder for directed polymers on supercritical percolation clusters
• Regularity for the stationary Navier-Stokes equations over bumpy boundaries and a local wall law
• Tiny fluctuations of the averaging process around its degenerate steady state
• Analytical and numerical study of a modified cell problem for the numerical homogenization of multiscale random fields
• Decay of semigroup for an infinite interacting particle system on continuum configuration spaces
• Regularity of a transmission problem and periodic homogenization
• Properties of the gradient squared of the discrete Gaussian free field
• A Nitsche Hybrid multiscale method with non-matching grids
• Scaling limit of the directional conductivity of random resistor networks on simple point processes
• A Carleman-Type Inequality in Elliptic Periodic Homogenization
• High-contrast random composites: homogenisation framework and new spectral phenomena
• Asymptotic behaviour and functional limit theorems for a time changed Wiener process
• Parabolic Homogenization with an Interface
• Coarsen entropy inequalities and its applications in quantitative stochastic homogenization
• Artificial boundary conditions for random ellitpic systems with correlated coefficient field
• On the duality between height functions and continuous spin models
• Homogenization theory of random walks among deterministic conductances
• Coupling between Brownian motion and random walks on the infinite percolation cluster