Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics

• Published in: Inventiones mathematicae Vol. 199; no. 2; pp. 455 - 515
• Main Authors: Gloria, A., Neukamm, S., Otto, F.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2015; Springer Nature B.V; Springer Verlag
• Subjects: Analysis of PDEs; Dynamics; Ergodic processes; Estimates; Homogenization; Homogenizing; Mathematical analysis; Mathematics; Mathematics and Statistics; Numerical Analysis; Optimization; Probability; Spectra; Stochasticity
• ISSN: 0020-9910; 1432-1297
• DOI: 10.1007/s00222-014-0518-z

We study quantitatively the effective large-scale behavior of discrete elliptic equations on the lattice Z d with random coefficients. The theory of stochastic homogenization relates the random, stationary, and ergodic field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinite-dimensional space of admissible coefficient fields. In this contribution we develop new quantitative methods for the corrector problem based on the assumption that ergodicity holds in the quantitative form of a Spectral Gap Estimate w.r.t. a Glauber dynamics on coefficient fields—as it is the case for independent and identically distributed coefficients. As a main result we prove an optimal decay in time of the semigroup associated with the corrector problem (i.e. of the generator of the process called “random environment as seen from the particle”). As a corollary we recover existence of stationary correctors (in dimensions d > 2 ) and prove new optimal estimates for regularized versions of the corrector (in dimensions d ≥ 2 ). We also give a self-contained proof of a new estimate on the gradient of the parabolic, variable-coefficient Green’s function, which is a crucial analytic ingredient in our approach. As an application of these results, we prove the first (and optimal) estimates for the approximation of the homogenized coefficients by the popular periodization method in case of independent and identically distributed coefficients.