Quantitative stochastic homogenization

Foams, grained materials, reinforced rubber are examples for random heterogeneous materials (RM). Those materials feature microstructural uncertainties: e.g. in a random composite the distribution, geometry, and constitutive parameters of the individual phases might be known on a statistical level only. RM are highly relevant for applications and thus it is desirable to understand their physical properties. In many cases, if the random material is statistically homogeneous, it displays on large length-scales an effective, deterministic physical behavior. This allows a tremendous reduction of complexity in the modeling of the RM: Instead of a model that resolves microscopic (and uncertain) properties (and thus invokes many degrees of freedoms), one can consider an effective, deterministic model describing a homogeneous material.

Stochastic homogenization (SH) provides rigorous analytical methods for the derivation
of effective “homogenized” models and allows to investigate the intimate connection between random microstructure and effective large-scale properties. The qualitative theory of SH goes back to the pioneering works by Papanicolaou & Varadhan, and Kozlov in 1979. Going beyond the qualitative theory by quantifying the size of the error (due to coarse graining) is of fundamental importance since it builds the analytic foundations for the construction of practical (i.e. computable) approximations of effective properties. It turns out that quantitative stochastic homogenization (QSH) requires substantially new ideas and input from regularity theory, probability theory and statistical mechanics.

• A. Gloria, S. Neukamm and F. Otto. Quantification of ergodicity in
  stochastic homogenization: optimal bounds via spectral gap on glauber dynamics.
  Inventiones Mathematicae, 199(2):455-515 2015
• A. Lamacz, S. Neukamm and F. Otto. Moment bounds for the corrector
  in stochastic homogenization of a percolation model. Electron. J. Probab., no. 106,
  1–30 2015
• J. Ben-Artzi, D. Marahrens and S. Neukamm. Moment bounds for
  the corrector in stochastic homogenization of discrete linear elasticity. Communi-
  cations in Partial Differential Equations, 42(2):179-234 2017
• S. Neukamm. An introduction to the qualitative and quantitative theory of ho-
  mogenization. Interdisciplinary Information Sciences (accepted) preprint:
  arXiv:1707.08992) 2017
• S. Neukamm. Quantitative stochastic homogenization of elliptic equations.
  MFO Report No. 35/2015.
• A. Gloria, S. Neukamm and F. Otto. A regularity theory for random
  elliptic operators. (revised and extended version of part I & II of arxiv:1409.2678v3)