## Section: Research Program

### Quantitative stochastic homogenization

Whereas the approximation of homogenized coefficients is an easy task in periodic homogenization, this is a highly nontrivial task for stochastic coefficients. This is in order to analyze numerical approximation methods of the homogenized coefficients that F. Otto (MPI for mathematics in the sciences, Leipzig, Germany) and A. Gloria obtained the first quantitative results in stochastic homogenization [3]. The development of a complete stochastic homogenization theory seems to be ripe for the analysis and constitutes the second major objective of this section.

In order to develop a quantitative theory of stochastic homogenization, one needs to quantitatively understand the corrector equation (3). Provided $A$ is stationary and ergodic, it is known that there exists a unique random field ${\phi}_{\xi}$ which is a distributional solution of (3) almost surely, such that $\nabla {\phi}_{\xi}$ is a stationary random field with bounded second moment $\u2329|\nabla {\phi}_{\xi}{|}^{2}\u232a<\infty $, and with $\phi \left(0\right)=0$. Soft arguments do not allow to prove that ${\phi}_{\xi}$ may be chosen stationary (this is wrong in dimension $d=1$). In [3], [4] F. Otto and A. Gloria proved that, in the case of discrete elliptic equations with iid conductances, there exists a unique stationary corrector ${\phi}_{\xi}$ with vanishing expectation in dimension $d>2$. Although it cannot be bounded, it has bounded finite moments of any order:

$\u2329|{\phi}_{\xi}{|}^{q}\u232a<\infty \phantom{\rule{4pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}q\ge 1.$ | (4) |

They also proved that the variance of spatial averages of the energy density $(\xi +\nabla {\phi}_{\xi})\xb7A(\xi +\nabla {\phi}_{\xi})$ on balls of radius $R$ decays at the rate ${R}^{-d}$ of the central limit theorem.
These are the *first optimal quantitative results* in stochastic homogenization.

The proof of these results, which is inspired by [49], is based on the insight that coefficients such as the Poisson random inclusions are special in the sense that the associated probability measure satisfies a spectral gap estimate. Combined with elliptic regularity theory, this spectral gap estimate quantifies ergodicity in stochastic homogenization. This systematic use of tools from statistical physics has opened the way to the quantitative study of stochastic homogenization problems, which we plan to fully develop.