Heterogeneous media are often naturally described in statistical terms. How to extract their effective behavior on large scales, like the large-scale conductivity, from the statistical specifications? A practioneers numerical approach is to sample the medium according to these specifications and to determine the permeability in the Cartesian directions by imposing simple boundary conditions. What is the error made in terms of the size of this ``representative volume element''?

Our interest in what is called ``stochastic homogenization'' grew out of this error analysis. In the course of developing such an error analysis, connections with the regularity theory of elliptic equations and with concepts from statistical mechanics have emerged in a clearer way.

We also gained an understanding of the structure of fluctuations of any solution: On large scales, the asymptotically Gaussian fluctuations are characterized by a single tensor-valued white noise. This characterization is path-wise and can also be extracted from the representative volume method. The latter is recent joint work with M. Duerinckx and A. Gloria.