Stochastic homogenization. Part 1 of 2.

It is well known from the theory of classical homogenization that the large scale behavior of linear elliptic equations with periodically oscillating coefficients can be described by a limiting equation (of the same type) with homogenized coefficients that are constant. Moreover, the homogenized coefficients are determined by a certain homogenization formula, the evaluation of which requires to solve a linear cell-problem. The same statement is true in the case of stochastic homogenization, e.g. for linear elliptic equations with random coefficients that are ergodic and statistically homogeneous in space.

Part 1 of the lecture is introductory. We recall basic notions, such as stationarity and ergodicity, present two (classical) proofs of the discrete, stochastic homogenization result and motivate the notion of correctors.

The content of part 2 is work in progress: In contrast to the deterministic case, in the stochastic case the cell problem has to be solved on an infinite domain. Hence, in practice, one has to make approximations. We are interested in the convergence rate to zero of the associated error for the approximation and study the case of discrete elliptic equations on the d-dimensional lattice. In particular, in the special case of random coefficients that are identically distributed and independent, we prove the optimal decay rate of the approximation-error. We explain that this observation can be used to setup and analyze approximation schemes that combine periodization, regularization and empirical averaging.