Approximations of Effective Coefficients in Stochastic Homogenization

This Note deals with approximations of homogenized coefficients of second order divergence form elliptic operators with random statistically homogeneous coefficients, by means of “periodization” and other ”cut-off” procedures. For instance in the case of periodic approximation, we consider a cubic sample (0, ) of the random medium, extend it periodically in and use the effective coefficients of the obtained periodic operators as an approximation of the effective coefficients of the original random operator. It is shown that these approximations converge a.s. as → ∞ and give back the effective coefficients of the original random operator. Moreover, under additional mixing conditions on the coefficients, the rate of convergence can be estimated by some negative power of which only depends on the dimension, the ellipticity constant and the rate of decay of the mixing coefficients. Similar results are established for approximations in terms of appropriate Dirichlet and Neumann problems localized in a cubic sample (0, )