AN OPTIMAL VARIANCE ESTIMATE IN STOCHASTIC HOMOGENIZATION OF DISCRETE ELLIPTIC EQUATIONS

We consider a discrete elliptic equation on the d-dimensional lattice Z(d) with random coefficients A of the simplest type: they are identically distributed and independent from edge to edge. On scales large w.r.t. the lattice spacing (i.e., unity), the solution operator is known to behave like the solution operator of a (continuous) elliptic equation with constant deterministic coefficients. This symmetric homogenized matrix A(hom) = a(hom) Id is characterized by xi . A(hom)xi = <(xi + del phi) . A(xi + Delta phi)> for any direction xi is an element of R-d, where the random field phi (the corrector) is the unique solution of --del* . A (xi + del phi) = 0 such that phi (0) = 0, del phi is stationary and (del phi) = 0, <.> denoting the ensemble average (or expectation). It is known (by ergodicity) that the above ensemble average of the energy density epsilon = (xi + del phi) . A(xi + del phi). which is a stationary random field, can be recovered by a system average. We quantify this by proving that the variance of a spatial average of epsilon on length scales L satisfies the optimal estimate, that is, var[Sigma epsilon eta(L) less than or similar to L-d, where the averaging function [i.e., Sigma eta(L) = l. supp(eta(L)) subset of {|x| <= L}] has to be smooth in the sense that |del eta(L)| less than or similar to L-1-d In two space dimensions (i.e., d = 2), there is a logarithmic correction. This estimate is optimal since it shows that smooth averages of the energy density epsilon decay in L as if E would be independent from edge to edge (which it is not for d > 1). This result is of practical significance, since it allows to estimate the dominant error when numerically computing a(hom).