Rate of convergence for correctors in almost periodic homogenization

In the homogenization of second order elliptic equations with periodic coefficients, it is well known that the rate of convergence of the zero order corrector u_n - u^hom in the L² norm is 1/n, the same as the scale of periodicity (see Jikov et al [6]). It is possible to have the same rate of convergence in the case of almost periodic coefficients under some stringent structural conditions on the coefficients (see Kozlov [7]). The goal of this note is to construct almost periodic media where the rate of convergence is lower than 1/n. To that aim, in the one dimensional setting, we introduce a family of random almost periodic coefficients for which we compute, using Fourier series analysis, the mean rate of convergence r_n (mean with respect to the random parameter). This allows us to present examples where we find r_n ≫ 1/n^r for every r > 0, showing a big contrast with the random case considered by Bourgeat and Piatnitski [2] where r_n ∼ 1/√n.