  We consider the variant of stochastic homogenization theory introduced in [X.
Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I 2006 and Journal de
Mathematiques Pures et Appliquees 2007]. The equation under consideration is a
standard linear elliptic equation in divergence form, where the highly
oscillatory coefficient is the composition of a periodic matrix with a
stochastic diffeomorphism. The homogenized limit of this problem has been
identified in [X. Blanc, C. Le Bris and P.-L. Lions, C. R. Acad. Sci. Serie I
2006].
  We first establish, in the one-dimensional case, a convergence result (with
an explicit rate) on the residual process, defined as the difference between
the solution to the highly oscillatory problem and the solution to the
homogenized problem.
  We next return to the multidimensional situation. As often in random
homogenization, the homogenized matrix is defined from a so-called corrector
function, which is the solution to a problem set on the entire space. We
describe and prove the almost sure convergence of an approximation strategy
based on truncated versions of the corrector problem.
