We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C.
Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006)
717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl.
88 (2007) 34–63.]. 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. Série I 343 (2006) 717–724.]. 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.