  We study the large scale behavior of elliptic systems with stationary random
coefficient that have only slowly decaying correlations. To this aim we analyze
the so-called corrector equation, a degenerate elliptic equation posed in the
probability space. In this contribution, we use a parabolic approach and
optimally quantify the time decay of the semigroup. For the theoretical point
of view, we prove an optimal decay estimate of the gradient and flux of the
corrector when spatially averaged over a scale R larger than 1. For the
numerical point of view, our results provide convenient tools for the analysis
of various numerical methods.
