Sublinear growth of the corrector in stochastic homogenization: Optimal stochastic estimates for slowly decaying correlations

We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and "essentially Gaussian" coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the corrector (caused by the quantitative decorrelation on larger scales) correctly. The result is based on estimates on the Malliavin derivative for certain functionals which are basically averages of the gradient of the corrector, on concentration of measure, and on a mean value property for $a$-harmonic functions.