  In 2018, Bourgain pioneered a novel perturbative harmonic-analytic approach
to the stochastic homogenization theory of discrete elliptic equations with
weakly random i.i.d. coefficients. The approach was subsequently refined to
show that homogenized approximations of ensemble averages can be derived to a
precision four times better than almost sure homogenized approximations, which
was unexpected by the state-of-the-art homogenization theory. In this paper, we
grow this budding theory in various directions: First, we prove that the
approach is robust by extending it to the continuum setting with exponentially
mixing random coefficients. Second, we give a new proof via Malliavin calculus
in the case of Gaussian coefficients, which avoids the main technicality of
Bourgain's original approach. This new proof also applies to strong Gaussian
correlations with power-law decay. Third, we extend Bourgain's approach to the
study of fluctuations by constructing weak correctors up to order $2d$, which
also clarifies the link between Bourgain's approach and the standard corrector
approach to homogenization. Finally, we draw several consequences from those
different results, both for quantitative homogenization of ensemble averages
and for asymptotic expansions of the annealed Green's function.
