  In the context of stochastic homogenization, the Bourgain-Spencer conjecture
states that the ensemble-averaged solution of a divergence-form linear elliptic
equation with random coefficients admits an intrinsic description in terms of
higher-order homogenized equations with an accuracy four times better than the
almost sure solution itself. While previous rigorous results were restricted to
a perturbative regime with small ellipticity ratio, we make the very first
progress in a non-perturbative setting, establishing half of the conjectured
optimal accuracy. The validity of the full conjecture remains an open question
and might in fact fail in general. Our approach involves the construction of a
new corrector theory in stochastic homogenization: while only a bounded number
of correctors can be constructed as stationary random fields in a strong sense,
we show that twice as many stationary correctors can be defined in a
Schwartz-like distributional sense on the probability space. We focus on the
Gaussian setting for the coefficient field, and the proof relies heavily on
Malliavin calculus.
