Non-perturbative approach to the Bourgain-Spencer conjecture in stochastic homogenization

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 prove the first half of this conjecture for the first time in a non-perturbative setting. 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 $L^2$ random fields, we show that twice as many stationary correctors can be defined in a Schwartz-like distributional sense on the probability space.