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

TL;DR

This paper advances the understanding of stochastic homogenization by establishing a non-perturbative partial proof of the Bourgain-Spencer conjecture, introducing a novel corrector theory using Malliavin calculus in the Gaussian setting.

Contribution

It provides the first non-perturbative progress on the Bourgain-Spencer conjecture by constructing a new corrector theory with distributional stationary correctors in stochastic homogenization.

Findings

Established half of the conjectured optimal accuracy in a non-perturbative regime.

Developed a new corrector theory with distributional stationary correctors.

Relied on Malliavin calculus in the Gaussian setting for the proof.

Abstract

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.