随机均匀化中Bourgain-Spencer猜想的非微扰方法
Non-perturbative approach to the Bourgain-Spencer conjecture in stochastic homogenization
AI总结：
针对随机均匀化中Bourgain-Spencer猜想此前仅能在小椭圆率微扰区域证明的问题，提出基于类Schwartz分布平稳校正子的非微扰方法，在高斯系数场下借助Malliavin演算证明了猜想一半的最优精度。
摘要：
在随机均匀化的研究背景下，Bourgain-Spencer猜想指出，带随机系数的散度型线性椭圆方程的系综平均解可以通过高阶均匀化方程进行内禀描述，其精度比几乎必然解本身高出四倍。此前的严格结果都局限于椭圆率较小的微扰区域，我们首次在非微扰框架下取得进展，证明了猜想中最优精度的一半。完整猜想的成立仍是一个开放问题，且实际上可能在一般情况下不成立。我们的方法涉及构建随机均匀化中的一种新的校正子理论：虽然在强意义下只能构造有限个平稳随机场形式的校正子，但我们证明，在概率空间上的类Schwartz分布意义下，可以定义数量为其两倍的平稳校正子。我们聚焦于系数场的高斯情形，证明过程大量依赖Malliavin（马利亚万）演算。
英文摘要：
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.
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