We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity, we prove that the $\mathrm{\Gamma }$-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies on results on the asymptotic behaviour of subadditive processes. The proof of the limit theorem uses a blow-up technique common for local energies, which can be extended to this ‘asymptotically local’ case. As a particular application, we derive a homogenization theorem on random perforated domains.

中文翻译：

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随机卷积能量的均匀化

我们证明了一类具有随机系数的二次卷积能量的同质化定理。在适当陈述的遍历性和平稳性假设下，我们证明$\mathrm{\Gamma }$这种能量的极限几乎肯定是确定性二次狄利克雷型积分泛函，其被积函数可以通过渐近公式来表征。这种表征的证明依赖于亚可加过程的渐近行为的结果。极限定理的证明使用了局部能量常用的爆炸技术，该技术可以扩展到这种“渐近局部”的情况。作为一个特殊的应用，我们推导出随机穿孔域的同质化定理。