Robustness of the pathwise structure of fluctuations in stochastic homogenization

• 成果类型: Article; Early Access
• 署名作者: Duerinckx, Mitia; Gloria, Antoine; Otto, Felix
• 署名单位: Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); Sorbonne Universite; Universite Paris Cite; Universite Libre de Bruxelles; Max Planck Society
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00983-w
• 发表日期: 2020
• 关键词: convergence; INTEGRALS

摘要

We consider a linear elliptic system in divergence form with random coefficients and study the random fluctuations of large-scale averages of the field and the flux of the solution operator. In the context of the random conductance model, we developed in a previous work a theory of fluctuations based on the notion of homogenization commutator: we proved that the two-scale expansion of this special quantity is accurate at leading order in the fluctuation scaling when averaged on large scales (as opposed to the two-scale expansion of the solution operator taken separately) and that the large-scale fluctuations of the field and the flux of the solution operator can be recovered from those of the commutator. This implies that the large-scale fluctuations of the commutator of the corrector drive all other large-scale fluctuations to leading order, which we refer to as thepathwise structureof fluctuations in stochastic homogenization. In the present contribution we extend this result in two directions: we treatcontinuumelliptic (possibly non-symmetric) systems and allow for stronglycorrelatedcoefficient fields (Gaussian-like with a covariance function that can display an arbitrarily slow algebraic decay at infinity). Our main result shows in this general setting that the two-scale expansion of the homogenization commutator is still accurate to leading order when averaged on large scales, which illustrates the robustness of the pathwise structure of fluctuations.