2022/01/28 by Leandro Chiarini, Chiarini, Leandro, Wioletta M. Ruszel +1 · 2 citations
Computer Science · Engineering · Mathematics · #35B27 #60G15 #60G20 #60G60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Nonlinear Partial Differential Equations #Primary: 60K37 #Probability (math.PR) #Secondary: 60J60
paper · pdf · doi:10.48550/arxiv.2201.12013
openalex publication_date 2022/01/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields Ξ^g,\bf a and bi-Laplacian fields Ξ^b,\bf a. They can be characterized as follows: for f=δ the solution u of ∇ ⋅ a ∇ u =f, \bf a is a uniformly elliptic random environment, is the covariance of Ξ^g,\bf a. When f is the white noise, the field Ξ^b,\bf a can be viewed as the distributional solution of the same elliptic equation. Our results characterize the scaling limit of such fields on both, a sufficiently regular domain D⊂ ℝd, or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator Δ, we will show that such families of fields converge to an appropriate multiple of the GFF resp. bi-Laplacian. The limiting fields are determined by their respective homogenized operator \ahom Δ, with constant \ahom depending on the law of the environment \bf a. The proofs are based on the results found in \citeArmstrong2019 and \citegloria2014optimal.