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• W4384200475 abstract "Abstract In this article, we study stochastic homogenization of non-homogeneous Gaussian free fields $$Xi ^{g,textbf{a}} $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>Ξ</mml:mi> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:msup> </mml:math> and bi-Laplacian fields $$Xi ^{b,textbf{a}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>Ξ</mml:mi> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:msup> </mml:math> . They can be characterized as follows: for $$f=delta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>=</mml:mo> <mml:mi>δ</mml:mi> </mml:mrow> </mml:math> the solution u of $$nabla cdot textbf{a} nabla u =f$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>∇</mml:mi> <mml:mo>·</mml:mo> <mml:mi>a</mml:mi> <mml:mi>∇</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> , $$textbf{a}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>a</mml:mi> </mml:math> is a uniformly elliptic random environment, is the covariance of $$Xi ^{g,textbf{a}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>Ξ</mml:mi> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:msup> </mml:math> . When f is the white noise, the field $$Xi ^{b,textbf{a}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msup> <mml:mi>Ξ</mml:mi> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>,</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> </mml:msup> </mml:math> 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 $$Dsubset mathbb {R}^d$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> </mml:math> , or on the discrete torus. Based on stochastic homogenization techniques applied to the eigenfunction basis of the Laplace operator $$Delta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Δ</mml:mi> </mml:math> , 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 $${{,mathrm{bar{textbf{a}}},}}Delta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mrow> <mml:mspace /> <mml:mover> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mspace /> </mml:mrow> <mml:mi>Δ</mml:mi> </mml:mrow> </mml:math> , with constant $${{,mathrm{bar{textbf{a}}},}}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mspace /> <mml:mover> <mml:mrow> <mml:mi>a</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>¯</mml:mo> </mml:mrow> </mml:mover> <mml:mspace /> </mml:mrow> </mml:math> depending on the law of the environment $$textbf{a}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>a</mml:mi> </mml:math> . The proofs are based on the results found in Armstrong et al. (in: Grundlehren der mathematischen Wissenschaften, Springer International Publishing, Cham, 2019) and Gloria et al. (ESAIM Math Model Numer Anal 48(2):325-346, 2014)." @default.
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• W4384200475 date "2023-07-13" @default.
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• W4384200475 title "Stochastic Homogenization of Gaussian Fields on Random Media" @default.
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