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Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

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Exact dimensionality and projection properties of Gaussian multiplicative chaos measures. / Falconer, Kenneth; Jin, Xiong.

In: Transactions of the American Mathematical Society, Vol. 372, No. 4, 15.08.2019, p. 2921-2957.

Research output: Contribution to journalArticle

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Falconer, K & Jin, X 2019, 'Exact dimensionality and projection properties of Gaussian multiplicative chaos measures', Transactions of the American Mathematical Society, vol. 372, no. 4, pp. 2921-2957. https://doi.org/10.1090/tran/7776

APA

Falconer, K., & Jin, X. (2019). Exact dimensionality and projection properties of Gaussian multiplicative chaos measures. Transactions of the American Mathematical Society, 372(4), 2921-2957. https://doi.org/10.1090/tran/7776

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Falconer K, Jin X. Exact dimensionality and projection properties of Gaussian multiplicative chaos measures. Transactions of the American Mathematical Society. 2019 Aug 15;372(4):2921-2957. https://doi.org/10.1090/tran/7776

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Falconer, Kenneth ; Jin, Xiong. / Exact dimensionality and projection properties of Gaussian multiplicative chaos measures. In: Transactions of the American Mathematical Society. 2019 ; Vol. 372, No. 4. pp. 2921-2957.

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@article{db68a0159c964953a3439278a888ee1d,
title = "Exact dimensionality and projection properties of Gaussian multiplicative chaos measures",
abstract = "Given a measure ν on a regular planar domain D, the Gaussian multiplicative chaos measure of ν studied in this paper is the random measure ^ν^ obtained as the limit of the exponential of the γ-parameter circle averages of the Gaussian free field on D weighted by ν. We investigate the dimensional and geometric properties of these random measures. We first show that if ν is a finite Borel measure on D with exact dimension α>0, then the associated GMC measure ^ν^ is non-degenerate and is almost surely exact dimensional with dimension α-γ2/2, provided γ2/2<α. We then show that if νt is a H{\"o}lder-continuously parameterized family of measures then the total mass of ^νt^ varies H{\"o}lder-continuously with t, provided that γ is sufficiently small. As an application we show that if γ<0.28, then, almost surely, the orthogonal projections of the γ-Liouville quantum gravity measure ^ν^ on a rotund convex domain D in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with H{\"o}lder continuous densities. Furthermore, ^ν^ has positive Fourier dimension almost surely.",
author = "Kenneth Falconer and Xiong Jin",
note = "Paper originally entitled 'H{\"o}lder continuity of the Liouville Quantum Gravity measure'",
year = "2019",
month = "8",
day = "15",
doi = "10.1090/tran/7776",
language = "English",
volume = "372",
pages = "2921--2957",
journal = "Transactions of the American Mathematical Society",
issn = "0002-9947",
publisher = "American Mathematical Society",
number = "4",

}

RIS (suitable for import to EndNote) - Download

TY - JOUR

T1 - Exact dimensionality and projection properties of Gaussian multiplicative chaos measures

AU - Falconer, Kenneth

AU - Jin, Xiong

N1 - Paper originally entitled 'Hölder continuity of the Liouville Quantum Gravity measure'

PY - 2019/8/15

Y1 - 2019/8/15

N2 - Given a measure ν on a regular planar domain D, the Gaussian multiplicative chaos measure of ν studied in this paper is the random measure ^ν^ obtained as the limit of the exponential of the γ-parameter circle averages of the Gaussian free field on D weighted by ν. We investigate the dimensional and geometric properties of these random measures. We first show that if ν is a finite Borel measure on D with exact dimension α>0, then the associated GMC measure ^ν^ is non-degenerate and is almost surely exact dimensional with dimension α-γ2/2, provided γ2/2<α. We then show that if νt is a Hölder-continuously parameterized family of measures then the total mass of ^νt^ varies Hölder-continuously with t, provided that γ is sufficiently small. As an application we show that if γ<0.28, then, almost surely, the orthogonal projections of the γ-Liouville quantum gravity measure ^ν^ on a rotund convex domain D in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with Hölder continuous densities. Furthermore, ^ν^ has positive Fourier dimension almost surely.

AB - Given a measure ν on a regular planar domain D, the Gaussian multiplicative chaos measure of ν studied in this paper is the random measure ^ν^ obtained as the limit of the exponential of the γ-parameter circle averages of the Gaussian free field on D weighted by ν. We investigate the dimensional and geometric properties of these random measures. We first show that if ν is a finite Borel measure on D with exact dimension α>0, then the associated GMC measure ^ν^ is non-degenerate and is almost surely exact dimensional with dimension α-γ2/2, provided γ2/2<α. We then show that if νt is a Hölder-continuously parameterized family of measures then the total mass of ^νt^ varies Hölder-continuously with t, provided that γ is sufficiently small. As an application we show that if γ<0.28, then, almost surely, the orthogonal projections of the γ-Liouville quantum gravity measure ^ν^ on a rotund convex domain D in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with Hölder continuous densities. Furthermore, ^ν^ has positive Fourier dimension almost surely.

UR - https://www.ams.org/journals/tran/0000-000-00/S0002-9947-2019-07776-0/

U2 - 10.1090/tran/7776

DO - 10.1090/tran/7776

M3 - Article

VL - 372

SP - 2921

EP - 2957

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 4

ER -

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