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Critical Brownian multiplicative chaos

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jats:titleAbstract</jats:title>jats:pBrownian multiplicative chaos measures, introduced in Jego (Ann Probab 48:1597–1643, 2020), Aïdékon et al. (Ann Probab 48(4):1785–1825, 2020) and Bass et al. (Ann Probab 22:566–625, 1994), are random Borel measures that can be formally defined by exponentiating jats:inline-formulajats:alternativesjats:tex-math$$\gamma $$</jats:tex-math><mml:math xmlns:mml=""> mml:miγ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> times the square root of the local times of planar Brownian motion. So far, only the subcritical measures where the parameter jats:inline-formulajats:alternativesjats:tex-math$$\gamma $$</jats:tex-math><mml:math xmlns:mml=""> mml:miγ</mml:mi> </mml:math></jats:alternatives></jats:inline-formula> is less than 2 were studied. This article considers the critical case where jats:inline-formulajats:alternativesjats:tex-math$$\gamma =2$$</jats:tex-math><mml:math xmlns:mml=""> mml:mrow mml:miγ</mml:mi> mml:mo=</mml:mo> mml:mn2</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, using three different approximation procedures which all lead to the same universal measure. On the one hand, we exponentiate the square root of the local times of small circles and show convergence in the Seneta–Heyde normalisation as well as in the derivative martingale normalisation. On the other hand, we construct the critical measure as a limit of subcritical measures. This is the first example of a non-Gaussian critical multiplicative chaos. We are inspired by methods coming from critical Gaussian multiplicative chaos, but there are essential differences, the main one being the lack of Gaussianity which prevents the use of Kahane’s inequality and hence a priori controls. Instead, a continuity lemma is proved which makes it possible to use tools from stochastic calculus as an effective substitute.</jats:p>


Funder: Österreichischen Akademie der Wissenschaften; doi:


4901 Applied Mathematics, 49 Mathematical Sciences, 4905 Statistics

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Probability Theory and Related Fields

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Springer Science and Business Media LLC