The Annals of Probability

Levy multiplicative chaos and star scale invariant random measures

Rémi Rhodes, Julien Sohier, and Vincent Vargas

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Abstract

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the structure of these measures, which reflects the constraints imposed by the continuous setting. In particular, we show that the continuous equation enjoys some specific properties that do not appear in the discrete star equation. To that purpose, we define a Lévy multiplicative chaos that generalizes the already existing constructions.

Article information

Source
Ann. Probab., Volume 42, Number 2 (2014), 689-724.

Dates
First available in Project Euclid: 24 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1393251300

Digital Object Identifier
doi:10.1214/12-AOP810

Mathematical Reviews number (MathSciNet)
MR3178471

Zentralblatt MATH identifier
1295.60064

Subjects
Primary: 60G57: Random measures
Secondary: 28A80: Fractals [See also 37Fxx] 60H25: Random operators and equations [See also 47B80] 60G15: Gaussian processes 60G18: Self-similar processes

Keywords
Random measure star equation scale invariance multiplicative chaos uniqueness infinitely divisible processes multifractal processes

Citation

Rhodes, Rémi; Sohier, Julien; Vargas, Vincent. Levy multiplicative chaos and star scale invariant random measures. Ann. Probab. 42 (2014), no. 2, 689--724. doi:10.1214/12-AOP810. https://projecteuclid.org/euclid.aop/1393251300


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References

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