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March 2014 Levy multiplicative chaos and star scale invariant random measures
Rémi Rhodes, Julien Sohier, Vincent Vargas
Ann. Probab. 42(2): 689-724 (March 2014). DOI: 10.1214/12-AOP810


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.


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Rémi Rhodes. Julien Sohier. Vincent Vargas. "Levy multiplicative chaos and star scale invariant random measures." Ann. Probab. 42 (2) 689 - 724, March 2014.


Published: March 2014
First available in Project Euclid: 24 February 2014

zbMATH: 1295.60064
MathSciNet: MR3178471
Digital Object Identifier: 10.1214/12-AOP810

Primary: 60G57
Secondary: 28A80 , 60G15 , 60G18 , 60H25

Keywords: infinitely divisible processes , Multifractal processes , multiplicative chaos , random measure , scale invariance , star equation , uniqueness

Rights: Copyright © 2014 Institute of Mathematical Statistics

Vol.42 • No. 2 • March 2014
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