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Levy multiplicative chaos and star scale invariant random measures

Sohier, Julien; Vargas, Vincent; Rhodes, Rémi (2014), Levy multiplicative chaos and star scale invariant random measures, Annals of Probability, 42, 2, p. 689-724. http://dx.doi.org/10.1214/12-AOP810

Type
Article accepté pour publication ou publié
External document link
https://hal.archives-ouvertes.fr/hal-00662375
Date
2014
Journal name
Annals of Probability
Volume
42
Number
2
Publisher
Institute of Mathematical Statistics
Pages
689-724
Publication identifier
http://dx.doi.org/10.1214/12-AOP810
Metadata
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Author(s)
Sohier, Julien
Vargas, Vincent
Rhodes, Rémi
Abstract (EN)
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 Levy multiplicative chaos that generalizes the already existing constructions.
Subjects / Keywords
multifractal processes; infinitely divisible processes; multiplicative chaos; uniqueness

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