Levy multiplicative chaos and star scale invariant random measures

Sohier, Julien; Vargas, Vincent; Rhodes, Rémi

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

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https://hal.archives-ouvertes.fr/hal-00662375

Date

2014

Nom de la revue

Annals of Probability

Volume

Numéro

Éditeur

Institute of Mathematical Statistics

Pages

689-724

Résumé (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.

Mots-clés

multifractal processes; infinitely divisible processes; multiplicative chaos; uniqueness