Levy multiplicative chaos and star scale invariant random measures

Rémi Rhodes (1) , Julien Sohier (1) , Vincent Vargas (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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: multiplicative chaos; uniqueness; infinitely divisible processes; multifractal processes

Domaines: Probabilités [math.PR]

Rémi Rhodes, Julien Sohier, Vincent Vargas. Levy multiplicative chaos and star scale invariant random measures. Annals of Probability, 2014, 42 (2), pp.689-724. ⟨hal-00662375⟩