Levy multiplicative chaos and star scale invariant random measures

TL;DR

This paper introduces Levy multiplicative chaos as a continuous analog of Mandelbrot's star equation, establishing existence, uniqueness, and structural properties of star scale invariant random measures with infinitely divisible weights.

Contribution

It provides the first explicit characterization of Levy multiplicative chaos and demonstrates its unique properties compared to the discrete star equation.

Findings

Existence and uniqueness of measures satisfying the continuous star equation

Explicit structural characterization of Levy multiplicative chaos

Identification of properties unique to the continuous setting

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 L\'{e}vy multiplicative chaos that generalizes the already existing constructions.