Lognormal scale invariant random measures

TL;DR

This paper studies a continuous version of Mandelbrot's star equation with lognormal weights, establishing existence, uniqueness, and properties of the resulting measures within Gaussian multiplicative chaos theory.

Contribution

It introduces a continuous analog of Mandelbrot's equation, proving existence, uniqueness, and characterizing the covariance structure of the measures.

Findings

Existence and uniqueness of solutions to the continuous Mandelbrot equation

Explicit covariance structure of the measures

Properties like long-range independence and isotropy derived from the equation

Abstract

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by J.P. Kahane in 1985 (or possibly extensions of this theory). As a by product, we also obtain an explicit characterization of the covariance structure of these measures. We also prove that qualitative properties such as long-range independence or isotropy can be read off the equation.