A Theory of Intermittency Renormalization of Gaussian Multiplicative Chaos Measures

TL;DR

This paper develops a comprehensive theory of intermittency differentiation for Gaussian Multiplicative Chaos measures, deriving exact functional equations and expansions that elucidate the distribution of the total mass, with applications to specific measures.

Contribution

It introduces a novel intermittency differentiation framework and provides explicit expansions and equations for Gaussian Multiplicative Chaos measures, extending to their dependence structures.

Findings

Derived a non-local functional equation for the measure's total mass derivative.

Computed the full intermittency expansion of the Mellin transform.

Conjectured the Morris integral distribution matches the total mass distribution.

Abstract

A theory of intermittency differentiation is developed for a general class of Gaussian Multiplicative Chaos measures including the measure of Bacry and Muzy on the interval and circle as special cases. An exact, non-local functional equation is derived for the derivative of a general functional of the total mass of the measure with respect to intermittency. The formal solution is given in the form of an intermittency expansion and proved to be a renormalized expansion in the centered moments of the total mass of the measure. The full intermittency expansion of the Mellin transform of the total mass is computed in terms of the corresponding expansion of log-moments. The theory is shown to extend to the dependence structure of the measure. For application, the intermittency expansion of the Bacry-Muzy measure on the circle is computed exactly, and the Morris integral probability distribution is shown to reproduce the moments of the total mass and the intermittency expansion, resulting in the conjecture that it is the distribution of the total mass. It is conjectured in general that the intermittency expansion captures the distribution of the total mass uniquely.