A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures

TL;DR

This paper develops a theoretical framework for intermittency differentiation in 1D infinitely divisible multiplicative chaos measures, deriving equations for the distribution of total mass and exploring invariance properties.

Contribution

It introduces a new theory of intermittency differentiation, establishes invariance properties, and derives a Feynman-Kac equation for the distribution of the total mass in these measures.

Findings

Derived a Feynman-Kac equation for total mass distribution

Established intermittency invariance of the underlying field

Analyzed moments and covariance structure of total mass

Abstract

A theory of intermittency differentiation is developed for a general class of 1D Infinitely Divisible Multiplicative Chaos measures. The intermittency invariance of the underlying infinitely divisible field is established and utilized to derive a Feynman-Kac equation for the distribution of the total mass of the limit measure by considering a stochastic flow in intermittency. The resulting equation prescribes the rule of intermittency differentiation for a general functional of the total mass and determines the distribution of the total mass and its dependence structure to the first order in intermittency. A class of non-local functionals of the limit measure extending the total mass is introduced and shown to be invariant under intermittency differentiation making the computation of the full high temperature expansion of the total mass distribution possible in principle. For application, positive integer moments and covariance structure of the total mass are considered in detail.