Trigonometric multiplicative chaos and Application to random distributions

TL;DR

This paper develops the theory of trigonometric multiplicative chaos to analyze the multifractal behavior of certain random trigonometric series on the torus, leading to new classes of random measures.

Contribution

It introduces the concept of trigonometric multiplicative chaos and applies it to study the multifractal properties of random trigonometric series on higher-dimensional tori.

Findings

Series are almost surely not Fourier-Stieljes but define pseudo-functions.

Partial sums exhibit multifractal behavior.

Theory extends to tori of dimension d ≥ 1.

Abstract

The random trigonometric series $\sum_{n=1}^\infty \rho_n \cos (nt +\omega_n)$ on the circle $\mathbb{T}$ are studied under the conditions $\sum |\rho_n|^2=\infty$ and $\rho_n\to 0$, where $\{\omega_n\}$ are iid and uniformly distributed on $\mathbb{T}$. They are almost surely not Fourier-Stieljes series but define pseudo-functions. This leads us to develop the theory of trigonometric multiplicative chaos, which are the limits of the exponentiations of partials sums. which produces a class of random measures. The behaviors of the partial sums of the above series are proved to be multifractal. Our theory holds on the torus $\mathbb{T}^d$ of dimension $d\ge 1$.