Trigonometric multiplicative chaos and Application to random distributions

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 $d\ge 1$.

48 pages, 2 figures

References in corpus (1)

• KPZ in one dimensional random geometry of multiplicative cascades