Absolute continuity of non-Gaussian and Gaussian multiplicative chaos measures

TL;DR

This paper demonstrates that non-Gaussian multiplicative chaos measures associated with log-correlated fields are almost surely mutually absolutely continuous with Gaussian multiplicative chaos measures, extending understanding of their measure-theoretic properties.

Contribution

The authors construct a coupling showing the absolute continuity between non-Gaussian and Gaussian multiplicative chaos measures, a novel connection in the study of log-correlated fields.

Findings

Non-Gaussian chaos measures are mutually absolutely continuous with GMC measures.

Coupling construction links non-Gaussian and Gaussian chaos measures.

Extension of measure-theoretic properties to non-Gaussian settings.

Abstract

In this article, we consider the multiplicative chaos measure associated to the log-correlated random Fourier series, or random wave model, with i.i.d. coefficients taken from a general class of distributions. This measure was shown to be non-degenerate when the inverse temperature is subcritical by Junnila (Int. Math. Res. Not. 2020 (2020), no. 19, 6169-6196). When the coefficients are Gaussian, this measure is an example of a Gaussian multiplicative chaos (GMC), a well-studied universal object in the study of log-correlated fields. In the case of non-Gaussian coefficients, the resulting chaos is not a GMC in general. However, we construct a coupling between the non-Gaussian multiplicative chaos measure and a GMC such that the two are almost surely mutually absolutely continuous.