Critical Brownian multiplicative chaos

TL;DR

This paper develops the first non-Gaussian critical multiplicative chaos measure by analyzing the limit of subcritical measures and exponentiating local times of planar Brownian motion at the critical parameter value.

Contribution

It introduces a novel construction of the critical Brownian multiplicative chaos measure using three approximation methods, overcoming the lack of Gaussian tools.

Findings

Convergence of measures in Seneta--Heyde and derivative martingale normalizations

Construction of the critical measure as a limit of subcritical measures

Establishment of a continuity lemma for stochastic calculus techniques

Abstract

Brownian multiplicative chaos measures, introduced in [Jeg20, AHS20, BBK94], are random Borel measures that can be formally defined by exponentiating $\gamma$ times the square root of the local times of planar Brownian motion. So far, only the subcritical measures where the parameter $\gamma$ is less than 2 were studied. This article considers the critical case where $\gamma =2$, using three different approximation procedures which all lead to the same universal measure. On the one hand, we exponentiate the square root of the local times of small circles and show convergence in the Seneta--Heyde normalisation as well as in the derivative martingale normalisation. On the other hand, we construct the critical measure as a limit of subcritical measures. This is the first example of a non-Gaussian critical multiplicative chaos. We are inspired by methods coming from critical Gaussian multiplicative chaos, but there are essential differences, the main one being the lack of Gaussianity which prevents the use of Kahane's inequality and hence a priori controls. Instead, a continuity lemma is proved which makes it possible to use tools from stochastic calculus as an effective substitute.