Critical Brownian multiplicative chaos

Abstract

Brownian multiplicative chaos measures, introduced in Jego (Ann Probab 48:1597–1643, 2020), Aïdékon et al. (Ann Probab 48(4):1785–1825, 2020) and Bass et al. (Ann Probab 22:566–625, 1994), are random Borel measures that can be formally defined by exponentiating $$\gamma $$ γ times the square root of the local times of planar Brownian motion. So far, only the subcritical measures where the parameter $$\gamma $$ γ is less than 2 were studied. This article considers the critical case where $$\gamma =2$$ γ = 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.

Multiplicative chaos of the Brownian loop soup

“…By now, Gaussian multiplicative chaos is a fundamental object in its own right which describes scaling limits arising naturally in many different contexts, including random matrices [11,25,37,49,66], the Riemann zeta function [56] and stochastic volatility models in finance [6] (see also [20]); see the surveys [50,53] and the book in preparation [10] for more context and references. More recently, it has been shown that an analogous theory can be developed in the case where ℎ describes (at least formally) the square root of the local time (that is, occupation field) of a Brownian trajectory; see [2,7,29,30,32]. The construction of the associated multiplicative chaos, a measure which we will denote in the following by  ℘ and which is now termed Brownian multiplicative chaos (following the terminology of [30]), is one of the first examples (together with [33] which studies random Fourier series with independent and identically distributed coefficients) of a multiplicative chaos in which the field ℎ is not Gaussian or approximately Gaussian.…”

“…More recently, it has been shown that an analogous theory can be developed in the case where $$h$$ describes (at least formally) the square root of the local time (that is, occupation field) of a Brownian trajectory; see [2, 7, 29, 30, 32]. The construction of the associated multiplicative chaos, a measure which we will denote in the following by $$\mathcal {M}^{\wp }$$ and which is now termed Brownian multiplicative chaos (following the terminology of [30]), is one of the first examples (together with [33] which studies random Fourier series with independent and identically distributed coefficients) of a multiplicative chaos in which the field $$h$$ is not Gaussian or approximately Gaussian.…”

“…Let us comment that the process () should actually possess a continuous modification. However, showing such a regularity is actually far from being simple (see [32, Proposition 1.2 and Remark 1.1]). Fortunately, this will not be needed in this article.…”

“…This generalisation has been studied in [29] and was key in order to characterise the law of Brownian multiplicative chaos. The current article focuses on the subcritical regime, but let us mention that Brownian chaos measures have also been constructed at criticality, that is, when $$a = 2$$ (equivalently, $$\gamma = 2$$); see [32].…”