June 2022 The multiplicative chaos of H=0 fractional Brownian fields
Paul Hager, Eyal Neuman
Author Affiliations +
Ann. Appl. Probab. 32(3): 2139-2179 (June 2022). DOI: 10.1214/21-AAP1730


We consider a family of fractional Brownian fields {BH}H(0,1) on Rd, where H denotes their Hurst parameter. We first define a rich class of normalizing kernels ψ and we rescale the normalised field by the square-root of the gamma function Γ(H), such that the covariance of


converges to the covariance of a log-correlated Gaussian field when H0. We then use Berestycki’s “good points” approach (Electron. Commun. Probab. 22 (2017) Paper No. 27) in order to derive the convergence of the exponential measure of the fractional Brownian field


towards a Gaussian multiplicative chaos, as H0 for all γ(0,γ(d)), where γ(d)>74d. As a corollary we establish the L2 convergence of MγH over the sets of “good points”, where the field XH has a typical behaviour. As a by-product of the convergence result, we prove that for log-normal rough volatility models with small Hurst parameter, the volatility process is supported on the sets of “good points” with probability close to 1. Moreover, on these sets the volatility converges in L2 to the volatility of multifractal random walks.


We are very grateful to Nathanael Berestycki whose numerous useful comments enabled us to significantly improve this paper. We are also thankful to Masaaki Fukasawa, to the Associate Editor and to the anonymous referee for careful reading of the manuscript, and for a number of useful suggestions.


Download Citation

Paul Hager. Eyal Neuman. "The multiplicative chaos of H=0 fractional Brownian fields." Ann. Appl. Probab. 32 (3) 2139 - 2179, June 2022. https://doi.org/10.1214/21-AAP1730


Received: 1 August 2020; Revised: 1 June 2021; Published: June 2022
First available in Project Euclid: 29 May 2022

MathSciNet: MR4430010
zbMATH: 1503.60046
Digital Object Identifier: 10.1214/21-AAP1730

Primary: 60G15 , 60G57 , 60G60
Secondary: 60G18

Keywords: fractional Brownian fields , Gaussian multiplicative chaos , Log-correlated Gaussian fields , multifractal random walk , Rough volatility

Rights: Copyright © 2022 Institute of Mathematical Statistics


This article is only available to subscribers.
It is not available for individual sale.

Vol.32 • No. 3 • June 2022
Back to Top