The multiplicative chaos of H=0 fractional Brownian fields

Abstract

We consider a family of fractional Brownian fields ${\{{\mathit{B}}^{\mathit{H}}\}}_{\mathit{H}\in (0,1)}$ on ${\mathbb{R}}^{\mathit{d}}$, 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 $\mathrm{\Gamma }(\mathit{H})$, such that the covariance of

$${\mathit{X}}^{\mathit{H}}(\mathit{x})=\mathrm{\Gamma }{(\mathit{H})}^{\frac{1}{2}}({\mathit{B}}^{\mathit{H}}(\mathit{x})-{\int }_{{\mathbb{R}}^{\mathit{d}}}{\mathit{B}}^{\mathit{H}}(\mathit{u})\mathit{\psi }(\mathit{u},\mathit{x})\mathit{d}\mathit{u}),$$

converges to the covariance of a log-correlated Gaussian field when $\mathit{H}\downarrow 0$. 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

$${\mathit{M}}_{\mathit{\gamma }}^{\mathit{H}}(\mathit{d}\mathit{x})={\mathit{e}}^{\mathit{\gamma }{\mathit{X}}^{\mathit{H}}(\mathit{x})-\frac{{\mathit{\gamma }}^2}{2}\mathit{E}[{\mathit{X}}^{\mathit{H}}{(\mathit{x})}^2]}\mathit{d}\mathit{x},$$

towards a Gaussian multiplicative chaos, as $\mathit{H}\downarrow 0$ for all $\mathit{\gamma }\in (0,{\mathit{\gamma }}^{\ast }(\mathit{d}))$, where ${\mathit{\gamma }}^{\ast }(\mathit{d})>\sqrt{\frac{7}{4}\mathit{d}}$. As a corollary we establish the ${\mathit{L}}^2$ convergence of ${\mathit{M}}_{\mathit{\gamma }}^{\mathit{H}}$ over the sets of “good points”, where the field ${\mathit{X}}^{\mathit{H}}$ 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 ${\mathit{L}}^2$ to the volatility of multifractal random walks.