The Multiplicative Chaos of $H=0$ Fractional Brownian Fields

Abstract

We consider a family of fractional Brownian fields $\{B^H{\}}_{H\in (0,1)}$ on ${\mathbb{R}}^d$, where $H$ denotes their Hurst parameter. We first define a rich class of normalizing kernels $\psi$ such that the covariance of $$X^H(x)=\Gamma (H{)}^{\frac{1}{2}}\left(B^H(x)-{\int }_{{\mathbb{R}}^d}{B}^H(u)\psi (u,x)du\right),$$ converges to the covariance of a log-correlated Gaussian field when $H\downarrow 0$. We then use Berestycki's ``good points'' approach in order to derive the limiting measure of the so-called multiplicative chaos of the fractional Brownian field $$M_{\gamma }^H(dx)=e^{\gamma X^H(x)-\frac{{\gamma }^2}{2}E[X^H(x{)}^2]}dx,$$ as $H\downarrow 0$ for all $\gamma \in (0,{\gamma }^{\ast }(d)]$, where ${\gamma }^{\ast }(d)>\sqrt{\frac{7}{4}d}$. As a corollary we establish the $L^2$ convergence of $M_{\gamma }^H$ over the sets of ``good points'', where the field $X^H$ 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 $L^2$ to the volatility of multifractal random walks.

48 pages, 1 figure