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 $$ M^{H}_\gamma(dx) = e^{\gamma X^{H}(x) - \frac{\gamma^{2}}{2} E[X^{H}(x)^{2}] }dx, $$ as $H\downarrow 0$ for all $\gamma \in (0,\gamma^{*}(d)]$, where $\gamma^{*}(d)>\sqrt{\frac{7}{4}d}$. As a corollary we establish the $L^{2}$ convergence of $M^{H}_\gamma$ 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.