The multiplicative chaos of H=0 fractional Brownian fields

We consider a family of fractional Brownian fields {BH}H?(0,1) on Rd, where H denotes their Hurst parameter. first define rich class normalizing kernels ? and we rescale the normalised field by square-root gamma function ?(H), such that covariance XH(x)=?(H)12(BH(x)??RdBH(u)?(u,x)du), converges to log-correlated Gaussian when H?0. then use Berestycki’s “good points” approach (Electron. Commun. Probab. 22 (2017) Paper No. 27) in order derive convergence exponential measure M?H(dx)=e?XH(x)??22E[XH(x)2]dx, towards multiplicative chaos, as H?0 for all ??(0,??(d)), ??(d)> 7 4d. As corollary establish L2 M?H over sets points”, XH has typical behaviour. by-product result, prove log-normal rough volatility models with small parameter, process is supported probability close 1. Moreover, these multifractal random walks.