We consider a family of fractional Brownian fields {BH}H∈(0,1) on R
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 Γ(H), such that the covariance of
XH(x) = Γ(H)
1
2

B
H(x) −
Z
Rd
B
H(u)ψ(u, x)du
,
converges to the covariance of a log-correlated Gaussian field when H ↓ 0.
We then use Berestycki’s “good points” approach [11] in order to derive the
convergence of the exponential measure of the fractional Brownian field
MH
γ
(dx) = e
γXH(x)−
γ
2
2
E[XH(x)
2
]
dx,
towards a Gaussian multiplicative chaos, as H ↓ 0 for all γ ∈ (0, γ∗
(d)], where
γ
∗
(d) >
q
7
4
d. As a corollary we establish the L
2
convergence of MH
γ over the
sets of “good points”, where the field XH has a typical behaviour. As a byproduct 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.