  We consider a class of stochastic processes with rough stochastic volatility,
examples of which include the rough Bergomi and rough Stein-Stein model, that
have gained considerable importance in quantitative finance. A basic question
for such (non-Markovian) models concerns efficient numerical schemes. While
strong rates are well understood (order $H$), we tackle here the intricate
question of weak rates.
  Our main result asserts that the weak rate, for a reasonably large class of
test function, is essentially of order $\min \{ 3H+\tfrac12, 1 \}$ where $H \in
(0,1/2]$ is the Hurst parameter of the fractional Brownian motion that
underlies the rough volatility process. Interestingly, the phase transitation
at $H=1/6$ is related to the correlation between the two driving factors, and
thus gives additional meaning to a quantity already of central importance in
stochastic volatility modelling. Our results are complemented by a lower bound
which show that the obtained weak rate is indeed optimal.
