Before revision: We  extend the scope of the vol-of-vol expansion given in 27 from finite dimensional stochastic volatility models to infinite dimensional (rough) forward variance modelsand provide newexplicit representations of the  push-down Malliavin weights that simplifies the computations and provides new insights into their structure. This will validate the Bergomi-Guyon expansion for a large class of forward variance models .
After revision: We extend the scope of the vol-of-vol expansion given in 27 from finite dimensional stochastic volatility models to infinite dimensional (rough) forward variance modelsand provide newexplicit representations of the push-down Malliavin weights that utilizes a precise understanding of the terms of this expansion .