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.
After revision: We introduce an asymptotic small noise expansion, a so called vol-of-vol expansion, for potentially infinite dimensional and rough stochastic volatility models. Thereby 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.