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 extend the scope of the vol-of-vol expansion given in 27 existing results for finite dimensional models and validate claims for infinite dimensional models. Furthermore we provide new, explicit (in the sense of non-recursive) representations of the push-down Malliavin weights that simplifies the computations and provides new insights into their structure.