  We show that introducing an exponential cut-off on a suitable Sobolev norm
facilitates the proof of quasi-invariance of Gaussian measures with respect to
Hamiltonian PDE flows and allows us to establish the exact Jacobi formula for
the density. We exploit this idea in two different contexts, namely the
periodic fractional Benjamin-Bona-Mahony (BBM) equation with dispersion~$\gamma
>1$ and the periodic one dimensional quintic defocussing nonlinear
Schr\"odinger equation (NLS). For the BBM equation we study the transport of
the cut-off Gaussian measures on fractional Sobolev spaces, while for the NLS
equation we study the measures based on the modified energies introduced by
Planchon-Visciglia and the third author. Moreover for the BBM equation we also
show almost sure global well-posedness for data in~$C^\a(\T)$ for arbitrarily
small~$\a>0$ and invariance of the Gaussian measure associated with the
$H^{\gamma/2}(\T)$ norm.
