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

Transport of Gaussian measures with exponential cut-off for Hamiltonian PDEs