  We study the transport properties of the Gaussian measures on Sobolev spaces
under the dynamics of the cubic fourth order nonlinear Schr\"odinger equation
on the circle. In particular, we establish an optimal regularity result for
quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main
new ingredient is an improved energy estimate established by performing an
infinite iteration of normal form reductions on the energy functional.
Furthermore, we show that the dispersion is essential for such a
quasi-invariance result by proving non quasi-invariance of the Gaussian
measures under the dynamics of the dispersionless model.
