  We study the transport property of Gaussian measures on Sobolev spaces of
periodic functions under the dynamics of the one-dimensional cubic fractional
nonlinear Schr\"{o}dinger equation. For the case of second-order dispersion or
greater, we establish an optimal regularity result for the quasi-invariance of
these Gaussian measures, following the approach by Debussche and Tsutsumi [15].
Moreover, we obtain an explicit formula for the Radon-Nikodym derivative and,
as a corollary, a formula for the two-point function arising in wave turbulence
theory. We also obtain improved regularity results in the weakly dispersive
case, extending those in [20]. Our proof combines the approach introduced by
Planchon, Tzvetkov and Visciglia [47] and that of Debussche and Tsutsumi [15].
