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We study the transport property of Gaussian measures on Sobolev spaces ofperiodic functions under the dynamics of the one-dimensional cubic fractionalnonlinear Schr\"{o}dinger equation. For the case of second-order dispersion orgreater, we establish an optimal regularity result for the quasi-invariance ofthese 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 turbulencetheory. We also obtain improved regularity results in the weakly dispersivecase, extending those in [20]. Our proof combines the approach introduced byPlanchon, Tzvetkov and Visciglia [47] and that of Debussche and Tsutsumi [15].

Transport of Gaussian measures under the flow of one-dimensional fractional nonlinear Schr\"{o}dinger equations