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

Abstract

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].