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

TL;DR

This paper investigates how Gaussian measures evolve under the flow of one-dimensional fractional nonlinear Schrödinger equations, establishing optimal regularity results and explicit formulas for Radon-Nikodym derivatives, with implications for wave turbulence.

Contribution

It provides the first optimal regularity results for Gaussian measure quasi-invariance under fractional NLS flows and derives explicit Radon-Nikodym derivatives, extending previous work to weakly dispersive cases.

Findings

Established optimal regularity for Gaussian measure quasi-invariance for second-order or higher dispersion.

Derived explicit Radon-Nikodym derivatives for the transported measures.

Extended regularity results to weakly dispersive fractional NLS cases.

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].