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arXiv · 2102.13398

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

Justin Forlano·Kihoon Seong
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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].

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Justin Forlano, Kihoon Seong. 2021-02-26. Transport of Gaussian measures under the flow of one-dimensional fractional nonlinear Schr\"{o}dinger equations. https://arxiv.org/abs/2102.13398

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