Transport of Gaussian measures under the flow of one-dimensional fractional nonlinear Schrödinger equations

2021/02/26 by Justin Forlano, Forlano, Justin, Kihoon Seong +1 · 3 citations
Mathematics · Economics, Econometrics and Finance · #Advanced Mathematical Physics Problems #Numerical methods in inverse problems #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2102.13398

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 Schrödinger 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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• Almost sure global nonlinear smoothing for the 2D NLS
• Quasi invariant Gaussian measures for the nonlinear Schrödinger equation on \mathbb T²
• Quantitative quasi-invariance of Gaussian measures below the energy level for the 1D generalized nonlinear Schrödinger equation and application to global well-posedness

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