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arXiv · 2502.17094

Qualitative quasi-invariance of low regularity Gaussian measures for the 1d quintic nonlinear Schr\"odinger equation

Alexis Knezevitch
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Abstract

We consider the 1d quintic nonlinear Schr\"odinger equation (NLS) on the torus with initial data distributed according to the Gaussian measures with covariance operator $(1-\Delta)^{-s}$, and denoted $\mu_s$. For the full range $s>\frac{9}{10}$, we prove that these Gaussian measures are quasi-invariant along the flow of (NLS), meaning that the law of the solution at any time is absolutely continuous with respect to the initial Gaussian measure. Moreover, the condition $s>\frac{9}{10}$ corresponds to the threshold where the Sobolev space $H^{\frac{2}{5}+}(\mathbb{T})$ is of $\mu_s$-full measure (it is of zero $\mu_s$-measure otherwise). This is the lower regularity Sobolev space where we currently know that (NLS) is globally well-posed, thanks to a work by LI-WU-XU. The present work extends the known threshold $s>\frac{3}{2}$ for the quasi-invariance down to $s>\frac{9}{10}$, but we do not obtain here quantitative results on the Radon-Nikodym derivatives. Our approach is based on a work of Sun-Tzvetkov, combining a Poincar\'e-Dulac normal form reduction with energy estimates. However, our main tool to obtain these energy estimates differs: we use the Bou\'e-Dupuis variational formula instead of Wiener Chaos.

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Alexis Knezevitch. 2025-02-24. Qualitative quasi-invariance of low regularity Gaussian measures for the 1d quintic nonlinear Schr\"odinger equation. https://doi.org/10.4171/aihpc%2F168

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