Qualitative quasi-invariance of low regularity Gaussian measures for the 1d quintic non-linear Schrödinger equation

Abstract

We consider the 1d quintic non-linear Schrödinger 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 $[t]H^{\frac{2}{5}+}(\mathbb{T})$ 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 the work of Li–Wu–Xu [J. Differential Equations 250 (2011) , 2715–2736]. The present work is partially an extension of [NoDEA Nonlinear Differential Equations Appl. 32 (2025), article no. 45]: we extend the quasi-invariance from the range $s>\frac{3}{2}$ to the range $s>\frac{9}{10}$, but we do not obtain here quantitative results on the Radon–Nikodym derivatives generated by the quasi-invariance. Our approach is based on the work of Sun–Tzvetkov [Comm. Pure Appl. Math. (2025)], combining a Poincaré–Dulac normal form reduction with energy estimates. However, our main tool to obtain these energy estimates differs: we use the Boué–Dupuis variational formula instead of Wiener chaos.