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

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.