Transport of low regularity Gaussian measures for the 1d quintic nonlinear Schr\"odinger equation

Abstract

We consider the 1d nonlinear Schr\"odinger equation (NLS) on the torus with initial data distributed according to the Gaussian measure with covariance operator $(1 - \Delta)^{-s}$, where $\Delta$ is the Laplace operator. We prove that the Gaussian measures are quasi-invariant along the flow of (NLS) for the full range $s > \frac{3}{2}$. This improves a previous result obtained by Planchon, Tzvetkov and Visciglia (in 2019), where the quasi-invariance is proven for $s=2k$, for all integers $k\geq 1$. In our approach, to prove the quasi-invariance, we directly establish an explicit formula for the Radon-Nikodym derivative $G_s(t,.)$ of the transported measures, which is obtained as the limit of truncated Radon-Nikodym derivatives $G_{s,N}(t,.)$ for transported measures associated with a truncated system. We also prove that the Radon-Nikodym derivatives belong to $L^p$, $p>1$, with respect to $H^1(\mathbb{T})$-cutoff Gaussian measures, relying on the introduction of weighted Gaussian measures produced by a normal form reduction, following a recent work by Sun and Tzvetkov (in 2023). Additionally, we prove that the truncated densities $G_{s,N}(t,.)$ converges to $G_s(t,.)$ in $L^p$ (with respect to the $H^1(\mathbb{T})$-cutoff Gaussian measures).