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

TL;DR

This paper proves the quasi-invariance of Gaussian measures under the 1d quintic NLS flow for a broad range of regularity levels, providing explicit formulas for Radon-Nikodym derivatives and extending previous results.

Contribution

It establishes quasi-invariance for all $s > 3/2$, improving prior results limited to integer regularities, and introduces explicit Radon-Nikodym derivatives for the measure transport.

Findings

Proves Gaussian measures are quasi-invariant for $s > 3/2$.

Derives explicit Radon-Nikodym derivatives for measure transport.

Shows convergence of truncated densities in $L^p$.

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).