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

TL;DR

This paper proves the quasi-invariance of certain Gaussian measures for the 1d quintic NLS at lower regularity levels than previously known, extending the understanding of measure behavior under the flow.

Contribution

It extends the quasi-invariance results for Gaussian measures of the 1d quintic NLS from regularity s>1.5 down to s>0.9 using a novel combination of normal form reduction and variational methods.

Findings

Quasi-invariance holds for Gaussian measures with covariance (1-Δ)^{-s} for s>0.9.

The measure is of full measure on H^{2/5+} space.

The approach uses Boué-Dupuis variational formula instead of Wiener Chaos.

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.