Quasi-invariance of Gaussian measures for the $3d$ energy critical nonlinear Schr\" odinger equation

TL;DR

This work proves a quasi-invariance result for Gaussian measures under the flow of the 3D energy-critical NLS on the torus, showing that for $s\ge 10$ the Gaussian measure $\mu_s$ with covariance $(1-\Delta)^{-s}$ is quasi-invariant under the NLS flow. The authors develop a normal-form reduction to construct a weighted, modified energy and a corresponding weighted Gaussian measure, paired with a careful frequency-truncation approximation and a detailed energy-estimate framework that handles resonant interactions. By establishing uniform $L^p$-bounds for the energy-divergence terms and passing to the limit in the truncation, they prove $(\Phi(t))_*\mu_s \ll \mu_s$ and $\mu_s \ll (\Phi(t))_*\mu_s$ for all real times, yielding qualitative quasi-invariance and enabling out-of-equilibrium statistical descriptions of the flow. The results extend prior 1D quasi-invariance to higher dimensions, avoid Gibbs renormalization, and yield corollaries on the absolute continuity of the law of solutions at a point and $L^1$-stability of transported densities, with potential connections to Malliavin calculus for further regularity insights.

Abstract

We consider the $3d$ energy critical nonlinear Schr\" odinger equation with data distributed according to the Gaussian measure with covariance operator $(1-Δ)^{-s}$, where $Δ$ is the Laplace operator and $s$ is sufficiently large. We prove that the flow sends full measure sets to full measure sets. We also discuss some simple applications. This extends a previous result by Planchon-Visciglia and the second author from $1d$ to higher dimensions.

Theorems & Definitions (51)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Proposition 2.1: Local existence of the weighted measure
• Proposition 2.2: Weighted energy estimate
• Proposition 3.1
• Proposition 3.2
• Lemma 3.3
• proof
• proof : Proof of Theorem \ref{['thm:main']}