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

TL;DR

This paper proves that the flow of the 3D energy-critical nonlinear Schrödinger equation preserves the measure of sets with full Gaussian measure, extending previous 1D results to higher dimensions.

Contribution

It extends the quasi-invariance of Gaussian measures for the nonlinear Schrödinger equation from 1D to 3D energy-critical case.

Findings

Flow preserves full measure sets under Gaussian measure.

Extension of measure invariance results from 1D to 3D.

Provides applications of the measure invariance property.

Abstract

We consider the $3d$ energy critical nonlinear Schr\" odinger equation with data distributed according to the Gaussian measure with covariance operator $(1-\Delta)^{-s}$, where $\Delta$ 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.