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

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arxiv 2308.12758 v3 pith:UHP4XP3M submitted 2023-08-24 math.AP math.PR

Chenmin Sun , Nikolay Tzvetkov This is my paper · ORCID
classification math.APmath.PR
keywords measurecriticaldeltaenergyequationfullgaussiannonlinear
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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.

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Cited by 3 Pith papers

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  1. Birkhoff normal form via decorated trees

    math.AP 2025-05 conditional novelty 6.0 of 10

    The Birkhoff normal form after any number of symplectic transformations is written exactly as a sum over decorated trees whose nodes encode resonant, non-resonant, and flow terms.

  2. New invariant surface measures for the cubic Schr\"odinger equation

    math.AP 2025-02 conditional novelty 6.0 of 10

    The authors construct invariant probability measures supported on level sets of the renormalized mass for the defocusing cubic nonlinear Schrodinger equation on the one- and two-dimensional torus.

  3. Improved quasi-invariance result for the periodic Benjamin-Ono-BBM equation

    math.AP 2025-01 accept novelty 6.0 of 10

    Quasi-invariance of Gaussian measures under the BO-BBM flow is established for the full global well-posedness range s > 1/2, improving the previous threshold s > 1.

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