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Transport of low regularity Gaussian measures for the 1d quintic nonlinear Schrödinger equation

Alexis Knezevitch

TL;DR

This work establishes quasi-invariance of the Gaussian measure with covariance $(1-\Delta)^{-s}$ under the 1D defocusing quintic NLS flow on the torus for all $s>\tfrac{3}{2}$. The authors construct an explicit Radon-Nikodym derivative for the transported measures by performing a Poincaré-Dulac normal form reduction, introducing an energy correction and a modified energy derivative, and then passing to the limit from truncated flows. They prove convergence of truncated densities to a limiting density and obtain $L^p$ integrability results for densities with respect to restricted Gaussian measures via weighted Gaussian measures that encode the nonlinearity. The approach combines deterministic multilinear estimates, dispersive Strichartz controls, and probabilistic Wiener chaos techniques to deliver quantitative quasi-invariance and convergence results, extending prior index-restricted results to a full range of regularity and providing explicit density formulas for the transported measures.

Abstract

We consider the 1d nonlinear Schrödinger equation (NLS) on the torus with initial data distributed according to the Gaussian measure with covariance operator $(1 - Δ)^{-s}$, where $Δ$ 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).

Transport of low regularity Gaussian measures for the 1d quintic nonlinear Schrödinger equation

TL;DR

This work establishes quasi-invariance of the Gaussian measure with covariance under the 1D defocusing quintic NLS flow on the torus for all . The authors construct an explicit Radon-Nikodym derivative for the transported measures by performing a Poincaré-Dulac normal form reduction, introducing an energy correction and a modified energy derivative, and then passing to the limit from truncated flows. They prove convergence of truncated densities to a limiting density and obtain integrability results for densities with respect to restricted Gaussian measures via weighted Gaussian measures that encode the nonlinearity. The approach combines deterministic multilinear estimates, dispersive Strichartz controls, and probabilistic Wiener chaos techniques to deliver quantitative quasi-invariance and convergence results, extending prior index-restricted results to a full range of regularity and providing explicit density formulas for the transported measures.

Abstract

We consider the 1d nonlinear Schrödinger equation (NLS) on the torus with initial data distributed according to the Gaussian measure with covariance operator , where is the Laplace operator. We prove that the Gaussian measures are quasi-invariant along the flow of (NLS) for the full range . This improves a previous result obtained by Planchon, Tzvetkov and Visciglia (in 2019), where the quasi-invariance is proven for , for all integers . In our approach, to prove the quasi-invariance, we directly establish an explicit formula for the Radon-Nikodym derivative of the transported measures, which is obtained as the limit of truncated Radon-Nikodym derivatives for transported measures associated with a truncated system. We also prove that the Radon-Nikodym derivatives belong to , , with respect to -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 converges to in (with respect to the -cutoff Gaussian measures).
Paper Structure (44 sections, 50 theorems, 420 equations)

This paper contains 44 sections, 50 theorems, 420 equations.

Table of Contents

  1. Introduction
  2. Description of the problem
  3. Formal computation and main results
  4. Organization of the paper
  5. Further remarks
  6. Acknowledgments
  7. Poincaré-Dulac normal form reduction and modified energy
  8. The truncated system
  9. Normal form reduction and modified energy
  10. Definition and deterministic properties of the energy correction
  11. Definition and deterministic properties of the modified energy derivative at 0
  12. Transport of Gaussian measures under the truncated flow
  13. A change-of-variable formula
  14. The Radon-Nikodym derivative for the truncated transported Gaussian measures
  15. Transport of Gaussian measures under the flow
  16. ...and 29 more sections

Key Result

Theorem 1.1

Let $s > \frac{3}{2}$ and $\sigma< s-\frac{1}{2}$ close enough to $s-\frac{1}{2}$. Let $t \in \mathbb{R}$. Then, for every $N \in \mathbb{N}$, the transported measure $\Phi_N(t)_\# \mu_s$ has a density $G_{s,N}(t,.)$ with respect to $\mu_s$ given by: Moreover, the transported measure $\Phi(t)_\# \mu_s$ has a density $G_{s}(t,.)$ with respect to $\mu_s$ given by: which is continuous on $H^{\sigma

Theorems & Definitions (112)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 2.2
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • Proposition 3.3
  • ...and 102 more