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Invariant Gibbs measures for the one-dimensional quintic nonlinear Schrödinger equation in infinite volume

Bjoern Bringmann, Gigliola Staffilani

TL;DR

The paper proves the invariance of the infinite-volume Gibbs measure for the defocusing 1D quintic nonlinear Schrödinger equation on the real line by extending Bourgain’s periodic-infinite-volume framework to $p=5$. A key advance is a growth estimate for the infinite-volume $\Phi^{p+1}_1$-measures obtained via stochastic quantization in the Hairer–Steele scheme, yielding uniform exponential integrability and $L^{\infty}$-control. The authors then construct a quantitative coupling between finite-volume Gibbs measures $\mu_L$ and the infinite-volume limit $\mu$, and use measure invariance together with uniform a priori estimates to show that periodic solutions converge to a global, invariant limit $u$ with Law$(u(t))=\mu$ for all times. The result broadens the invariant Gibbs-measure program to higher nonlinearity in one dimension and highlights the utility of stochastic quantization and quantitative Skorokhod couplings in infinite-volume dispersive PDEs.

Abstract

We prove the invariance of the Gibbs measure for the defocusing quintic nonlinear Schrödinger equation on the real line. This builds on earlier work by Bourgain, who treated the cubic nonlinearity. The key new ingredient is a growth estimate for the infinite-volume $Φ^{p+1}_1$-measures, which is proven via the stochastic quantization method.

Invariant Gibbs measures for the one-dimensional quintic nonlinear Schrödinger equation in infinite volume

TL;DR

The paper proves the invariance of the infinite-volume Gibbs measure for the defocusing 1D quintic nonlinear Schrödinger equation on the real line by extending Bourgain’s periodic-infinite-volume framework to . A key advance is a growth estimate for the infinite-volume -measures obtained via stochastic quantization in the Hairer–Steele scheme, yielding uniform exponential integrability and -control. The authors then construct a quantitative coupling between finite-volume Gibbs measures and the infinite-volume limit , and use measure invariance together with uniform a priori estimates to show that periodic solutions converge to a global, invariant limit with Law for all times. The result broadens the invariant Gibbs-measure program to higher nonlinearity in one dimension and highlights the utility of stochastic quantization and quantitative Skorokhod couplings in infinite-volume dispersive PDEs.

Abstract

We prove the invariance of the Gibbs measure for the defocusing quintic nonlinear Schrödinger equation on the real line. This builds on earlier work by Bourgain, who treated the cubic nonlinearity. The key new ingredient is a growth estimate for the infinite-volume -measures, which is proven via the stochastic quantization method.

Paper Structure

This paper contains 15 sections, 30 theorems, 248 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Further comments
  4. Preliminaries
  5. Harmonic Analysis
  6. Probability theory
  7. The Gibbs measure in infinite volume
  8. The Hairer-Steele argument
  9. Gaussian estimates
  10. Proof of Theorem \ref{['intro:thm-measure']}
  11. Coupling
  12. Uniform estimates for the nonlinear Schrödinger equations
  13. Difference estimates for the nonlinear Schrödinger equations
  14. Proof of the main theorem
  15. Quantitative Skorokhod representation theorem

Key Result

Theorem 1.1

Let $3 \leq p\leq 5$ and let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. Furthermore, let $\phi_L,\phi \colon (\Omega \times \mathbb{R}, \mathcal{F}\times \mathcal{B}(\mathbb{R}))\rightarrow \mathbb{C}$, where $L\in 2^{\mathbb{N}_0}$ and $\mathcal{B}(\mathbb{R})$ is the Borel $\sigma$- Finally, for all $L\in 2^{\mathbb{N}_0}$, let $u_L$ be the unique global solution of intro:eq-NLS w

Theorems & Definitions (65)

  • Theorem 1.1: Dynamics
  • Remark 1.2: Couplings
  • Theorem 1.3: Measures
  • Remark 1.4
  • Lemma 2.1: Local Bernstein-estimate
  • proof : Proof of Lemma \ref{['prelim:lem-bernstein']}:
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • ...and 55 more