Revisiting Bourgain's probabilistic construction of solutions to the 2-d cubic NLS

2025/05/30 by Tadahiro Oh, Yuzhao Wang, Oh, Tadahiro +1 · 3 citations
Economics, Econometrics and Finance · Mathematics · #35Q55 #60H15 #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Probability (math.PR) #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.2505.24271

openalex publication_date 2025/05/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a seminal paper (1996), Bourgain proved invariance of the Gibbs measure for the defocusing cubic nonlinear Schrödinger equation on the two-dimensional torus by constructing local-in-time solutions in a probabilistic manner. In this note, we revisit and streamline his argument, using the random tensor estimate developed by Deng, Nahmod, and Yue (2022).

Cited by

• Almost sure global nonlinear smoothing for the 2D NLS
• Invariant Gibbs dynamics for the nonlinear Schrödinger equations on the disc
• Sharp unconditional well-posedness of the 2-d periodic cubic hyperbolic nonlinear Schrödinger equation

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