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

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arxiv 2505.24271 v1 pith:GMCGYEBW submitted 2025-05-30 math.AP math.PR

Tadahiro Oh , Yuzhao Wang This is my paper
classification math.APmath.PR
keywords bourgaincubicprobabilisticsolutionsargumentconstructingconstructiondefocusing
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In a seminal paper (1996), Bourgain proved invariance of the Gibbs measure for the defocusing cubic nonlinear Schr\"odinger 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).

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Cited by 4 Pith papers

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  1. Fourier restriction norm method adapted to controlled paths: stochastic wave equations

    math.AP 2026-07 accept novelty 8.0 of 10

    Pathwise local well-posedness of stochastic nonlinear wave equations with multiplicative noise is established in optimal regularity ranges by unifying Fourier restriction norm methods with rough path integration.

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    Irregular modulation of dispersion yields pathwise regularization-by-noise for multiplicative Young noise on stochastic KdV, giving local well-posedness in every Hs.

  3. Sharp unconditional well-posedness of the 2-$d$ periodic cubic hyperbolic nonlinear Schr\"odinger equation

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  4. A remark on pathwise well-posedness of the 1-$d$ stochastic heat equation

    math.AP 2026-07 accept novelty 6.0 of 10

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