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.
Revisiting Bourgain's probabilistic construction of solutions to the 2-$d$ cubic NLS
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abstract
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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Irregular modulation of dispersion yields pathwise regularization-by-noise for multiplicative Young noise on stochastic KdV, giving local well-posedness in every Hs.
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Fourier restriction norm method adapted to controlled paths: stochastic wave equations
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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Nonlinear PDEs with modulated dispersion III: multiplicative noises
Irregular modulation of dispersion yields pathwise regularization-by-noise for multiplicative Young noise on stochastic KdV, giving local well-posedness in every Hs.