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We obtain probabilistic local well-posedness in quasilinear regimes for theSchr\"odinger half-wave equation with a cubic nonlinearity. We need to use arefined ansatz because of the lack of probabilistic smoothing in the Picard'siterations, which is due to the high-low-low frequency interactions. The proofis an adaptation of the method of Bringmann on the derivative nonlinear waveequation to Schr\"odinger-type equations. In addition, we discuss ill-posednessresults for this equation.

Refined probabilistic local well-posedness for a cubic Schr\"odinger half-wave equation