2026/08/05 by Yongming Luo
Mathematics · #math.AP #math.PR
arxiv created 2026/08/05 · arxiv updated 2026/08/06
We study the Cauchy problem for the nonlinear Schrödinger equation on \mathbb Td with random initial data and a general non-algebraic power-type nonlinearity. We establish almost sure local well-posedness in every spatial dimension and for the whole mass-supercritical range allowed by the natural condition 0<s\mathrm c<1+a. The main new ingredient is a frequency-gaining probabilistic refinement of the Galilean bilinear estimates recently developed by Kwak and Kwon \citeKwakKwon. In the random setting, the gauge decomposition gives rise to three new types of terms: a mean-free coefficient, an opposite-phase interaction, and a scalar remainder. We control them by new resonance counting and large deviation arguments, and close the local theory through a phase-adapted two-component contraction. In the energy-critical case, our result extends the low-dimensional algebraic theories of Nahmod--Staffilani \citeNahmodStaffilani15 and Yue \citeYue21 to every dimension d≥3, including the higher-dimensional non-algebraic models.