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arXiv · 2606.30223

Invariant Gibbs measures and global dynamics for fractional cubic Schr\"odinger equations on the torus

Yuzhao Wang·Haitian Yue·Chenyuan Zhang·Lifeng Zhao
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Abstract

We consider the defocusing Wick-ordered cubic fractional nonlinear Schr\"odinger equation on the two-dimensional torus with dispersion relation $\omega(k)=|k|^\alpha$. In the weakly dispersive regime $\frac{29}{15}<\alpha<2$, we construct global dynamics for almost every initial datum with respect to the associated Gibbs measure as the limit of the finite-dimensional truncated flows and prove invariance of the Gibbs measure. The core of the proof is an almost sure local theory based on the method of random averaging operators (arXiv:1910.08492v2). The main new ingredients are fractional lattice counting estimates and localized random tensor bounds, which exploit the geometric structure of the fractional phase in place of the classical number-theoretic tools available for quadratic dispersion.

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Yuzhao Wang, Haitian Yue, Chenyuan Zhang, Lifeng Zhao. 2026-06-29. Invariant Gibbs measures and global dynamics for fractional cubic Schr\"odinger equations on the torus. https://arxiv.org/abs/2606.30223

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