Random tensors, propagation of randomness, and nonlinear dispersive equations

• 成果类型: Article
• 署名作者: Deng, Yu; Nahmod, Andrea R.; Yue, Haitian
• 署名单位: University of Southern California; University of Massachusetts System; University of Massachusetts Amherst; ShanghaiTech University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01084-8
• 发表日期: 2022
• 页码: 539-686
• 关键词: schrodinger-equation; invariant-measures; well-posedness; triviality; DYNAMICS

摘要

We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work (Deng et al. in: Invariant Gibbs measures and global strong solutions for 4009 nonlinear Schrodinger equations in dimension two, ), to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we establish almost-sure local well-posedness for semilinear Schrodinger equations in the full subcritical range relative to the probabilistic scaling (Theorem 1.1). The solution we construct has an explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients. As a byproduct we also obtain new results concerning regular data and long-time solutions, in particular Theorem 1.6, which provides long-time control for random homogeneous data, demonstrating the highly nontrivial fact that the first energy cascade happens at a much later time than in the deterministic setting. In the random setting, the probabilistic scaling is the natural scaling for dispersive equations, and is different from the natural scaling for parabolic equations. Our theory of random tensors can be viewed as the dispersive counterpart of the existing parabolic theories (regularity structures, para-controlled calculus and renormalization group techniques).