Random tensors, propagation of randomness, and nonlinear dispersive equations
Random tensors, propagation of randomness, and nonlinear dispersive equations
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DOI：
10.1007/s00222-021-01084-8
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发表时间：
2020-06
影响因子：
3.1
通讯作者：
Yu Deng;A. Nahmod;H. Yue
中科院分区：
文献类型：
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作者：
Yu Deng;A. Nahmod;H. Yue
我们引入了随机张量理论，它自然地扩展了我们早期工作中的随机平均算子的方法(邓等人)。In：二维非线性薛定谔方程的不变Gibbs测度和整体强解，Arxiv：1910.08492)，研究了非线性色散方程下的随机性传播。通过应用这一理论，我们建立了半线性薛定谔方程在相对于概率尺度的全亚临界范围内的几乎必然的局部适定性(定理1.1)。我们构造的解具有关于具有自适应随机张量系数的多线性高斯的显式展开式。作为副产品，我们还得到了关于规则数据和长时间解的新结果，特别是定理91.6，它为随机均匀数据提供了长时间控制，证明了第一能量级联发生在比确定性背景下要晚得多的时间这一非常重要的事实。在随机情况下，概率标度是色散方程的自然标度，不同于抛物方程的自然标度。我们的随机张量理论可以看作是现有抛物线理论(正则性结构、准控制微积分和重整化群技术)的色散对应。
We introduce thetheory of random tensors, which naturally extends the method ofrandom averaging operatorsin our earlier work (Deng et al. in: Invariant Gibbs measures and global strong solutions for the nonlinear Schrödinger equations in dimension two, arXiv:1910.08492 ), to study the propagation of randomness under nonlineardispersiveequations. By applying this theory we establish almost-sure local well-posedness for semilinear Schrödinger equations in thefull subcriticalrange relative to theprobabilistic 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 isdifferentfrom 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).