论文标题
随机张量，随机性的传播和非线性分散方程
Random tensors, propagation of randomness, and nonlinear dispersive equations
论文作者
论文摘要
抽象的。本文的目的是双重的。我们介绍了随机张量的理论，该理论自然地扩展了我们较早的作品ARXIV：1910.08492的随机平均操作员的方法，以研究非线性分散方程中随机性的传播。通过应用该理论，我们还解决了Arxiv：1910.08492中的猜想1.7，并在概率缩放中属于亚临界的空间中建立了几乎纯净的局部局部良好性。我们发现的解决方案在具有适应的随机张量系数的多线性高斯人方面具有明确的扩展。 在随机环境中，概率缩放是分散方程的自然缩放缩放，并且与抛物线方程的自然缩放尺度不同。我们的理论涵盖了概率缩放中完整的亚临界体制，可以看作是现有抛物线理论的分散性对应物（规则性结构，para控制的算术和重新归一化组技术）。
Abstract. The purpose of this paper is twofold. We introduce the theory of random tensors, which naturally extends the method of random averaging operators in our earlier work arXiv:1910.08492, to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we also solve Conjecture 1.7 in arXiv:1910.08492, and establish almost-sure local well-posedness for semilinear Schrödinger equations in spaces that are subcritical in the probabilistic scaling. The solution we find has an explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients. 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, which covers the full subcritical regime in the probabilistic scaling, can be viewed as the dispersive counterpart of the existing parabolic theories (regularity structure, para-controlled calculus and renormalization group techniques).
