Random tensors, propagation of randomness, and nonlinear dispersive equations

Abstract

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).

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Cited by in corpus (9)

• On the derivation of the wave kinetic equation for NLS
• Derivation of the homogeneous kinetic wave equation: longer time scales
• Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three
• Probabilistic local well-posedness of the cubic nonlinear wave equation in negative Sobolev spaces
• Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise
• Refined probabilistic global well-posedness for the weakly dispersive NLS
• Propagation of chaos and the higher order statistics in the wave kinetic theory
• Almost sure global well-posedness for the energy supercritical NLS on the unit ball of ${\mathbb{R}}^3$
• On long time behavior for stochastic nonlinear Schrödinger equations with a multiplicative noise