Random tensors, propagation of randomness, and nonlinear dispersive equations

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).

Automated summary

The theory of random tensors is introduced to study the propagation of randomness under nonlinear dispersive equations, extending the method of random averaging operators. By applying this theory, almost-sure local well-posedness is established for semilinear Schrodinger equations, with solutions having an explicit expansion in terms of multilinear Gaussians. New results are obtained regarding regular data and long-time solutions, demonstrating that the first energy cascade occurs much later in the random setting compared to the deterministic setting.