Random tensors, propagation of randomness, and nonlinear dispersive equations

Abstract

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 the nonlinear Schrödinger equations in dimension two, arXiv:1910.08492), to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we establish almost-sure local well-posedness for semilinear Schrödinger 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-

Invariant Gibbs measure and global strong solutions for the Hartree NLS equation in dimension three

“…This leads to the notion of the probabilistic scaling critical index s pr := −1/(p − 1) which is much lower than the classical scaling critical index s cr := (d/2) − 2/(p − 1) in the case of p-th power nonlinearity in d dimensions. In [19] we proved that almost-sure local well-posedness indeed holds in H s in the full probabilistic subcritical range when s > s pr , in any dimensions and for any (odd) power nonlinearity.…”

“…For the case of (1.1), a similar argument as in [18,19] yields that the probabilistic scaling critical index for (1.1) is s pr = (−1 − β)/2 which is lower than −1/2, so it is reasonable to think that almost-sure well-posedness would be true. However the situation here is somewhat different from [18,19] due to the asymmetry of the nonlinearity (1.1) compared to the power one, which leads to interesting modifications of the methods in these previous works, as we will discuss in Section 1.3 below.…”

“…In this theory, the linear operators are extended to multilinear operators which are represented by tensors, and whole algebraic and analytic theories are then developed for these random tensors. For NLS equations with odd power nonlinearity, this theory leads to the proof of optimal almostsure well-posedness results, see [19]. We remark that, while the theory of random tensors is more powerful than random averaging operators, the latter has a simpler structure, is less notationheavy, and is already sufficient in many situations (especially if one is not very close to probabilistic criticality).…”

“…On the other hand, it is known since the pioneering work of Bourgain [6] that with random initial data, one can go below the classical scaling critical threshold and obtain almost-sure well-posedness results. In the recent works [18,19] of the authors, an intuitive probabilistic scaling argument was performed. This leads to the notion of the probabilistic scaling critical index s pr := −1/(p − 1) which is much lower than the classical scaling critical index s cr := (d/2) − 2/(p − 1) in the case of p-th power nonlinearity in d dimensions.…”

“…In [19], the random averaging operators is extended to the more general theory of random tensors. In this theory, the linear operators are extended to multilinear operators which are represented by tensors, and whole algebraic and analytic theories are then developed for these random tensors.…”