We provide the rigorous derivation of the wave kinetic equation from the cubic nonlinear Schrödinger (NLS) equation at the kinetic timescale, under a particular scaling law that describes the limiting process. This solves a main conjecture in the theory of wave turbulence, i.e. the kinetic theory of nonlinear wave systems. Our result is the wave analog of Lanford's theorem on the derivation of the Boltzmann kinetic equation from particle systems, where in both cases one takes the thermodynamic limit as the size of the system diverges to infinity, and as the interaction strength of waves/radius of particles vanishes to 0, according to a particular scaling law (Boltzmann-Grad in the particle case).More precisely, in dimensions d ≥ 3, we consider the (NLS) equation in a large box of size L with a weak nonlinearity of strength α. In the limit L → ∞ and α → 0, under the scaling law α ∼ L −1 , we show that the long-time behavior of (NLS) is statistically described by the wave kinetic equation, with well justified approximation, up to times that are O(1) (i.e independent of L and α) multiples of the kinetic timescale T kin ∼ α −2 . This is the first result of its kind for any nonlinear dispersive system.

In this paper, we consider the defocusing Hartree nonlinear Schrödinger equations on T3 with real-valued and even potential V and Fourier multiplier decaying such as |k|−β. By relying on the method of random averaging operators [Deng et al., arXiv:1910.08492 (2019)], we show that there exists β0, which is less than but close to 1, such that for β > β0, we have invariance of the associated Gibbs measure and global existence of strong solutions in its statistical ensemble. In this way, we extend Bourgain’s seminal result [J. Bourgain, J. Math. Pures Appl. 76, 649–702 (1997)], which requires β > 2 in this case.

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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-