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.

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-

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 [32], to study the propagation of randomness under nonlinear dispersive equations. By applying this theory we also solve Conjecture 1.7 in [32], 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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We prove the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation, which is also known as the hyperbolic Φ 4 3 -model. This result is the hyperbolic counterpart to seminal works on the parabolic Φ 4 3 -model by Hairer [Hai14] and Hairer-Matetski [HM18]. The heart of the matter lies in establishing local in time existence and uniqueness of solutions on the statistical ensemble, which is achieved by using a para-controlled Ansatz for the solution, the analytical framework of the random tensor theory, and the combinatorial molecule estimates.The singularity of the Gibbs measure with respect to the Gaussian free field brings out a new caloric representation of the Gibbs measure and a synergy between the parabolic and hyperbolic theories embodied in the analysis of heat-wave stochastic objects. Furthermore from a purely hyperbolic standpoint our argument relies on key new ingredients that include a hidden cancellation between sextic stochastic objects and a new bilinear random tensor estimate. 3.5. Main estimates 3.6. Proof of local well-posedness 4. Global well-posedness and invariance 4.1. Global bounds 4.2. Stability theory 4.3. Proof of global well-posedness and invariance 5. Integer lattice counting and basic tensors estimates 5.1. Lattice point counting estimates 5.2. Tensors and tensor norms 5.3. Base tensors estimates 5.4. The cubic tensor † Corresponding author.