Randomization and the Gross-Pitaevskii hierarchy

Abstract

We study the Gross-Pitaevskii hierarchy on the spatial domain ${\mathbb{T}}^3$. By using an appropriate randomization of the Fourier coefficients in the collision operator, we prove an averaged form of the main estimate which is used in order to contract the Duhamel terms that occur in the study of the hierarchy. In the averaged estimate, we do not need to integrate in the time variable. An averaged spacetime estimate for this range of regularity exponents then follows as a direct corollary. The range of regularity exponents that we obtain is $\alpha >\frac{3}{4}$. It was shown in our previous joint work with Gressman that the range $\alpha >1$ is sharp in the corresponding deterministic spacetime estimate. This is in contrast to the non-periodic setting, which was studied by Klainerman and Machedon, in which the spacetime estimate is known to hold whenever $\alpha \ge 1$. The goal of our paper is to extend the range of $\alpha$ in this class of estimates in a \emph{probabilistic sense}. We use the new estimate and the ideas from its proof in order to study randomized forms of the Gross-Pitaevskii hierarchy. More precisely, we consider hierarchies similar to the Gross-Pitaevskii hierarchy, but in which the collision operator has been randomized. For these hierarchies, we show convergence to zero in low regularity Sobolev spaces of Duhamel expansions of fixed deterministic density matrices. We believe that the study of the randomized collision operators could be the first step in the understanding of a nonlinear form of randomization.

51 pages. Revised version

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• Correlation structures, Many-body Scattering Processes and the Derivation of the Gross-Pitaevskii Hierarchy
• The Rigorous Derivation of the 2D Cubic Focusing NLS from Quantum Many-body Evolution
• The Derivation of the ${\mathbb{T}}^3$ Energy-critical NLS from Quantum Many-body Dynamics
• On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy
• Focusing Quantum Many-body Dynamics II: The Rigorous Derivation of the 1D Focusing Cubic Nonlinear Schrödinger Equation from 3D
• The Gross-Pitaevskii hierarchy on general rectangular tori
• Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space
• Invariant Gibbs Measure for 3D NLW in Infinite Volume
• The unconditional uniqueness for the energy-supercritical NLS
• Uniqueness of Solutions to the Spectral Hierarchy in Kinetic Wave Turbulence Theory
• The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on ${\mathbb{T}}^4$
• Invariance of Gibbs measures under the flows of Hamiltonian equations on the real line
• Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel
• On the mean-field and semiclassical limit from quantum $N$-body dynamics