Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation

• 成果类型: Article
• 署名作者: Bringmann, Bjoern; Deng, Yu; Nahmod, Andrea R.; Yue, Haitian
• 署名单位: Institute for Advanced Study - USA; Princeton University; University of Southern California; University of Massachusetts System; University of Massachusetts Amherst; ShanghaiTech University
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01254-4
• 发表日期: 2024
• 页码: 1133-1411
• 关键词: global well-posedness; long-time behavior; schrodinger-equation; sure scattering; SCALING LIMITS; triviality

摘要

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 Phi 34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Phi <^>{4}_{3}$\end{document}-model. This result is the hyperbolic counterpart to seminal works on the parabolic Phi 34\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Phi <^>{4}_{3}$\end{document}-model by Hairer (Invent. Math. 198(2):269-504, 2014) and Hairer-Matetski (Ann. Probab. 46(3):1651-1709, 2018).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.