论文标题
三维立方非线性波方程的不变吉布斯度量
Invariant Gibbs measures for the three dimensional cubic nonlinear wave equation
论文作者
论文摘要
我们在三维立方波方程的动力学下证明了吉布斯度量的不变性，这也称为双曲线$φ^4_3 $ -MODEL。该结果是与抛物线$φ^4_3 $ -MODEL的开创性作品的双曲线对应物，由Hairer '14和Hairer-Matetski '18。物质的核心在于在统计集合中建立局部存在和解决方案的独特性，这是通过使用PARA对照的ANSATZ进行溶液，随机张量理论的分析框架和组合分子估计来实现的。吉布斯（Gibbs）相对于高斯自由场进行测量的奇异性带来了吉布斯度量的新热量表示，以及在热波随机物体分析中体现的抛物线和双曲线理论之间的协同作用。此外，从纯粹的双曲线角度来看，我们的论点依赖于关键的新成分，这些成分包括六句随机对象和新的双线性随机张量估计中的隐藏取消。
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 '14 and Hairer-Matetski '18. 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.
