论文标题
具有Hartree非线性I：测量的三维波方程的不变吉布斯度量
Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures
论文作者
论文摘要
在这个两纸系列中，我们证明了具有Hartree非线性的三维波方程的Gibbs的不变性。主要的新颖性是吉布斯（Gibbs）相对于高斯自由场的奇异性。奇异性在我们的论点的理论和动态方面都有几种后果。 在本文中，我们构建和研究了吉布斯的测量。我们的方法基于Barashkov和Gubinelli的早期工作，以$φ^4_3 $ -MODEL。最重要的是，我们的截短的吉布斯度量是针对该系列第二部分的动态方面量身定制的。此外，我们开发了针对Hartree互动非本地性的新工具。我们还确定了吉布斯测量的奇异性和绝对连续性之间的确切阈值，具体取决于相互作用电位的规律性。
In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The main novelty is the singularity of the Gibbs measure with respect to the Gaussian free field. The singularity has several consequences in both measure-theoretic and dynamical aspects of our argument. In this paper, we construct and study the Gibbs measure. Our approach is based on earlier work of Barashkov and Gubinelli for the $Φ^4_3$-model. Most importantly, our truncated Gibbs measures are tailored towards the dynamical aspects in the second part of the series. In addition, we develop new tools dealing with the non-locality of the Hartree interaction. We also determine the exact threshold between singularity and absolute continuity of the Gibbs measure depending on the regularity of the interaction potential.
