Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity II: Dynamics

TL;DR

This paper establishes the invariance of a singular Gibbs measure for a 3D wave equation with Hartree nonlinearity, focusing on the global dynamics using advanced analytical techniques.

Contribution

It introduces a new globalization method to handle the measure's singularity and extends the understanding of the equation's dynamics.

Findings

Proved invariance of the Gibbs measure for the 3D wave equation

Developed a new globalization argument for singular measures

Analyzed the equation's global dynamics with advanced harmonic analysis

Abstract

In this two-paper series, we prove the invariance of the Gibbs measure for a three-dimensional wave equation with a Hartree nonlinearity. The novelty lies in the singularity of the Gibbs measure with respect to the Gaussian free field. In this paper, we focus on the dynamical aspects of our main result. The local theory is based on a para-controlled approach, which combines ingredients from dispersive equations, harmonic analysis, and random matrix theory. The main contribution, however, lies in the global theory. We develop a new globalization argument, which addresses the singularity of the Gibbs measure and its consequences.