Invariant Gibbs measures for the three-dimensional wave equation with a Hartree nonlinearity I: Measures

TL;DR

This paper constructs and analyzes a singular Gibbs measure for a 3D wave equation with Hartree nonlinearity, highlighting measure-theoretic and dynamical challenges and establishing the measure's properties based on the interaction potential.

Contribution

It introduces a novel construction of the Gibbs measure for the 3D wave-Hartree model, addressing its singularity and developing new tools for non-local interactions.

Findings

Gibbs measure constructed and studied for the 3D wave-Hartree equation.

Identified the threshold between singularity and absolute continuity.

Developed new analytical tools for non-local Hartree interactions.

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 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 $\Phi^4_3$-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.