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

In this talk, we discuss the construction and invariance of the Gibbs measure for a threedimensional wave equation with a Hartree-nonlinearity.

In the rst part of the talk, we construct the Gibbs measure and examine its properties. We discuss the mutual singularity of the Gibbs measure and the so-called Gaussian free eld. In contrast, the Gibbs measure for one or two-dimensional wave equations is absolutely continuous with respect to the Gaussian free eld.

In the second part of the talk, we discuss the probabilistic well-posedness of the corresponding nonlinear wave equation, which is needed in the proof of invariance. At the moment, this is the only theorem proving the invariance of any singular Gibbs measure under a dispersive equation.