In this talk, we study the invariance of the Gibbs measure for Schrödinger-wave system.

In the first half of this talk, we discuss the construction and properties of the Gibbs measure. Following that, we connect our theorem with classical and recent developments in constructive QFT, stochastic PDEs, and dispersive PDEs.

In the second half of this talk, we look into the dynamical aspect, specifically focusing on the probabilistic well-posedness in support of the Gibbs measure. This part combines ingredients from dispersive equations, harmonic analysis, and probability theory.