Gibbs measures were first introduced in statistical mechanics, where they have been used to describe statistical equilibria of physical systems. In recent years, there has been significant interest in the dynamics of Gibbs measures under heat, wave, and Schrödinger equations. In the first part of the talk, we will discuss the invariance of Gibbs measures under the cubic nonlinear wave equation. In the second part of the talk, we will then discuss the global well-posedness of the stochastic Yang--Mills heat flow.

The talk is based on joint works with Y. Deng, A. Nahmod, and H. Yue, and separately with S. Cao, M. Hairer, and W. Zhao.