Transport properties of Gibbs and Gaussian measures under the flow of Hamiltonian PDEs

Transport properties of Gibbs measures under several dynamical systems have been studied in probability theory. In this talk, we study the Gibbs measure with log-correlated base Gaussian fields on functions/distributions. We first discuss the (non-)construction of the Gibbs measure and then study invariance, and quasi-invariance of the measures along the flow of dispersive Hamiltonian PDEs (like Schr¨odinger and wave equations).