Personal overview of Random data theory for Nonlinear Dispersive Equations: Old and New

The studies of Hamiltonian PDEs with random initial data have several aspects.  From the PDE point of view, randomization provides a possibility to analyze solutions with data of supercritical regularity. From the dynamical systems theory point of view, the motivation is to give macroscopic descriptions of the flow of these equations, including transport properties of Gaussian measures, extending recurrence properties, and understanding the wave turbulence phenomenon. 

The goal of this talk is to give an (personal) overview of some development along these lines for defocusing nonlinear wave equations and Schr\"odinger equations with periodic boundary conditions. More precisely, I will discuss three aspects: The invariant measure problem, the structure of solutions at supercritical regularities, and the statistical description of solutions outside the equilibrium.