Probabilistic scaling, propagation of randomness and invariant Gibbs measures

In this talk, we will start by describing how classical tools from probability

offer a robust framework to understand the dynamics of waves via appropriate ensembles

on phase space rather than particular microscopic dynamical trajectories. We will continue

by explaining the fundamental shift in paradigm that arises from the “correct” scaling in this

context and how it opened the door to unveil the random structures of nonlinear waves that

live on high frequencies and fine scales as they propagate. We will then discuss how these 

ideas broke the logjam in the study of the Gibbs measures associated to nonlinear 

Schrödinger equations in the context of equilibrium statistical mechanics and of the 

hyperbolic $\Phi^4_3$ model in the context of constructive quantum field theory. 

We will end with some open challenges about the long-time propagation of randomness 

and out-of-equilibrium dynamics.