Appears in collection : Partial Differential Equations, Analysis and Geometry

I will talk about some recent work on the problem of establishing rigorously a wave turbulence theory for water waves systems. This is a classical problem in Mathematical Physics, going back to pioneering work of Hasselmann. To address it we propose a new mechanism, based on a combination of two main ingredients: (1) deterministic energy estimates for all solutions that are small in $L^\infty$-based norms, and (2) probabilistic arguments aimed at understanding propagation of randomness on long time intervals. This is joint work with Yu Deng and Fabio Pusateri.