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.

More precisely, I will discuss recent results on the long-time existence of solutions of water waves systems and on the well-posedness theory of the associated kinetic equations. To address the quasi-linear nature of the problems 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.