On the derivation of the kinetic wave equation
I will present recent work, in collaboration with Charles Collot, on the derivation of the kinetic wave equation. This equation is believed to describe (weakly) nonlinear dispersive equations in a turbulent regime – it is very similar to the Boltzmann equation, with particles replaced by phonons, or linear waves. Until recently, little was known rigorously on the derivation of the kinetic wave equation from nonlinear dispersive models. We show that the kinetic wave equation provides the correct description on long time scales, almost up to the expected kinetic time scale. The proof combines ideas from random PDEs, harmonic analysis, and graph theory (through Feynman graphs).