Program

Overview of (wave) turbulence and statistical mechanics. Formal derivation of Wave Kinetic Equations from semi-linear, dispersive PDEs. Random initial data RP and RPA. Resonance vs quasi-resonances, counting lemmas and scaling laws.

Introduction to Gaussian random variables and Gaussian Hilbert Spaces. Isserlis’/Wick’s theorem, Feynmann diagrams and combinatorics. Wick products, Wiener chaos decomposition and Gaussian hypercontractivity estimates.

The cubic NLS equation with random initial data. Picard iteration scheme. Closed formulas for Picard iterates in terms of trees. Control of the remainder terms and high-order TT* argument.

Probabilistic bounds on Picard iterates, two and three-vector counting and general counting algorithms.

Derivation of the Wave Kinetic equation from Picard iteration: Poisson summation, stationary phase argument, Gauss sums and Hua’s lemma.