Decorated trees, arborification for cancellations in wave turbulence

In this work, we review part of the results obtained in [12] for computing cancellations for dispersive PDEs with random initial data. The idea is to get a new combinatorial perspective on the cancellations discovered by Deng and Hani (see [16]) in the context of Wave Turbulence when one wants to derive rigorously wave-kinetic equations. This new perspective is based on decorated trees developed for low regularity schemes, together with a well-chosen arborification map that rewrites these trees into linear combinations of words. With this new combinatorial basis, one develops graphical rules to compute cancellations.