Cancellations for dispersive PDEs with random initial data

TL;DR

This paper introduces a combinatorial formalism using arborification to analyze cancellations in dispersive PDEs with random initial data, aiding in understanding wave turbulence and Gibbs measure invariance.

Contribution

It presents a novel arborification-based approach as an alternative to existing methods for handling cancellations in dispersive PDEs with randomness.

Findings

Formalism enables computation of cancellations in wave turbulence.

Proves invariance of Gibbs measure for 3D cubic wave equation.

Provides an alternative to molecule-based methods.

Abstract

In this work, we provide a combinatorial formalism for dealing with the cancellations that have appeared recently in the context of dispersive PDEs with random initial data. The main idea is to transform iterated integrals encoded by decorated trees into words via an arborification map. This provides a formalism alternative to the one of molecules introduced by Deng and Hani (2023). It allows us to compute the cancellations coming from Wave turbulence and the proof of the invariance of the Gibbs measure under the dynamics of the three-dimensional cubic wave equation.