Abstract integration, Combinatorics of Trees and Differential Equations

TL;DR

This review explores the interplay between rough path theory, combinatorics of rooted trees, and differential equations, highlighting algebraic structures and applications to complex infinite-dimensional systems like Navier-Stokes and KdV equations.

Contribution

It synthesizes recent developments connecting rough path theory, Hopf algebras, and combinatorics, providing a unified framework for analyzing irregular differential equations.

Findings

Connections between rough paths and rooted trees are elucidated.

Applications to Navier-Stokes and KdV equations demonstrate practical relevance.

Abstract algebraic approach to integration over irregular paths is outlined.

Abstract

This is a review paper on recent work about the connections between rough path theory, the Connes-Kreimer Hopf algebra on rooted trees and the analysis of finite and infinite dimensional differential equation. We try to explain and motivate the theory of rough paths introduced by T. Lyons in the context of differential equations in presence of irregular noises. We show how it is used in an abstract algebraic approach to the definition of integrals over paths which involves a cochain complex of finite increments. In the context of such abstract integration theories we outline a connection with the combinatorics of rooted trees. As interesting examples where these ideas apply we present two infinite dimensional dynamical systems: the Navier-Stokes equation and the Korteweg-de-Vries equation.