Date
2011
Dewey
Analyse
Sujet
Equations différentielles; Mathematical Physics
Book title
Combinatorics and Physics
Author
Kurusch Ebrahimi-Fard, Matilde Marcolli, Walter D. van Suijlekom
Publisher
American Mathematical Society
Publisher city
Providence, R.I.
Year
2011
ISBN
978-0-8218-5329-0
Author
Gubinelli, Massimiliano
Type
Chapitre d'ouvrage
Item number of pages
135-152
Abstract (EN)
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.