Solving Rough Differential Equations with the Theory of Regularity Structures
The purpose of this article is to solve rough differential equations with the theory of regularity structures. These new tools recently developed by Martin Hairer for solving semi-linear partial differential stochastic equations were inspired by the rough path theory. We take a pedagogical approach to facilitate the understanding of this new theory. We recover results of the rough path theory with the regularity structure framework. Hence, we show how to formulate a fixed point problem in the abstract space of modelled distributions to solve the rough differential equations. We also give a proof of the existence of a rough path lift with the theory of regularity structure.
I am very grateful to Laure Coutin and Antoine Lejay for their availability, help and their careful rereading.
I deeply thank Peter Friz for suggesting me this topic during my master thesis and for welcoming me at the Technical University of Berlin for 4 months.
- 2.I. Bailleul, F. Bernicot, D. Frey, Higher order paracontrolled calculus, 3d-PAM and multiplicative burgers equations (2015). Preprint arXiv:1506.08773Google Scholar
- 5.A.M. Davie, Differential equations driven by rough paths: an approach via discrete approximation. Appl. Math. Res. eXpress 2008, abm009 (2008)Google Scholar
- 9.M. Gubinelli, P. Imkeller, N. Perkowski, Paracontrolled distributions and singular PDEs. Preprint arXiv:1210.2684 (2012)Google Scholar
- 14.T. Lyons, Differential equations driven by rough signals (I): an extension of an inequality of L.C. Young. Math. Res. Lett 1(4), 451–464 (1994)Google Scholar