# Solving Rough Differential Equations with the Theory of Regularity Structures

• Antoine Brault
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2252)

## Abstract

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.

## Notes

### Acknowledgements

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.

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