Solving rough differential equations with the theory of regularity structures

TL;DR

This paper demonstrates how the theory of regularity structures can be used to solve rough differential equations, providing a new framework that recovers rough path results and offers a fixed point formulation.

Contribution

It introduces a novel application of regularity structures to solve rough differential equations, bridging the gap with rough path theory and providing a pedagogical exposition.

Findings

Recovered rough path theory results within regularity structures

Formulated fixed point problem in the space of modelled distributions

Proved existence of a rough path lift using regularity structures

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.