On Sobolev rough paths

TL;DR

This paper develops a Sobolev space framework for rough paths, establishing the continuity of solutions to rough differential equations driven by these paths in low regularity settings.

Contribution

It introduces Sobolev rough paths and controlled Sobolev paths, and proves the local Lipschitz continuity of the solution map in this new setting.

Findings

Sobolev rough path space is well-defined for low regularity

Solution map for rough differential equations is Lipschitz continuous in Sobolev space

Framework extends rough path theory to lower regularity regimes

Abstract

We introduce the space of rough paths with Sobolev regularity and the corresponding concept of controlled Sobolev paths. Based on these notions, we study rough path integration and rough differential equations. As main result, we prove that the solution map associated to differential equations driven by rough paths is a locally Lipschitz continuous map on the Sobolev rough path space for any arbitrary low regularity $\alpha$ and integrability $p$ provided $\alpha >1/p$.