A Sobolev rough path extension theorem via regularity structures

TL;DR

This paper extends the rough path theory to Sobolev paths with specific regularity and integrability conditions, utilizing a generalized Hairer reconstruction theorem to establish a Lipschitz continuous lifting map.

Contribution

It introduces a Sobolev rough path extension theorem using regularity structures, generalizing Hairer's reconstruction theorem to Sobolev models and distributions.

Findings

Sobolev paths can be lifted to Sobolev rough paths under certain conditions

The lifting map is locally Lipschitz continuous in the Sobolev metric

The approach broadens rough path theory to include Sobolev regularity settings

Abstract

We show that every $\mathbb{R}^d$-valued Sobolev path with regularity $\alpha$ and integrability $p$ can be lifted to a Sobolev rough path provided $\alpha < 1/p<1/3$. The novelty of our approach is its use of ideas underlying Hairer's reconstruction theorem generalized to a framework allowing for Sobolev models and Sobolev modelled distributions. Moreover, we show that the corresponding lifting map is locally Lipschitz continuous with respect to the inhomogeneous Sobolev metric.