On Lipschitz maps and the H\"older regularity of flows

TL;DR

This paper explores fundamental properties of Lipschitz maps and their flows, extending classical results to the Lipschitz setting to enhance understanding and applications in rough path analysis and numerical methods.

Contribution

It generalizes classical smooth map results to Lipschitz maps, introduces almost Lipschitz maps for better flow control, and discusses implications for rough path theory.

Findings

Classical properties of smooth maps are extended to Lipschitz maps.

Introduction of almost Lipschitz maps for sharper flow control.

Potential applications in numerical methods for rough paths.

Abstract

This paper regroups some of the basic properties of Lipschitz maps and their flows. Many of the results presented here are classical in the case of smooth maps. We prove them here in the Lipschitz case for a better understanding of the Lipschitz geometry and for a quantification of the related properties, which would be of use to the development of numerical methods for rough paths for example. We also introduce the notion of almost Lipschitz maps, which provide a sharper control and description of flows of Lipschitz vector fields and local inverses of Lipschitz injective immersions