A new definition of rough paths on manifolds

TL;DR

This paper proposes a minimal and locally focused framework for defining rough paths on manifolds, overcoming limitations of classical approaches that rely on smooth structures and local solutions.

Contribution

It introduces a less rigid, locally emphasizing definition of rough paths on manifolds, extending the classical theory and enabling broader applications.

Findings

A minimal framework for rough paths on manifolds is proposed.

The new approach emphasizes local behavior of rough paths.

It extends to defining coloured paths on manifolds.

Abstract

Smooth manifolds are not the suitable context for trying to generalize the concept of rough paths on a manifold. Indeed, when one is working with smooth maps instead of Lipschitz maps and trying to solve a rough differential equation, one loses the quantitative estimates controlling the convergence of the Picard sequence. Moreover, even with a definition of rough paths in smooth manifolds, ordinary and rough differential equations can only be solved locally in such case. In this paper, we first recall the foundations of the Lipschitz geometry, introduced in "Rough Paths on Manifolds" (Cass, T., Litterer, C. & Lyons, T.), along with the main findings that encompass the classical theory of rough paths in Banach spaces. Then we give what we believe to be a minimal framework for defining rough paths on a manifold that is both less rigid than the classical one and emphasized on the local behaviour of rough paths. We end by explaining how this same idea can be used to define any notion of coloured paths on a manifold.