Controlled Rough Paths on Manifolds I

TL;DR

This paper develops a foundational theory of controlled rough paths on manifolds, defining integrals and solving rough differential equations while addressing geometric structures like parallelisms.

Contribution

It introduces equivalent definitions of manifold-valued controlled rough paths and establishes an integration theory independent of additional geometric structures.

Findings

Defined manifold-valued controlled rough paths and their equivalences

Developed an integration theory using parallelisms and proved independence from them

Proved existence and uniqueness of solutions to rough differential equations on manifolds

Abstract

In this paper, we build the foundation for a theory of controlled rough paths on manifolds. A number of natural candidates for the definition of manifold valued controlled rough paths are developed and shown to be equivalent. The theory of controlled rough one-forms along such a controlled path and their resulting integrals are then defined. This general integration theory does require the introduction of an additional geometric structure on the manifold which we refer to as a "parallelism." The transformation properties of the theory under change of parallelisms is explored. Using these transformation properties, it is shown that the integration of a smooth one-form along a manifold valued controlled rough path is in fact well defined independent of any additional geometric structures. We present a theory of push-forwards and show how it is compatible with our integration theory. Lastly, we give a number of characterizations for solving a rough differential equation when the solution is interpreted as a controlled rough path on a manifold and then show such solutions exist and are unique.