Controlled rough paths on manifolds I

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”. A choice of parallelism allows us to compare nearby tangent spaces on the manifold which is necessary to fully discuss controlled rough one-forms. The transformation properties of the theory under change of parallelisms is explored. Although the integration of a general controlled one-form along a rough path does depend on the choice of parallelism, we show for a special class of controlled one-forms – those which are the restriction of smooth one-forms on the manifold – the resulting path integral is in fact independent of any choice of parallelism. 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.