UC San Diego

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. 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.

We develop the notion of parallel translation along a controlled rough path. This lends itself to a theory of rolling and unrolling maps for not only controlled rough paths but

controlled rough one-forms.