2015/04/13 by Driver, Bruce K., Semko, Jeremy S.
#58J65 #60H30 #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1504.03308
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.