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arXiv · 1504.03308

Controlled Rough Paths on Manifolds I

Bruce K. Driver·Jeremy S. Semko
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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.

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Bruce K. Driver, Jeremy S. Semko. 2015-04-13. Controlled Rough Paths on Manifolds I. https://arxiv.org/abs/1504.03308

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