Revista Matemática Iberoamericana
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Published online: 2017-10-02
Controlled rough paths on manifolds IBruce K. Driver and Jeremy S. Semko (1) University of California at San Diego, La Jolla, USA
(2) University of California at San Diego, La Jolla, USA
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.
Keywords: Dynamical systems, rough paths
Driver Bruce, Semko Jeremy: Controlled rough paths on manifolds I. Rev. Mat. Iberoam. 33 (2017), 885-950. doi: 10.4171/RMI/959