Controlled rough paths on manifolds I
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Bruce K. Driver [1] ; Jeremy S. Semko [1]
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[1] University of California at San Diego
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 33, Nº 3, 2017 (Ejemplar dedicado a: Yves Meyer), págs. 885-950
- Idioma: inglés
- DOI: 10.4171/RMI/959
- Texto completo no disponible (Saber más ...)
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Resumen
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.
