2020/07/14 by John Armstrong, Damiano Brigo, Armstrong, John +5
Computer Science · Engineering · Medicine · #60L20 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Medical Imaging Techniques and Applications #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2007.06970
openalex publication_date 2020/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a theory of manifold-valued rough paths of bounded 3 > p-variation, which we do not assume to be geometric. Rough paths are defined in charts, and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of RDEs driven by a rough path valued in another manifold, are given. When the path is the realisation of semimartingale we recover the theory of Itô integration and SDEs on manifolds [É89]. We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in [CDL15] to the setting of non-geometric rough paths and controlled integrands more general than 1-forms. In the last section we turn to parallel transport and Cartan development: the lack of geometricity leads us to make the choice of a connection on the tangent bundle of the manifold TM, which figures in an Itô correction term in the parallelism RDE; such connection, which is not needed in the geometric/Stratonovich setting, is required to satisfy properties which guarantee well-definedness, linearity, and optionally isometricity of parallel transport. We conclude by providing numerous examples, some accompanied by numerical simulations, which explore the additional subtleties introduced by our change in perspective.