Non-Geometric Rough Paths on Manifolds
John Armstrong, Damiano Brigo, Thomas Cass, Emilio Ferrucci

TL;DR
This paper develops a comprehensive theory of non-geometric rough paths on manifolds, extending rough path analysis beyond geometric constraints and incorporating connection-dependent integrals and parallel transport.
Contribution
It introduces a coordinate-free, connection-dependent framework for non-geometric rough paths on manifolds, including new definitions of rough integrals and RDEs, extending existing theories.
Findings
Unified local and extrinsic formulations of rough paths on manifolds.
Extension of Itô calculus to non-geometric rough paths on manifolds.
Numerical examples illustrating the impact of non-geometricity.
Abstract
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 It\^o integration and SDEs on manifolds [\'E89]. 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…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Advanced Numerical Analysis Techniques · Medical Imaging Techniques and Applications
