Non-Geometric Rough Paths on Manifolds

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 It\^o 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.