Non‐geometric rough paths on manifolds

Abstract

We provide a theory of manifold-valued rough paths of bounded 3 > p $3 > p$ -variation, which we do not assume to be geometric. Rough paths are defined in charts, relying on the vector space-valued theory of Friz and Hairer (A course on rough paths, 2014), and coordinate-free (but connection-dependent) definitions of the rough integral of cotangent bundle-valued controlled paths, and of rough differential equations 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 stochastic differential equations on manifolds (Émery, Stochastic calculus in manifolds, 1989). We proceed to present the extrinsic counterparts to our local formulae, and show how these extend the work in Cass et al. (Proc. Lond. Math. Soc. (3) 111 (2015) 1471–1518) 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 T M $TM$ , which figures in an Itô correction term in the parallelism rough differential equation; 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 a few examples that explore the additional subtleties introduced by our change in perspective.