Abstract
We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold valued rough paths. Finally, this framework is used to show there is a one to one correspondence between rough paths on a d-dimensional manifold and rough paths on d-dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eeels and Elworthy and Malliavin.
Highlights
In the series of papers [24,25,26], Terry Lyons introduced and began the development of the theory of rough paths on a Banach space W
Among the many applications arising from the interplay of rough paths and stochastic analysis are the study of solutions to stochastic differential equations driven by Gaussian signals see e.g. [4], [3], [5], [16], [7] and the analysis of broad classes of stochastic partial differential equations (SPDEs) [1], [9], [21], [20]
In Example 4.12, we use constrained rough differential equations (RDEs) to give examples of weakly geometric rough paths on M and in Theorem 4.18 we show that all X ∈W Gp (M ) arise as in Example 4.12
Summary
In the series of papers [24,25,26], Terry Lyons introduced and began the development of the theory of rough paths on a Banach space W. The key starting point of this paper is Definition 3.15 which basically states that a weakly geometric rough path X = (xs, Xst) ∈ E ⊕ E ⊗ E is constrained to M iff 1 – forms on M can consistently be integrated along X We expect that the results of this paper will help lay the foundation of future work that explores the properties of manifold valued solutions to stochastic differential equations driven by Gaussian processes such as fractional Brownian motions. In Appendix A we gather together some needed results of the Banach space-valued rough path theory while Appendix B explains a few details on how to view O (M ) as an embedded submanifold which are needed in Section 6 of the paper
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