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Published online by Cambridge University Press: 08 December 2022
In this chapter, we will discuss the (infinite-dimensional) geometric framework for rough paths and their signature. Rough path theory originated in the 1990s with the work of T. Lyons. It seeks to establish a theory of integrals and differential equations driven by rough signals. For example, one is interested in controlled ordinary differential equations driven by a rough signal. Here, a rough signal is a Hölder continuous path of potentially low Hölder regularity. Numerical methods for equations with more regularity suggest that iterated integrals of the rough signal against itself are needed to construct solutions. However due to Youngs theorem, these iterated integrals do not exist. To compensate this problem, the notion of a rough path was developed. After a qucik introduction to the theory of rough paths, we shall see that rough paths of various flavours can be understood as certain continuous paths taking values in infinite-dimensional Lie groups. The main focus of the chapter is to present an introduction to this geometric side of the theory.
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