A combinatorial approach to geometric rough paths and their controlled paths

TL;DR

This paper introduces a combinatorial framework for transforming weakly geometric rough paths and their controlled paths, enabling explicit expressions and fundamental identities without relying on smooth approximations, and extends rough path theory to manifolds.

Contribution

It provides a novel combinatorial approach to rough paths and controlled paths, deriving explicit formulas and extending the theory to manifold settings for arbitrary p-variation.

Findings

Explicit combinatorial expression for rough path lift

Fundamental identities like associativity and change of variables established

Extension of rough path theory to manifolds for any p-variation

Abstract

We develop the structure theory for transformations of weakly geometric rough paths of bounded $1 < p$-variation and their controlled paths. Our approach differs from existing approaches as it does not rely on smooth approximations. We derive an explicit combinatorial expression for the rough path lift of a controlled path, and use it to obtain fundamental identities such as the associativity of the rough integral, the adjunction between pushforwards and pullbacks, and a change of variables formula for rough differential equations (RDEs). As applications we define rough paths, rough integration and RDEs on manifolds, extending the results of [CDL15] to the case of arbitrary $p$.