Differential Structure and Flow equations on Rough Path Space

TL;DR

This paper develops a differential structure on the space of weakly geometric p rough paths, enabling the definition of derivatives and flow equations in a more general setting than classical Wiener space, with applications to analysis on rough path space.

Contribution

It introduces a novel differential structure on rough path space, allowing derivatives and flow equations without requiring Wiener measure quasi-invariance.

Findings

Defined a tangent space for rough paths with a clear interpretation.

Established a method to differentiate functionals on rough path space.

Solved flow equations for vector fields satisfying Lipschitz conditions.

Abstract

We introduce a differential structure for the space of weakly geometric p rough paths over a Banach space V for 2<p<3. We begin by considering a certain natural family of smooth rough paths and differentiating in the truncated tensor series. The resulting object has a clear interpretation, even for non-smooth rough paths, which we take to be an element of the tangent space. We can associate it uniquely to an equivalence class of curves, with equivalence defined by our differential structure. Thus, for a functional on rough path space, we can define the derivative in a tangent direction analogous to defining the derivative in a Cameron-Martin direction of a functional on Wiener space. Our tangent space contains many more directions than the Cameron-Martin space and we do not require quasi-invariance of Wiener measure. In addition we also locally (globally) solve the associated flow equation for a class of vector fields satisfying a local (global) Lipshitz type condition.