Differential structure and flow equations on rough path space

Author: 

Qian, Z
Tudor, J

Publication Date: 

1 September 2011

Journal: 

Bulletin des Sciences Mathematiques

Last Updated: 

2020-05-15T02:03:44.177+01:00

Issue: 

6-7

Volume: 

135

DOI: 

10.1016/j.bulsci.2011.07.011

page: 

695-732

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. © 2011 Elsevier Masson SAS.

Symplectic id: 

186421

Submitted to ORA: 

Not Submitted

Publication Type: 

Journal Article