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Terry Lyons and Zhongmin Qian

Print publication date: 2002

Print ISBN-13: 9780198506485

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198506485.001.0001

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VECTOR FIELDS AND FLOW EQUATIONS

Chapter:
(p.181) 7 VECTOR FIELDS AND FLOW EQUATIONS
Source:
System Control and Rough Paths
Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198506485.003.0007

This chapter studies spaces of paths as geometric objects in their own right, and particularly looks at the possibility of constructing interesting and mathematically meaningful vector fields on these infinite dimensional spaces. A natural example of a vector field on path space can be obtained by solving a differential equation along the path. In this way, one can obtain a new direction at every point along the path and one might try to deform the path in these directions. Formally, such vector fields on path space are called Driver vector fields. Because the Itô functional is so discontinuous, these vector fields have very bad properties when one takes a classical perspective on path space. However, the Driver vector fields are far more regular when regarded as vector fields on rough path space. This chapter proves that one can actually solve the ordinary differential equations that they define and produce flows on path space corresponding to Driver vector fields. By constructing these flows, one is implicitly constructing a solution to a rough non linear equation. One should not be surprised that, in general, the solutions blow up after a short time interval.

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