Rectifiable paths with polynomial log‐signature are straight lines

Author:

Friz Peter K.1,Lyons Terry2,Seigal Anna3

Affiliation:

1. Institut für Mathematik Technische Universität Berlin and WIAS Berlin Berlin Germany

2. Mathematical Institute University of Oxford Oxford UK

3. John A. Paulson School of Engineering and Applied Sciences Harvard University Cambridge Massachusetts USA

Abstract

AbstractThe signature of a rectifiable path is a tensor series in the tensor algebra whose coefficients are definite iterated integrals of the path. The signature characterizes the path up to a generalized form of reparameterization. It is a classical result of Chen that the log‐signature (the logarithm of the signature) is a Lie series. A Lie series is polynomial if it has finite degree. We show that the log‐signature is polynomial if and only if the path is a straight line up to reparameterization. Consequently, the log‐signature of a rectifiable path either has degree one or infinite support. Though our result pertains to rectifiable paths, the proof uses rough path theory, in particular that the signature characterizes a rough path up to reparameterization.

Funder

Deutsche Forschungsgemeinschaft

Engineering and Physical Sciences Research Council

Lloyd's Register Foundation

Government of the United Kingdom

Office for National Statistics

National Science Foundation

Publisher

Wiley

同舟云学术

1.学者识别学者识别

2.学术分析学术分析

3.人才评估人才评估

"同舟云学术"是以全球学者为主线,采集、加工和组织学术论文而形成的新型学术文献查询和分析系统,可以对全球学者进行文献检索和人才价值评估。用户可以通过关注某些学科领域的顶尖人物而持续追踪该领域的学科进展和研究前沿。经过近期的数据扩容,当前同舟云学术共收录了国内外主流学术期刊6万余种,收集的期刊论文及会议论文总量共计约1.5亿篇,并以每天添加12000余篇中外论文的速度递增。我们也可以为用户提供个性化、定制化的学者数据。欢迎来电咨询!咨询电话:010-8811{复制后删除}0370

www.globalauthorid.com

TOP

Copyright © 2019-2024 北京同舟云网络信息技术有限公司
京公网安备11010802033243号  京ICP备18003416号-3