Journal article
Rectifiable paths with polynomial log‐signature are straight lines
- Abstract:
- The 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.
- Publication status:
- Published
- Peer review status:
- Peer reviewed
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- Files:
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(Preview, Accepted manuscript, pdf, 328.1KB, Terms of use)
-
- Publisher copy:
- 10.1112/blms.13110
Authors
+ Lloyd's Register Foundation
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- Funder identifier:
- https://ror.org/057q4mw47
- Grant:
- G0095
+ Deutsche Forschungsgemeinschaft
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- Funder identifier:
- https://ror.org/018mejw64
- Grant:
- EXC‐2046/1
+ Engineering and Physical Sciences Research Council
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- Funder identifier:
- https://ror.org/0439y7842
- Grant:
- EP/N510129/1
- EP/S026347/1
- Publisher:
- Wiley
- Journal:
- Bulletin of the London Mathematical Society More from this journal
- Volume:
- 56
- Issue:
- 9
- Pages:
- 2922-2934
- Publication date:
- 2024-07-04
- Acceptance date:
- 2024-05-27
- DOI:
- EISSN:
-
1469-2120
- ISSN:
-
0024-6093
- Language:
-
English
- Keywords:
- Pubs id:
-
2014102
- Local pid:
-
pubs:2014102
- Source identifiers:
-
W4400368033
- Deposit date:
-
2026-06-18
- ARK identifier:
Terms of use
- Copyright holder:
- Friz et al.
- Copyright date:
- 2024
- Rights statement:
- © 2024 The Author(s). The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
- Notes:
- This is the accepted manuscript version of the article. The final version is available online from Wiley at https://dx.doi.org/10.1112/blms.13110
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