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 characterises the path up to a generalised form of reparametrisation. It is a classical result of K. T. 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 reparametrisation. 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 results from rough path theory, in particular that the signature characterises a rough path up to reparametrisation.
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Cite
@article{arxiv.2305.19210,
title = {Rectifiable paths with polynomial log-signature are straight lines},
author = {Peter K. Friz and Terry Lyons and Anna Seigal},
journal= {arXiv preprint arXiv:2305.19210},
year = {2023}
}
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11 pages