Boedihardjo, H;
Ni, H;
Qian, Z;
(2014)
Uniqueness of signature for simple curves.
Journal of Functional Analysis
, 267
(6)
pp. 1778-1806.
10.1016/j.jfa.2014.06.006.
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Abstract
We propose a topological approach to the problem of determining a curve from its iterated integrals. In particular, we prove that a family of terms in the signature series of a two dimensional closed curve with finite p variation, 1≤p<21≤p<2, are in fact moments of its winding number. This relation allows us to prove that the signature series of a class of simple non-smooth curves uniquely determine the curves. This implies that outside a Chordal SLEκSLEκ null set, where 0<κ≤40<κ≤4, the signature series of curves uniquely determine the curves. Our calculations also enable us to express the Fourier transform of the n-point functions of SLE curves in terms of the expected signature of SLE curves. Although the techniques used in this article are deterministic, the results provide a platform for studying SLE curves through the signatures of their sample paths.
| Type: | Article |
|---|---|
| Title: | Uniqueness of signature for simple curves |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1016/j.jfa.2014.06.006 |
| Publisher version: | https://doi.org/10.1016/j.jfa.2014.06.006 |
| Language: | English |
| Additional information: | This version is the author accepted manuscript. For information on re-use, please refer to the publisher’s terms and conditions. |
| Keywords: | Rough path theory; Uniqueness of signature problem; SLE curves |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/1555643 |
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