TY  - JOUR
Y1  - 2014/09/15/
UR  - https://doi.org/10.1016/j.jfa.2014.06.006
ID  - discovery1555643
N2  - 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 SLE?SLE? null set, where 0<??40<??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.
SP  - 1778
VL  - 267
JF  - Journal of Functional Analysis
A1  - Boedihardjo, H
A1  - Ni, H
A1  - Qian, Z
AV  - public
TI  - Uniqueness of signature for simple curves
SN  - 0022-1236
EP  - 1806
IS  - 6
N1  - This version is the author accepted manuscript. For information on re-use, please refer to the publisher?s terms and conditions.
KW  - Rough path theory; 
Uniqueness of signature problem; 
SLE curves
ER  -