@article{discovery1555643, year = {2014}, title = {Uniqueness of signature for simple curves}, number = {6}, journal = {Journal of Functional Analysis}, note = {This version is the author accepted manuscript. For information on re-use, please refer to the publisher's terms and conditions.}, pages = {1778--1806}, volume = {267}, month = {September}, issn = {0022-1236}, keywords = {Rough path theory; Uniqueness of signature problem; SLE curves}, url = {https://doi.org/10.1016/j.jfa.2014.06.006}, 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{$\leq$}p{\ensuremath{<}}21{$\leq$}p{\ensuremath{<}}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{\ensuremath{\kappa}}SLE{\ensuremath{\kappa}} null set, where 0{\ensuremath{<}}{\ensuremath{\kappa}}{$\leq$}40{\ensuremath{<}}{\ensuremath{\kappa}}{$\leq$}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.}, author = {Boedihardjo, H and Ni, H and Qian, Z} }