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Uniqueness of signature for simple curves. (English) Zbl 1294.60063
Summary: 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, \(1 \leq p < 2\), 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 \(\text{SLE}_\kappa\) null set, where \(0 < \kappa \leq 4\), 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.

60G17 Sample path properties
60H05 Stochastic integrals
26A18 Iteration of real functions in one variable
26A42 Integrals of Riemann, Stieltjes and Lebesgue type
Full Text: DOI
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