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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.

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