Uniqueness for the signature of a path of bounded variation and the reduced path group

Abstract

We introduce the notions of tree-like path and tree-like equivalence between paths and prove that the latter is an equivalence relation for paths of finite length.We show that the equivalence classes form a group with some similarity to a free group, and that in each class there is a unique path that is tree reduced.The set of these paths is the Reduced Path Group.It is a continuous analogue of the group of reduced words.The signature of the path is a power series whose coefficients are certain tensor valued definite iterated integrals of the path.We identify the paths with trivial signature as the tree-like paths, and prove that two paths are in tree-like equivalence if and only if they have the same signature.In this way, we extend Chen's theorems on the uniqueness of the sequence of iterated integrals associated with a piecewise regular path to finite length paths and identify the appropriate extended meaning for parametrisation in the general setting.It is suggestive to think of this result as a noncommutative analogue of the result that integrable functions on the circle are determined, up to Lebesgue null sets, by their Fourier coefficients.As a second theme we give quantitative versions of Chen's theorem in the case of lattice paths and paths with continuous derivative, and as a corollary derive results on the triviality of exponential products in the tensor algebra.

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