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

By Ben M. Hambly, Terry Lyons. 2005

329 citations recorded by the source corpus.

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 one special tree reduced path. The set of these paths is the Reduced Path Group. It is a continuous analogue to the group of reduced words. The signature of the path is a power series whose coefficients are 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 reparameterisation in the general setting. It is suggestive to think of this result as a non-commutative 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.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/539086ab9010af9e70f73e14ec07d60a6c021843

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

Citations in the corpus

Neural Controlled Differential Equations for Irregular Time Series (2020, background): “Time as a channel CDEs exhibit a tree-like invariance property [18].”
Sparse arrays of signatures for online character recognition (2013, methodology): “More recently, path signatures have been used to solve diﬀerential equations driven by rough paths [11, The signature extracts enough information from the path to solve any linear diﬀerential equation and uniquely characterizes paths of ﬁnite length [5].”
Characteristic functions of measures on geometric rough paths (2013, background): “The exact geometric equivalence of paths of bounded variation possessing the same signature was first described by Hambly and Lyons [19], and recently extended to all geometric rough paths [2].”
The Signature of a Rough Path: Uniqueness (2014, methodology): “Our proof does not make use of Hambly–Lyons’ result and hence in the case p = 1, we have given a new and simple proof of Hambly–Lyons’ theorem in [17].”
Discretely sampled signals and the rough Hoff process (2013, background): “The rough path lift X, also known as the signature, is a well-defined transform in the sense than no information is lost (see [14]).”
General Rough integration, Levy Rough paths and a Levy--Kintchine type formula (2012, background): “[7, 16, 22, 5] Recall that for a smooth path X : [0, T ] → R, its signature S = S (X) is given by the group-like element”
Optimal stopping with signatures (2021, background): “To better understand the importance of the signature, let us ﬁrst recall that the signature X ∞ determines the underlying path X (up to “tree-like excursions”), which was ﬁrst proved in [HL10] for paths X of bounded variation and later extended to less regular paths.”