## UNIQUENESS FOR THE SIGNATURE OF A PATH OF BOUNDED VARIATION AND THE REDUCED PATH GROUP (2006)

Citations: | 6 - 1 self |

### BibTeX

@MISC{Hambly06uniquenessfor,

author = {B. M. Hambly and Terry and J. Lyons},

title = {UNIQUENESS FOR THE SIGNATURE OF A PATH OF BOUNDED VARIATION AND THE REDUCED PATH GROUP},

year = {2006}

}

### OpenURL

### Abstract

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 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. 1.

### Citations

102 |
degenerations of hyperbolic structures
- Valuations
(Show Context)
Citation Context ... until the first time they separate. (This idea dates back at least to Kolmogorov and his introduction of filtrations). Another popular and equivalent approach to continuous trees is through R-trees (=-=[10]-=- p425 and the references there). Interestingly, analysts and probabilists have generally rejected the abstract tree as too wild an object, and usually add extra structure, essentially a second topolog... |

32 | Differential equations driven by rough paths (Lectures from the - Lyons, Caruana, et al. - 2007 |

30 |
Brownian excursions, trees and measure-valued branching processes
- GALL
- 1991
(Show Context)
Citation Context ...rem to the bounded variation setting. For this second goal we need a notion of tree-like path, our definition codes Rtrees by positive continuous functions on the line, as developed, for instance, in =-=[5]-=-. Definition 1.3. Xt, t ∈ [0, T] is a tree-like path in V if there exists a positive real valued continuous function h defined on [0, T] such that h (0) = h (T) = 0 and such that ‖Xt − Xs‖V ≤ h (s) + ... |

12 | Hyperbolic Geometry
- Cannon, Floyd, et al.
- 1997
(Show Context)
Citation Context ...then {y|Id (y, x) = 0} is the tangent space to H in R d+1 and moreover Id (z, z) is positive definite on z ∈ {y|Id (y, x) = 0} and so this inner product is a Riemannian structure on H). In fact, (see =-=[1]-=-, p83) distances in H can be calculated using Id (3.1) − coshd(x, y) = Id (x, y) If SO (Id) denotes the group of matrices with positive determinant preserving 3 the quadratic form Id then one can prov... |

8 |
Integration of paths—a faithful representation of paths by non-commutative formal power series
- Chen
- 1958
(Show Context)
Citation Context ... an equivalence relation and identify the sense in which the signature of a path determines the path. The first detailed studies of the iterated integrals of paths are due to K. T. Chen. In fact Chen =-=[2]-=- proves the following theorems which are clear precursors to our own results: Chen Theorem 1: Let dγ1, · · · , dγd be the canonical 1-forms on Rd . If α, β ∈ [a, b] → Rd are irreducible piecewise regu... |

3 | A Non-quasiconvex Subgroup of a Hyperbolic Group with an Exotic Limit Set
- Kapovich
- 1995
(Show Context)
Citation Context ...hs of paths5 between the two vertices a, b in the graph. Then g is a geodesic metric on V . Trees are exactly the graphs that give rise to 0-hyperbolic metrics in the sense of Gromov (see for example =-=[4]-=-). (4). There are many ways to enumerate the edges and nodes of a finite rooted tree. One way is to think of a family tree recording the descendants of a single individual (the root). Start with the r... |

2 |
Smoothness of Itô maps and diffusion processes on path spaces
- Li, Lyons
(Show Context)
Citation Context ...nd of the development of γ into Hyperbolic space. Amongst these paths γ of fixed length, straight lines maximise ̺ as the developments are geodesics. The function ̺ is a smooth function on path space =-=[6]-=-. Therefore one would expect that for some constant K ̺ (γ) ≥ l − Kε 2 whenever γ is in the ε-neighbourhood (for the appropriate norm) of a straight line. We will make this precise using Taylor’s theo... |

2 |
Rough paths and system control
- Lyons, Qian
- 2003
(Show Context)
Citation Context ...map the signature map and sometimes denote it by S : X → S (X) when this helps our presentation. The signature of X is a natural object to study. The map X → X is a homomorphism (c.f. Chen’s identity =-=[7]-=-) from the monoid of paths with concatenation to (a group embedded in) the algebra T (V ). The signature X(= X0,T) can be computed by solving the differential equation = X0,u ⊗ dXu X0,0 = (1, 0, 0, . ... |

2 | On the radius of convergence of the logarithmic signature
- Lyons, Sidorova
(Show Context)
Citation Context ...ve estimates to finite length paths. In order to do this we need to discuss the development of a path into a suitable version of hyperbolic space - a technique that has more recently proved useful in =-=[8]-=-. Using this idea we obtain a quantitative estimate on the difference between the length of the developed path and its chord in terms of the modulus of continuity of the derivative of the path. This a... |

2 |
Dirichlet problems on Riemann surfaces and conformal mappings
- Ohtsuka
- 1951
(Show Context)
Citation Context ...γ in V and let the occupation measure µ on (V, B(V )) be denoted µ (A) = |{s < T |γ (s) ∈ A}| , A ⊂ V. Let n (x) be the number of points on [0, T] corresponding under γ to x ∈ E. By the area formulae =-=[11]-=- p125-126, one has the total variation, or length, of the path γ is given by ∫ (5.1) V ar (γ) = n (x)Λ1 (dx) , where Λ1 is one dimensional Hausdorff measure. Moreover, for any continuous function f ∫ ... |