Comptes Rendus
no. 2
Mathematical analysis/Functional analysis
A non-vanishing property for the signature of a path
[Une propriété de non-nullité pour la signature d'un chemin]

Présenté par : the Editorial Board

Horatio Boedihardjo1  ; Xi Geng2
1 Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom
2 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States
Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 120-129.

Nous montrons que la signature d'un chemin continu, de longueur finie, dans un espace de Banach réel, ne peut pas avoir une infinité de composantes nulles, à moins d'être de type arbre. En particulier, cela nous permet de renforcer un théorème limite pour la signature, récemment obtenu par Chang, Lyons et Ni. Notre démonstration repose sur un argument de complexification et des résultats profonds d'approximations polynomiales holomorphes de la théorie de plusieurs variables complexes.

We prove that a continuous path with finite length in a real Banach space cannot have infinitely many zero components in its signature unless it is tree-like. In particular, this allows us to strengthen a limit theorem for signature recently proved by Chang, Lyons, and Ni. What lies at the heart of our proof is a complexification idea together with deep results from holomorphic polynomial approximations in the theory of several complex variables.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.12.006
Horatio Boedihardjo 1 ; Xi Geng 2

1 Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom
2 Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States 
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Horatio Boedihardjo; Xi Geng. A non-vanishing property for the signature of a path. Comptes Rendus. Mathématique, Volume 357 (2019) no. 2, pp. 120-129. doi : 10.1016/j.crma.2018.12.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.12.006/

[1] H. Boedihardjo; X. Geng Tail asymptotics of the Brownian signature, Trans. Amer. Math. Soc. (2016) (in press) | arXiv | DOI

[2] H. Boedihardjo; X. Geng; T. Lyons; D. Yang The signature of a rough path: uniqueness, Adv. Math., Volume 293 (2016), pp. 720-737

[3] J. Chang; T. Lyons; H. Ni Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length, C. R. Acad. Sci. Paris, Ser. I, Volume 356 (2018) no. 1, pp. 720-724

[4] J. Chang, T. Lyons, H. Ni, Corrigendum to “Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length”, 2018.

[5] J. Diestel; J.J. Uhl Vector Measures, American Mathematical Society, 1977

[6] B. Hambly; T. Lyons Uniqueness for the signature of a path of bounded variation and the reduced path group, Ann. of Math. (2), Volume 171 (2010) no. 1, pp. 109-167

[7] N. Levenberg Approximation in CN, Surv. Approx. Theory, Volume 2 (2006), pp. 92-140

[8] J.C. Rosales; P.A. García-Sánchez Numerical Semigroups, Springer, 2009

[9] A.E. Taylor Analysis in complex Banach spaces, Bull. Amer. Math. Soc., Volume 49 (1943), pp. 652-659

[10] G. van Zyl Complexification of the projective and injective tensor products, Stud. Math., Volume 189 (2008) no. 2, pp. 105-112

[11] B.M. Weinstock On the polynomial convexity of the union of two maximal totally real subspaces of Cn, Math. Ann., Volume 282 (1988), pp. 131-138

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