Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length

Jiawei Chang; Terry Lyons; Hao Ni

doi:10.1016/j.crma.2018.05.010

Mathematical analysis

Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length

Presented by: the Editorial Board

Jiawei Chang ¹; Terry Lyons ^1,²; Hao Ni ^3,²

¹ Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG, United Kingdom
² The Alan Turing Institute, British Library, 96 Euston Road, London NW1 2DB, United Kingdom
³ Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom

Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 720-724.

Abstract
Résumé

For a path of length $L > 0$ , if for all $n \geq 1$ , we multiply the n-th term of the signature by $n! L^{- n}$ , we say that the resulting signature is ‘normalised’. It has been established (T. J. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths, Springer, 2007) that the norm of the n-th term of the normalised signature of a bounded-variation path is bounded above by 1. In this article, we discuss the super-multiplicativity of the norm of the signature of a path with finite length, and prove by Fekete's lemma the existence of a non-zero limit of the n-th root of the norm of the n-th term in the normalised signature as n approaches infinity.

Pour une trajectoire de longueur $L > 0$ , si l'on multiplie le n-ième terme de la signature par $n! L^{- n}$ pour tout $n \geq 1$ , la signature ainsi obtenue est dite « normalisée ». Il a été établi (T. J. Lyons, M. Caruana, T. Lévy, Differential equations driven by rough paths, Springer, 2007) que la norme du n-ième terme de la signature normalisée d'une trajectoire à variation bornée est majorée par 1. Dans cet article, nous étudions la super-multiplicativité de la norme de la signature d'une trajectoire de longueur finie, et nous démontrons, à l'aide du lemme de Fekete, l'existence d'une limite non nulle lorsque n tend l'infini pour la racine n-ième de la norme du n-ième terme de la signature normalisée.

Received: 2018-05-11
Accepted: 2018-05-17
Published online: 2018-05-22

DOI: 10.1016/j.crma.2018.05.010

Author's affiliations:

Jiawei Chang ¹; Terry Lyons ^{1, 2}; Hao Ni ^{3, 2}

¹ Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Rd, Oxford OX2 6GG, United Kingdom
² The Alan Turing Institute, British Library, 96 Euston Road, London NW1 2DB, United Kingdom
³ Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom

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     author = {Jiawei Chang and Terry Lyons and Hao Ni},
     title = {Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {720--724},
     publisher = {Elsevier},
     volume = {356},
     number = {7},
     year = {2018},
     doi = {10.1016/j.crma.2018.05.010},
     language = {en},
}

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Jiawei Chang; Terry Lyons; Hao Ni. Super-multiplicativity and a lower bound for the decay of the signature of a path of finite length. Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 720-724. doi : 10.1016/j.crma.2018.05.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.05.010/

References
Cited by

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[2] 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

[3] T.J. Lyons; M. Caruana; T. Lévy Differential Equations Driven by Rough Paths, Springer, 2007

[4] R.A. Ryan Introduction to Tensor Products of Banach Spaces, Springer Science & Business Media, 2013

[5] J.M. Steele Probability Theory and Combinatorial Optimization, vol. 69, Society for Industrial and Applied Mathematics (SIAM), 1997

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