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.

  • [cfkp] J. W. Cannon, W. J. Floyd, R. Kenyon, and W. R. Parry, "Hyperbolic Geometry," in Flavors of Geometry, Cambridge: Cambridge Univ. Press, 1997, pp. 59-115.
    @incollection {cfkp, MRKEY = {1491098},
      AUTHOR = {Cannon, James W. and Floyd, William J. and Kenyon, Richard and Parry, Walter R.},
      TITLE = {Hyperbolic Geometry},
      BOOKTITLE = {Flavors of Geometry},
      SERIES = {Math. Sci. Res. Inst. Publ.},
      NUMBER = {31},
      PAGES = {59--115},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1997},
      MRCLASS = {57M50 (57N10)},
      MRNUMBER = {99c:57036},
      MRREVIEWER = {Lee Mosher},
      ZBLNUMBER = {0899.51012},
      }
  • [KTCHENone] Go to document K. Chen, "Integration of paths—a faithful representation of paths by non-commutative formal power series," Trans. Amer. Math. Soc., vol. 89, pp. 395-407, 1958.
    @article {KTCHENone, MRKEY = {0106258},
      AUTHOR = {Chen, Kuo-sai},
      TITLE = {Integration of paths---a faithful representation of paths by non-commutative formal power series},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {89},
      YEAR = {1958},
      PAGES = {395--407},
      ISSN = {0002-9947},
      MRCLASS = {22.00},
      MRNUMBER = {21 \#4992},
      MRREVIEWER = {J. C. Moore},
      DOI = {10.2307/1993193},
      ZBLNUMBER = {0097.25803},
      }
  • [fawcett] T. A. Fawcett, Ph.D. thesis.
    @misc{fawcett,
      author={Fawcett, T.A.},
      TITLE={Ph.D. thesis},
      NOTE={Mathematical Institute, University of Oxford, 2002},
      }
  • [HLnotes] B. M. Hambly and T. J. Lyons, Some notes on trees and paths.
    @misc{HLnotes,
      author={Hambly, B.M. and Lyons, T.J.},
      TITLE={Some notes on trees and paths},
      NOTE={http://arxiv.org/abs/0809.1365},
      }
  • [snake] J. Le Gall, "Brownian excursions, trees and measure-valued branching processes," Ann. Probab., vol. 19, iss. 4, pp. 1399-1439, 1991.
    @article {snake, MRKEY = {1127710},
      AUTHOR = {Le Gall, Jean-Fran{ç}ois},
      TITLE = {Brownian excursions, trees and measure-valued branching processes},
      JOURNAL = {Ann. Probab.},
      FJOURNAL = {The Annals of Probability},
      VOLUME = {19},
      YEAR = {1991},
      NUMBER = {4},
      PAGES = {1399--1439},
      ISSN = {0091-1798},
      CODEN = {APBYAE},
      MRCLASS = {60J80 (60G57 60J55 60J65)},
      MRNUMBER = {93b:60195},
      MRREVIEWER = {Klaus Fleischmann},
      ZBLNUMBER = {0753.60078},
      }
  • [LixLy2006] Go to document X. Li and T. J. Lyons, "Smoothness of Itô maps and diffusion processes on path spaces. I," Ann. Sci. École Norm. Sup., vol. 39, iss. 4, pp. 649-677, 2006.
    @article {LixLy2006, MRKEY = {2290140},
      AUTHOR = {Li, Xiang-Dong and Lyons, Terry J.},
      TITLE = {Smoothness of {I}tô maps and diffusion processes on path spaces. {I}},
      JOURNAL = {Ann. Sci. École Norm. Sup.},
      FJOURNAL = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
      VOLUME = {39},
      YEAR = {2006},
      NUMBER = {4},
      PAGES = {649--677},
      ISSN = {0012-9593},
      CODEN = {ASENAH},
      MRCLASS = {60G17 (58J65 60J65)},
      MRNUMBER = {2008a:60096},
      MRREVIEWER = {Isamu D{ô}ku},
      DOI = {10.1016/j.ansens.2006.07.001},
      ZBLNUMBER = {1127.60033},
      }
  • [lyons] T. J. Lyons and Z. Qian, "System control and rough paths," in Oxford Math. Monogr., U.K.: Oxford University Press, 2002.
    @incollection {lyons,
      author={Lyons, Terry J. and Qian, Z.},
      TITLE={System control and rough paths},
      BOOKTITLE={Oxford Math. Monogr.},
      PUBLISHER={Oxford University Press},
      ADDRESS={U.K.},
      YEAR={2002},
      MRNUMBER={2005f:93001},
      ZBLNUMBER={1029.93001},
      }
  • [LySiDoob] T. J. Lyons and N. Sidorova, "On the radius of convergence of the logarithmic signature," Illinois J. Math., vol. 50, iss. 1-4, pp. 763-790, 2006.
    @article {LySiDoob, MRKEY = {2247845},
      AUTHOR = {Lyons, Terry J. and Sidorova, Nadia},
      TITLE = {On the radius of convergence of the logarithmic signature},
      JOURNAL = {Illinois J. Math.},
      FJOURNAL = {Illinois Journal of Mathematics},
      VOLUME = {50},
      YEAR = {2006},
      NUMBER = {1-4},
      PAGES = {763--790},
      ISSN = {0019-2082},
      CODEN = {IJMTAW},
      MRCLASS = {60H10 (34A34 34F05 93C35)},
      MRNUMBER = {2007m:60165},
      MRREVIEWER = {Habib Ouerdiane},
      ZBLNUMBER = {1103.60060},
      }
  • [LyStFlour] T. J. Lyons, M. Caruana, and T. Lévy, Differential Equations Driven by Rough Paths, New York: Springer-Verlag, 2007.
    @book {LyStFlour, MRKEY = {2314753},
      AUTHOR = {Lyons, Terry J. and Caruana, Michael and L{é}vy, Thierry},
      TITLE = {Differential Equations Driven by Rough Paths},
      SERIES = {Lecture Notes in Mathematics},
      NUMBER = {1908},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2007},
      PAGES = {xviii+109},
      ISBN = {978-3-540-71284-8; 3-540-71284-4},
      MRCLASS = {60H10 (60-02 60H07)},
      MRNUMBER = {2009c:60156},
      MRREVIEWER = {Bohdan Maslowski},
      ZBLNUMBER = {05161505},
      }
  • [ohtsuka] Go to document M. Ohtsuka, "Dirichlet problems on Riemann surfaces and conformal mappings," Nagoya Math. J., vol. 3, pp. 91-137, 1951.
    @article {ohtsuka, MRKEY = {0045830},
      AUTHOR = {Ohtsuka, Makoto},
      TITLE = {Dirichlet problems on {R}iemann surfaces and conformal mappings},
      JOURNAL = {Nagoya Math. J.},
      FJOURNAL = {Nagoya Mathematical Journal},
      VOLUME = {3},
      YEAR = {1951},
      PAGES = {91--137},
      ISSN = {0027-7630},
      MRCLASS = {30.0X},
      MRNUMBER = {13,642f},
      MRREVIEWER = {R. de Possel},
      URL = {http://projecteuclid.org/getRecord?id=euclid.nmj/1118799225},
      ZBLNUMBER = {0043.30004},
      }
  • [Pis] G. Pisier, Factorization of Linear Operators and Geometry of Banach Spaces, Providence, RI: Amer. Math. Soc., 1986.
    @book {Pis, MRKEY = {829919},
      AUTHOR = {Pisier, Gilles},
      TITLE = {Factorization of Linear Operators and Geometry of {B}anach Spaces},
      SERIES = {CBMS Regional Conference Series in Mathematics},
      NUMBER = {60},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS={Providence, RI},
      YEAR = {1986},
      PAGES = {x+154},
      ISBN = {0-8218-0710-2},
      MRCLASS = {47A68 (46B20 47B10)},
      MRNUMBER = {88a:47020},
      MRREVIEWER = {Bernd Rosenberger},
      ZBLNUMBER = {0588.46010},
      }
  • [Sh] G. R. Shorack, "Convergence of reduced empirical and quantile processes with application to functions of order statistics in the non-I.I.D. case," Ann. Statist., vol. 1, pp. 146-152, 1973.
    @article {Sh, MRKEY = {0336776},
      AUTHOR = {Shorack, Galen R.},
      TITLE = {Convergence of reduced empirical and quantile processes with application to functions of order statistics in the non-{I}.{I}.{D}. case},
      JOURNAL = {Ann. Statist.},
      FJOURNAL = {The Annals of Statistics},
      VOLUME = {1},
      YEAR = {1973},
      PAGES = {146--152},
      ISSN = {0090-5364},
      MRCLASS = {60B10 (62G30)},
      MRNUMBER = {49 \#1549},
      MRREVIEWER = {G. E. Noether},
      ZBLNUMBER={0255.62044},
      }

Authors

Ben Hambly

Mathematical Institute
Oxford University
24-29 St. Giles
Oxford OX1 3LB
England

Terry Lyons

Mathematical Institute
Oxford University
24-29 St. Giles
Oxford OX1 3LB
England