Journal of the Mathematical Society of Japan

The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations

Terry J. LYONS and Danyu YANG

Full-text: Open access


We give an overview of the recent approach to the integration of rough paths that reduces the problem to an inhomogeneous analogue of the classical Young integration [13]. As an application, we extend an argument of Schwartz [11] to rough differential equations, and prove the existence, uniqueness and continuity of the solution, which is applicable when the driving path takes values in nilpotent Lie group or Butcher group.

Article information

J. Math. Soc. Japan, Volume 67, Number 4 (2015), 1681-1703.

First available in Project Euclid: 27 October 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H99: None of the above, but in this section
Secondary: 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

rough paths theory Young integral integrable one-forms universal limit theorem


LYONS, Terry J.; YANG, Danyu. The theory of rough paths via one-forms and the extension of an argument of Schwartz to rough differential equations. J. Math. Soc. Japan 67 (2015), no. 4, 1681--1703. doi:10.2969/jmsj/06741681.

Export citation


  • J. C. Butcher, An algebraic theory of integration methods, Math. Comp., 26 (1972), 79–106.
  • A. Connes and D. Kreimer, Hopf algebras, renormalization and noncommutative geometry, In Quantum field theory: perspective and prospective, Springer, 1999, 59–109.
  • M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86–140.
  • M. Gubinelli, Ramification of rough paths, Journal of Differential Equations, 248 (2010), 693–721.
  • K. Hara and M. Hino, Fractional order Taylor's series and the neo-classical inequality, Bull. Lond. Math. Soc., 42 (2010), 467–477.
  • T. Lyons, Differential equations driven by rough signals, I, An extension of an inequality of L. C. Young, Math. Res. Lett., 1 (1994), 451–464.
  • T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215–310.
  • T. J. Lyons, M. Caruana and T. Lévy, Differential equations driven by rough paths, Springer, 2007.
  • T. J. Lyons and D. Yang, Integration of time-varying cocyclic one-forms against rough paths, arXiv preprint arXiv:1408.2785, 2014.
  • C. Reutenauer, Free lie algebras, Handbook of algebra, 3, North-Holland, Amsterdam, 2003, 887–903.
  • L. Schwartz, La convergence de la série de Picard pour les EDS (Equations Différentielles Stochastiques), In Séminaire de Probabilités XXIII, Springer, 1989, 343–354.
  • E. M. Stein, Singular integrals and differentiability properties of functions, 2, Princeton university press, 1970.
  • L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Mathematica, 67 (1936), 251–282.