Flows Driven by Banach Space-Valued Rough Paths

  • Ismaël Bailleul
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)


We show in this note how the machinery of \(\mathcal{C}^{1}\)-approximate flows devised in the work Flows driven by rough paths, and applied there to reprove and extend most of the results on Banach space-valued rough differential equations driven by a finite dimensional rough path can be used to deal with rough differential equations driven by an infinite dimensional Banach space-valued weak geometric Hölder p-rough paths, for any p > 2, giving back Lyons’ theory in its full force in a simple way.


Flows Infinite dimensional rough paths Rough differential equations 



This research was partially supported by an ANR grant “Retour post-doctorant”. The author warmly thanks the UniversitÃl’ de Bretagne Occidentale where part of this work was written.


  1. 1.
    T. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoam. 14(2), 215–310 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    M. Gubinelli, Controlling rough paths. J. Funct. Anal. 216(1), 86–140 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    A. Deya, M. Gubinelli, S. Tindel, Non-linear rough heat equations. Probab. Theory Relat. Fields 153(1–2), 97–147 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    M. Gubinelli, Rough solutions for the periodic Korteweg–de Vries equation. Commun. Pure Appl. Anal. 11(2), 709–733 (2012)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    F. Flandoli, M. Gubinelli, E. Priola, Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180(1), 1–53 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    M. Gubinelli, S. Tindel, Rough evolution equations. Ann. Probab. 38(1), 1–75 (2010)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    M. Hairer, Solving the KPZ equation. Ann. Maths 178(2), 559–664 (2013)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    M. Hairer, A theory of regularity structures. Preprint (2013)Google Scholar
  9. 9.
    P. Friz, N. Victoir, Euler estimates for rough differential equations. J. Differ. Equ. 244(2), 388–412 (2008)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    P. Friz, N. Victoir, Multidimensional Stochastic Processes as Rough Paths. CUP, Encyclopedia of Mathematics and its Applications, vol. 89. (2010)Google Scholar
  11. 11.
    A.M. Davie, Differential equations driven by rough paths: an approach via discrete approximation. Appl. Math. Res. Express AMRX 2, 40 (2007)MathSciNetGoogle Scholar
  12. 12.
    I. Bailleul, Flows driven by rough paths. Preprint. arXiv:1203.0888 (2012)Google Scholar
  13. 13.
    Y. Boutaib, L.G. Gyurko, T. Lyons, D. Yang, Dimension-free estimates of rough differential equations. Preprint (2013)Google Scholar
  14. 14.
    T.J. Lyons, M. Caruana, Th. Lévy, Differential Equations Driven by Rough Paths. Lecture Notes in Mathematics, vol. 1908 (Springer, New York, 2007)Google Scholar
  15. 15.
    D. Feyel, A. de La Pradelle, Curvilinear integrals along enriched paths. Electron. J. Probab. 11, 860–892 (2006)CrossRefMathSciNetGoogle Scholar
  16. 16.
    D. Feyel, A. de La Pradelle, G. Mokobodzki, A non-commutative sewing lemma. Electron. Commun. Probab. 13, 24–34 (2008)MathSciNetzbMATHGoogle Scholar
  17. 17.
    I. Bailleul, Path-dependent rough differential equations. Preprint. arXiv:1309.1291 (2013)Google Scholar
  18. 18.
    T. Lyons, Z. Qian, System Control and Rough Paths. Oxford Mathematical Monographs. (2002)Google Scholar
  19. 19.
    R. Azencott, Formule de Taylor stochastique et développements asymptotiques d’intégrales de Feynman. Séminaire de Probabilités, vol. XVI (1982)Google Scholar
  20. 20.
    G. Ben Arous, Flots et séries de Taylor stochastiques. Prob. Theory Relat. Fields 81, 29–77, 1989.CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    F. Castell, Asymptotic expansion of stochastic flows. Prob. Theory Relat. Fields 96, 225–239 (1993)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    A. Lejay, On rough differential equations. Electron. J. Probab. 12, 341–364 (2009)MathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.IRMARRennesFrance

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