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From random walks to rough paths

Authors: Emmanuel Breuillard, Peter Friz and Martin Huesmann
Journal: Proc. Amer. Math. Soc. 137 (2009), 3487-3496
MSC (2000): Primary 60B15
Published electronically: May 20, 2009
MathSciNet review: 2515418
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Abstract: Donsker's invariance principle is shown to hold for random walks in rough path topology. As an application, we obtain Donsker-type weak limit theorems for stochastic integrals and differential equations.

References [Enhancements On Off] (What's this?)

  • 1. E. Breuillard.
    Random walks on lie groups.
    Preprint, 2007.
  • 2. Kuo-Tsai Chen.
    Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula.
    Ann. of Math. (2), 65:163-178, 1957. MR 0085251 (19:12a)
  • 3. Pierre Crépel and Albert Raugi.
    Théorème central limite sur les groupes nilpotents.
    Ann. Inst. H. Poincaré Sect. B (N.S.), 14(2):145-164, 1978. MR 507730 (82a:60016)
  • 4. Peter K. Friz and Nicolas B. Victoir.
    Multidimensional stochastic processes as rough paths: theory and applications.
    Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2009,
    to appear. Draft available at
  • 5. Ulf Grenander.
    Probabilities on algebraic structures.
    John Wiley & Sons Inc., New York, 1963. MR 0259969 (41:4598)
  • 6. John Lamperti.
    On convergence of stochastic processes.
    Trans. Amer. Math. Soc., 104:430-435, 1962. MR 0143245 (26:804)
  • 7. Terry Lyons.
    Differential equations driven by rough signals.
    Rev. Mat. Iberoamericana, 14(2):215-310, 1998. MR 1654527 (2000c:60089)
  • 8. Terry Lyons.
    Systems controlled by rough paths.
    In European Congress of Mathematics, pages 269-281. Eur. Math. Soc., Zürich, 2005. MR 2185750 (2006g:34008)
  • 9. Terry Lyons and Zhongmin Qian.
    System Control and Rough Paths. Oxford Mathematical Monographs,
    Oxford University Press, 2002. MR 2036784 (2005f:93001)
  • 10. Terry J. Lyons, Michael Caruana, and Thierry Lévy.
    Differential equations driven by rough paths, volume 1908 of Lecture Notes in Mathematics.
    Springer, Berlin, 2007.
    Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6-24, 2004, with an introduction by Jean Picard. MR 2314753 (2009c:60156)
  • 11. Daniel Revuz and Marc Yor.
    Continuous martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].
    Springer-Verlag, Berlin, third edition, 1999. MR 1725357 (2000h:60050)
  • 12. Daniel W. Stroock and S. R. S. Varadhan.
    Limit theorems for random walks on Lie groups.
    Sankhyā Ser. A, 35(3):277-294, 1973. MR 0517406 (58:24457)
  • 13. Joseph C. Watkins.
    Donsker's invariance principle for Lie groups.
    Ann. Probab., 17(3):1220-1242, 1989. MR 1009453 (90m:60013)
  • 14. Donald Wehn.
    Probabilities on Lie groups.
    Proc. Nat. Acad. Sci. U.S.A., 48:791-795, 1962. MR 0153042 (27:3011)

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Additional Information

Emmanuel Breuillard
Affiliation: Laboratoire de Mathematiques, Universite Paris 11, 91405 Orsay, France

Peter Friz
Affiliation: Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge CB3 0WB, United Kingdom

Martin Huesmann
Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany

Keywords: Donskers's theorem, rough paths
Received by editor(s): October 16, 2008
Published electronically: May 20, 2009
Communicated by: Peter A. Clarkson
Article copyright: © Copyright 2009 American Mathematical Society

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