## From random walks to rough paths

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- by Emmanuel Breuillard, Peter Friz and Martin Huesmann PDF
- Proc. Amer. Math. Soc.
**137**(2009), 3487-3496 Request permission

## 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

- E. Breuillard. Random walks on lie groups. Preprint, 2007.
- Kuo-Tsai Chen,
*Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula*, Ann. of Math. (2)**65**(1957), 163–178. MR**85251**, DOI 10.2307/1969671 - Pierre Crépel and Albert Raugi,
*Théorème central limite sur les groupes nilpotents*, Ann. Inst. H. Poincaré Sect. B (N.S.)**14**(1978), no. 2, 145–164 (French, with English summary). MR**507730** - 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 http://www.statslab.cam.ac.uk/~peter/RoughPathsBook/. - Ulf Grenander,
*Probabilities on algebraic structures*, 2nd ed., Almqvist & Wiksell, Stockholm; John Wiley & Sons Inc., New York-London, 1968. MR**0259969** - John Lamperti,
*On convergence of stochastic processes*, Trans. Amer. Math. Soc.**104**(1962), 430–435. MR**143245**, DOI 10.1090/S0002-9947-1962-0143245-1 - Terry J. Lyons,
*Differential equations driven by rough signals*, Rev. Mat. Iberoamericana**14**(1998), no. 2, 215–310. MR**1654527**, DOI 10.4171/RMI/240 - Terry Lyons,
*Systems controlled by rough paths*, European Congress of Mathematics, Eur. Math. Soc., Zürich, 2005, pp. 269–281. MR**2185750** - Terry Lyons and Zhongmin Qian,
*System control and rough paths*, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2002. Oxford Science Publications. MR**2036784**, DOI 10.1093/acprof:oso/9780198506485.001.0001 - Terry J. Lyons, Michael Caruana, and Thierry Lévy,
*Differential equations driven by rough paths*, Lecture Notes in Mathematics, vol. 1908, Springer, Berlin, 2007. Lectures from the 34th Summer School on Probability Theory held in Saint-Flour, July 6–24, 2004; With an introduction concerning the Summer School by Jean Picard. MR**2314753** - Daniel Revuz and Marc Yor,
*Continuous martingales and Brownian motion*, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR**1725357**, DOI 10.1007/978-3-662-06400-9 - Daniel W. Stroock and S. R. S. Varadhan,
*Limit theorems for random walks on Lie groups*, Sankhyā Ser. A**35**(1973), no. 3, 277–294. MR**517406** - Joseph C. Watkins,
*Donsker’s invariance principle for Lie groups*, Ann. Probab.**17**(1989), no. 3, 1220–1242. MR**1009453** - Donald Wehn,
*Probabilities on Lie groups*, Proc. Nat. Acad. Sci. U.S.A.**48**(1962), 791–795. MR**153042**, DOI 10.1073/pnas.48.5.791

## 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
- MR Author ID: 656436
- Email: P.K.Friz@statslab.cam.ac.uk
**Martin Huesmann**- Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Germany
- Received by editor(s): October 16, 2008
- Published electronically: May 20, 2009
- Communicated by: Peter A. Clarkson
- © Copyright 2009 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**137**(2009), 3487-3496 - MSC (2000): Primary 60B15
- DOI: https://doi.org/10.1090/S0002-9939-09-09930-4
- MathSciNet review: 2515418