Abstract
We use simple sub-Riemannian techniques to prove that every weak geometric p-rough path (a geometric p-rough path in the sense of [20]) is the limit in sup-norm of a sequence of canonically lifted smooth paths, uniformly bounded in p-variation, thus clarifying the two different definitions of a geometric p-rough path. Our proofs are sufficiently general to include the case of Hölder- and modulus-type regularity. This allows us to extend a few classical results on Hölder-spaces and p -variation spaces to the non-commutative setting necessary for the theory of rough paths. As an application, we give a precise description of the support of Enhanced Fractional Brownian Motion, and prove a conjecture by Ledoux et al.
Article PDF
Similar content being viewed by others
References
Ciesielski, Z.: On the isomorphisms of the spaces H α and m. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 8, 217–222 (1960)
Coutin, L., Qian Z.: Stochastic analysis, rough path analysis and fractional Brownian motions Probab. Theory Relat. Fields 122, 108–140 (2002)
Decreusefond, L., Üstünel, A.S.: Stochastic Analysis of the Fractional Brownian Motion, Potential Analysis 10, 177–214 (1997)
Dudley, R.M., Norvaisa, R.: An introduction to p-variation and Young integrals. Lecture notes.
Folland, G.B., Stein, E.M.: Hardy spaces on homogeneous groups. Mathematical Notes, 28. Princeton University Press, 1982
De La Pradelle, A., Feyel, D.: Curvilinear Integrals along Rough Paths. Preprint
Friz, P.: Continuity of the Ito-map for Hoelder rough paths with applications to the support theorem in Hölder norm, Probability and PDEs in Modern Applied Mathematics, The IMA Volumes in Mathematics and its Applications, Vol. 140, 2005
Friz, P., Lyons, T., Stroock, D.: Lévy's area under conditioning, Annales de l'Institut Henri Poincare (B), Probability and Statistics, 2005. To appear
Friz, P., Victoir, N.: Approximations of the Brownian Rough Path with Applications to Stochastic Analysis, Annales de l'Institut Henri Poincare (B), Probability and Statistics, Volume 41, Issue 4, 2005, pp. 703–724
Goodman, R.: Filtrations and Asymptotic Automorphisms on Nilpotent Lie Groups. J.Diff.Geometry 12, 183–196 (1977)
Haynes, G.W., Hermes H.: Nonlinear Controllability via Lie Theory. SIAM J. Control Optim. 8 (4), 450–460 (1970)
Kurzweil, J., Jarnik, J.: Limit Process in Ordinary Differential Equations''. J. Appl. Math. Phys. 38, 241–256 (1987)
Kurzweil, J., Jarnik, J.: Iterated Lie Brackets in Limit Processes in Ordinary Differential Equations. Results in Mathematics 14, 125–137 (1988)
Kurzweil, J., Jarnik, J.: A Convergence Effect in Ordinary Differential Equations. Asym. Meth. Math. Physics (Russian), 301, 'Naukova Pumka', Kiev, 1989
Lejay, A.: Introduction to Rough Paths, Séminaire de Probabilité, Springer. To appear
Ledoux, M., Qian, Z., Zhang, T.: Large deviations and support theorem for diffusion processes via rough paths. Stochastic Process. Appl. 102 (2), 265–283 (2002)
Liu, W.: An approximation algorithm for nonholonomic systems. SIAM J. Control Optim. 35 (4), 1328–1365 (1997)
Liu, W.S.: Averaging Theorems for Highly Oscillatory Differential Equations and Iterated Lie Brackets.'' SIAM J. Control Optim. 35 (6), (1997)
Liu, W., Sussman, H.: Shortest paths for sub-Riemannian metrics on rank-two distributions. Mem. Amer. Math. Soc. 118 (564), (1995)
Lyons, T.: Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (2), 215–310 (1998)
Lyons, T., Qian, Z.: System Control and Rough Paths, Oxford University Press, 2002
Lyons, T., Victoir, N.: An Extension Theorem to Rough Path. Preprint
Malliavin, P.: Stochastic Analysis, Springer, 1997
Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications. Math.Surveys and Monographs 91. AMS, 2002
Musielak, J., Semadeni, Z.: Some classes of Banach spaces depending on a parameter. Studia Math. 20, 271–284 (1961)
Sussmann, H.J., Liu, W.: Limits of highly oscillatory controls and the approximation of general paths by admissible trajectories.'' IEEE Publications, New York, 1991, pp. 437–442
Wiener, N.: The quadratic variation of a function and its Fourier coefficients. J. Math. and Phys. 3, 72–94 (1924)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Friz, P., Victoir, N. A note on the notion of geometric rough paths. Probab. Theory Relat. Fields 136, 395–416 (2006). https://doi.org/10.1007/s00440-005-0487-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-005-0487-7