Journal of the Mathematical Society of Japan

Rough flows

Ismaël BAILLEUL and Sebastian RIEDEL

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We introduce in this work a concept of rough driver that somehow provides a rough path-like analogue of an enriched object associated with time-dependent vector fields. We use the machinery of approximate flows to build the integration theory of rough drivers and prove well-posedness results for rough differential equations on flows and continuity of the solution flow as a function of the generating rough driver. We show that the theory of semimartingale stochastic flows developed in the 80's and early 90's fits nicely in this framework, and obtain as a consequence some strong approximation results for general semimartingale flows and provide a fresh look at large deviation theorems for ‘Gaussian’ stochastic flows.


The first author was partly supported by the ANR project “Retour Post-doctorant”, no. 11–PDOC-0025. He also thanks the U.B.O. for their hospitality, part of this work was written there. The second author was partly supported by the DFG Research Unit FOR 2402.

Article information

J. Math. Soc. Japan, Volume 71, Number 3 (2019), 915-978.

Received: 12 March 2018
First available in Project Euclid: 28 May 2019

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Mathematical Reviews number (MathSciNet)

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G44: Martingales with continuous parameter 34F05: Equations and systems with randomness [See also 34K50, 60H10, 93E03]

stochastic flows rough flows semimartingale velocity fields


BAILLEUL, Ismaël; RIEDEL, Sebastian. Rough flows. J. Math. Soc. Japan 71 (2019), no. 3, 915--978. doi:10.2969/jmsj/80108010.

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  • [Bai15] I. Bailleul, Flows driven by rough paths, Rev. Mat. Iberoamericana, 2015.
  • [Bai18] I. Bailleul, On the definition of a solution to a rough differential equation, arXiv: 1803.06479, 2018.
  • [Bax80] P. Baxendale, Wiener processes on manifolds of maps, Proc. Royal Soc. Edinburgh, 87 (1980), 127–152.
  • [BC17] I. Bailleul and R. Catellier, Rough flows and homogenization in stochastic turbulence, J. Differential Equations, 263 (2017), 4894–4928.
  • [BDM08] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390–1420.
  • [BDM10] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for stochastic flows of diffeomorphisms, Bernoulli, 16 (2010), 234–257.
  • [BF61] Y. N. Blagovescenskii and M. Freidlin, Some properties of diffusion processes depending on a parameter, Soviet Math., 2 (1961), 633–636.
  • [BG17] I. Bailleul and M. Gubinelli, Unbounded rough drivers, Ann. Fac. Sci. Toulouse Math. (6), 26 (2017), 795–830.
  • [BGN04] W. Bertram, H. Glöckner and K.-H. Neeb, Differential calculus over general base fields and rings, Expo. Math., 22 (2004), 213–282.
  • [Bis81] J.-M. Bismut, Mécanique aléatoire, Lecture Notes in Math., 866, Springer-Verlag, Berlin-New York, 1981.
  • [Bog98] V. I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, 62, Amer. Math. Soc., Providence, RI, 1998.
  • [BRS17] I. Bailleul, S. Riedel and M. Scheutzow, Random dynamical systems, rough paths and rough flows, J. Differential Equations, 262 (2017), 5792–5823.
  • [CK91] E. Carlen and P. Krée, $L^p$ estimates on iterated stochastic integrals, Ann. Probab., 19 (1991), 354–368.
  • [CW17] T. Cass and M. Weidner, Tree algebras over topological vector spaces in rough path theory, arXiv:1604.07352v2, 2017.
  • [Dav07] A. M. Davie, Differential equations driven by rough paths: an approach via discrete approximation, Appl. Math. Res. Express, AMRX, Art. ID abm009, 2007.
  • [DD12] S. Dereich and G. Dimitroff, A support theorem and a large deviation principle for Kunita flows, Stoch. Dyn., 12 (2012), no. 3, 1150022.
  • [Der10] S. Dereich, Rough paths analysis of general Banach space-valued Wiener processes, J. Funct. Anal., 258 (2010), 2910–2936.
  • [DS89] J.-D. Deuschel and D. W. Stroock, Large deviations, Pure and Applied Mathematics, 137, Academic Press, Inc., Boston, MA, 1989.
  • [DZ98] A. Dembo and O. Zeitouni, Large deviations techniques and applications, Applications of Mathematics, 38, Springer-Verlag, New York, 1998.
  • [Elw78] K. D. Elworthy, Stochastic dynamical systems and their flows, In: Stochastic analysis, Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1978, Academic Press, New York-London, 1978, 79–95.
  • [Elw82] K. D. Elworthy, Stochastic differential equations on manifolds, London Mathematical Society Lecture Notes Series, Cambridge Univ. Press, Cambridge-New York, 1982.
  • [FdLP06] D. Feyel and A. de La Pradelle, Curvilinear integrals along enriched paths, Electron. J. Probab., 11 (2006), 860–892.
  • [FdLPM08] D. Feyel, A. de La Pradelle and G. Mokoboski, A non-commutative sewing lemma, Elec. Comm. Probab., 13 (2008), 24–34.
  • [FH14] P. K. Friz and M. Hairer, A Course on Rough Paths with an introduction to regularity structures, Universitext, XIV, Springer, Cham, 2014.
  • [FV07] P. Friz and N. Victoir, Large deviation principle for enhanced Gaussian processes, Ann. Inst. H. Poincaré Probab. Statist., 43 (2007), 775–785.
  • [FV10] P. K. Friz and N. B. Victoir, Multidimensional stochastic processes as rough paths, Theory and applications, Cambridge Studies in Advanced Mathematics, 120, Cambridge University Press, Cambridge, 2010.
  • [Gub04] M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86–140.
  • [Har81] T. Harris, Brownian motions on the homeomorphisms of the plane, Ann. Prob., 9 (1981), 232–254.
  • [IA72] I. Gihman and A. Skorohod, Stochastic differential equations, Springer, Berlin, 1972.
  • [IW81] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, North-Holland Mathematical Library, 24, North-Holland Publishing Co., Amsterdam-New York, Kodansha, Ltd., Tokyo, 1981.
  • [Kal02] O. Kallenberg, Foundations of modern probability, Probability and its Applications, Springer-Verlag, New York, 2002.
  • [Kun81] H. Kunita, On the decomposition of solutions of stochastic differential equations, In: Stochastic integrals, Proc. Sympos., Univ. Durham, Durham, 1980, Lecture Notes in Math., 851, Springer, Berlin-New York, 1981, 213–255.
  • [Kun86a] H. Kunita, Convergence of stochastic flows connectd with stochastic ordinary differential equations, Stochastics, 17 (1986), 215–251.
  • [Kun86b] H. Kunita, Lectures on stochastic flows and applications, Tata Institue of Fundamental Research, Bombay, 1986.
  • [Kun90] H. Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990.
  • [Led96] M. Ledoux, Isoperimetry and Gaussian analysis, In: Lectures on probability theory and statistics, Saint-Flour, 1994, Lecture Notes in Math., 1648 (1996), 165–294.
  • [LJ82] Y. Le Jan, Flots de diffusion dans $\mathbf{R}^{d}$, C. R. Acad. Sci. Paris Sér. I Math., 294 (1982), 697–699.
  • [LJ85] Y. Le Jan, On isotropic Brownian motions, Z. Wahrsch. Verw. Gebiete, 70 (1985), 609–620.
  • [LJW84] Y. Le Jan and S. Watanabe, Stochastic flows of diffeomorphisms, In: Stochastic analysis, Katata/Kyoto, 1982, North-Holland Math. Library, 32, North-Holland, Amsterdam, 1984, 307–332.
  • [LLQ02] M. Ledoux, T. Lyons and Z. Qian, Lévy area of Wiener processes in Banach spaces, Ann. Probab., 30 (2002), 546–578.
  • [LQZ02] M. Ledoux, Z. Qian and T. Zhang, Large deviations and support theorem for diffusion processes via rough paths, Stochastic Process. Appl., 102 (2002), 265–283.
  • [Lyo98] T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215–310.
  • [MSS06] A. Millet and M. Sanz-Solé, Large deviations for rough paths of the fractional Brownian motion, Ann. Inst. H. Poincaré Probab. Statist., 42 (2006), 245–271.
  • [Rya02] R. A. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Math., Springer-Verlag London, Ltd., London, 2002.
  • [Sch09] M. Scheutzow, Chaining techniques and their application to stochastic flows, In: Trends in stochastic analysis, London Math. Soc. Lecture Note Ser., 353, Cambridge Univ. Press, Cambridge, 2009, 35–63.