Rough flows. (English) Zbl 1480.60151

Summary: 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.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter
34F05 Ordinary differential equations and systems with randomness
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[1] [Bax80] P. Baxendale, Wiener processes on manifolds of maps, Proc. Royal Soc. Edinburgh, 87 (1980), 127-152. · Zbl 0456.60074
[2] [BC17] I. Bailleul and R. Catellier, Rough flows and homogenization in stochastic turbulence, J. Differential Equations, 263 (2017), 4894-4928. · Zbl 1372.60078
[3] [BDM08] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for infinite dimensional stochastic dynamical systems, Ann. Probab., 36 (2008), 1390-1420. · Zbl 1155.60024
[4] [BDM10] A. Budhiraja, P. Dupuis and V. Maroulas, Large deviations for stochastic flows of diffeomorphisms, Bernoulli, 16 (2010), 234-257. · Zbl 1452.60021
[5] [BF61] Y. N. Blagovescenskii and M. Freidlin, Some properties of diffusion processes depending on a parameter, Soviet Math., 2 (1961), 633-636. · Zbl 0114.07801
[6] [BG17] I. Bailleul and M. Gubinelli, Unbounded rough drivers, Ann. Fac. Sci. Toulouse Math. (6), 26 (2017), 795-830. · Zbl 1391.60150
[7] [BGN04] W. Bertram, H. Glöckner and K.-H. Neeb, Differential calculus over general base fields and rings, Expo. Math., 22 (2004), 213-282. · Zbl 1099.58006
[8] [Bis81] J.-M. Bismut, Mécanique aléatoire, Lecture Notes in Math., 866, Springer-Verlag, Berlin-New York, 1981.
[9] [Bog98] V. I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, 62, Amer. Math. Soc., Providence, RI, 1998.
[10] [BRS17] I. Bailleul, S. Riedel and M. Scheutzow, Random dynamical systems, rough paths and rough flows, J. Differential Equations, 262 (2017), 5792-5823. · Zbl 1382.37050
[11] [CK91] E. Carlen and P. Krée, \(L^p\) estimates on iterated stochastic integrals, Ann. Probab., 19 (1991), 354-368.
[12] [CW17] T. Cass and M. Weidner, Tree algebras over topological vector spaces in rough path theory, arXiv:1604.07352v2, 2017.
[13] [DD12] S. Dereich and G. Dimitroff, A support theorem and a large deviation principle for Kunita flows, Stoch. Dyn., 12 (2012), no. 3, 1150022. · Zbl 1247.60083
[14] [Der10] S. Dereich, Rough paths analysis of general Banach space-valued Wiener processes, J. Funct. Anal., 258 (2010), 2910-2936. · Zbl 1204.60007
[15] [DS89] J.-D. Deuschel and D. W. Stroock, Large deviations, Pure and Applied Mathematics, 137, Academic Press, Inc., Boston, MA, 1989.
[16] [DZ98] A. Dembo and O. Zeitouni, Large deviations techniques and applications, Applications of Mathematics, 38, Springer-Verlag, New York, 1998. · Zbl 0896.60013
[17] [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.
[18] [FdLP06] D. Feyel and A. de La Pradelle, Curvilinear integrals along enriched paths, Electron. J. Probab., 11 (2006), 860-892. · Zbl 1110.60031
[19] [FdLPM08] D. Feyel, A. de La Pradelle and G. Mokoboski, A non-commutative sewing lemma, Elec. Comm. Probab., 13 (2008), 24-34. · Zbl 1186.26009
[20] [FH14] P. K. Friz and M. Hairer, A Course on Rough Paths with an introduction to regularity structures, Universitext, XIV, Springer, Cham, 2014. · Zbl 1327.60013
[21] [FV07] P. Friz and N. Victoir, Large deviation principle for enhanced Gaussian processes, Ann. Inst. H. Poincaré Probab. Statist., 43 (2007), 775-785. · Zbl 1172.60306
[22] [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. · Zbl 1193.60053
[23] [Gub04] M. Gubinelli, Controlling rough paths, J. Funct. Anal., 216 (2004), 86-140. · Zbl 1058.60037
[24] [Har81] T. Harris, Brownian motions on the homeomorphisms of the plane, Ann. Prob., 9 (1981), 232-254. · Zbl 0457.60013
[25] [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.
[26] [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.
[27] [Kun86a] H. Kunita, Convergence of stochastic flows connectd with stochastic ordinary differential equations, Stochastics, 17 (1986), 215-251.
[28] [Kun90] H. Kunita, Stochastic flows and stochastic differential equations, Cambridge Studies in Advanced Mathematics, 24, Cambridge University Press, Cambridge, 1990.
[29] [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. · Zbl 0855.00026
[30] [LJ82] Y. Le Jan, Flots de diffusion dans \(\mathbf{R}^d\), C. R. Acad. Sci. Paris Sér. I Math., 294 (1982), 697-699. · Zbl 0497.60070
[31] [LJ85] Y. Le Jan, On isotropic Brownian motions, Z. Wahrsch. Verw. Gebiete, 70 (1985), 609-620. · Zbl 0576.60072
[32] [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.
[33] [LLQ02] M. Ledoux, T. Lyons and Z. Qian, Lévy area of Wiener processes in Banach spaces, Ann. Probab., 30 (2002), 546-578. · Zbl 1016.60071
[34] [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. · Zbl 1075.60510
[35] [Lyo98] T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310. · Zbl 0923.34056
[36] [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. · Zbl 1087.60035
[37] [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. · Zbl 1180.37077
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