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Densities for rough differential equations under Hörmander’s condition. (English) Zbl 1205.60105
Considering a differential equation driven by a random Gaussian rough path on a Euclidean space, the authors prove under some assumptions, that if the vector fields of the equation satisfy the Hörmander condition, then the solution has a density with respect to the Lebesgue measure. As an example, equations driven by fractional Brownian motions with Hurst parameter \(H>1/4\) enter the framework of this article. The proof of this beautiful result uses some other results about rough paths (small time expansion, description of the support of Gaussian rough paths) jointly with techniques of the Malliavin calculus (differentiability with respect to Cameron-Martin perturbations, Malliavin matrix).

MSC:
60H07 Stochastic calculus of variations and the Malliavin calculus
60G17 Sample path properties
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[1] F. Baudoin and M. Hairer, ”A version of Hörmander’s theorem for the fractional Brownian motion,” Probab. Theory Related Fields, vol. 139, iss. 3-4, pp. 373-395, 2007. · Zbl 1123.60038 · doi:10.1007/s00440-006-0035-0
[2] D. R. Bell, The Malliavin Calculus, Mineola, NY: Dover Publ., 2006. · Zbl 1099.60041
[3] J. Bismut, Large Deviations and the Malliavin Calculus, Boston, MA: Birkhäuser, 1984. · Zbl 0537.35003
[4] T. Cass, P. Friz, and N. Victoir, ”Non-degeneracy of Wiener functionals arising from rough differential equations,” Trans. Amer. Math. Soc., vol. 361, iss. 6, pp. 3359-3371, 2009. · Zbl 1175.60034 · doi:10.1090/S0002-9947-09-04677-7 · arxiv:0707.0154
[5] P. Cattiaux and L. Mesnager, ”Hypoelliptic non-homogeneous diffusions,” Probab. Theory Related Fields, vol. 123, iss. 4, pp. 453-483, 2002. · Zbl 1009.60058 · doi:10.1007/s004400100194
[6] K. Chen, ”Integration of paths, geometric invariants and a generalized Baker-Hausdorff formula,” Ann. of Math., vol. 65, pp. 163-178, 1957. · Zbl 0077.25301 · doi:10.2307/1969671 · www.jstor.org
[7] L. Coutin and Z. Qian, ”Stochastic analysis, rough path analysis and fractional Brownian motions,” Probab. Theory Related Fields, vol. 122, iss. 1, pp. 108-140, 2002. · Zbl 1047.60029 · doi:10.1007/s004400100158
[8] L. Decreusefond, ”Stochastic integration with respect to Volterra processes,” Ann. Inst. H. Poincaré Probab. Statist., vol. 41, iss. 2, pp. 123-149, 2005. · Zbl 1071.60040 · doi:10.1016/j.anihpb.2004.03.004 · numdam:AIHPB_2005__41_2_123_0 · eudml:77839 · arxiv:math/0302047
[9] F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage, New York: Springer-Verlag, 2006. · Zbl 1106.91031
[10] R. M. Dudley, ”Sample functions of the Gaussian process,” Ann. Probability, vol. 1, iss. 1, pp. 66-103, 1973. · Zbl 0261.60033 · doi:10.1214/aop/1176997026
[11] D. Feyel and A. de La Pradelle, ”Curvilinear integrals along enriched paths,” Electron. J. Probab., vol. 11, p. no. 34, 860-892, 2006. · Zbl 1110.60031 · eudml:127241
[12] D. Filipović and J. Teichmann, ”On the geometry of the term structure of interest rates,” in Stochastic Analysis with Applications to Mathematical Finance, , 2004, pp. 129-167. · Zbl 1048.60045 · doi:10.1098/rspa.2003.1238
[13] P. Friz, T. Lyons, and D. Stroock, ”Lévy’s area under conditioning,” Ann. Inst. H. Poincaré Probab. Statist., vol. 42, iss. 1, pp. 89-101, 2006. · Zbl 1099.60054 · doi:10.1016/j.anihpb.2005.02.003 · numdam:AIHPB_2006__42_1_89_0 · eudml:77889
[14] P. Friz and N. Victoir, ”Approximations of the Brownian rough path with applications to stochastic analysis,” Ann. Inst. H. Poincaré Probab. Statist., vol. 41, iss. 4, pp. 703-724, 2005. · Zbl 1080.60021 · doi:10.1016/j.anihpb.2004.05.003 · numdam:AIHPB_2005__41_4_703_0 · eudml:77863 · arxiv:math/0308238
[15] P. Friz and N. Victoir, ”A note on the notion of geometric rough paths,” Probab. Theory Related Fields, vol. 136, iss. 3, pp. 395-416, 2006. · Zbl 1108.34052 · doi:10.1007/s00440-005-0487-7 · arxiv:math/0403115
[16] P. Friz and N. Victoir, ”A variation embedding theorem and applications,” J. Funct. Anal., vol. 239, iss. 2, pp. 631-637, 2006. · Zbl 1114.46022 · doi:10.1016/j.jfa.2005.12.021 · arxiv:math/0511520
[17] P. Friz and N. Victoir, Differential equations driven by Gaussian signals, to appear in Ann. Inst. H. Poincaré Probab. Statist.. · Zbl 1202.60058 · doi:10.1214/09-AIHP202 · eudml:241550 · arxiv:0707.0313
[18] P. Friz and N. Victoir, ”Euler estimates for rough differential equations,” J. Differential Equations, vol. 244, iss. 2, pp. 388-412, 2008. · Zbl 1140.60037 · doi:10.1016/j.jde.2007.10.008 · arxiv:math/0602345
[19] P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge: Cambridge Univ. Press, 2010, vol. 120. · Zbl 1193.60053 · doi:10.1017/CBO9780511845079
[20] M. Hairer and J. C. Mattingly, ”Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing,” Ann. of Math., vol. 164, iss. 3, pp. 993-1032, 2006. · Zbl 1130.37038 · doi:10.4007/annals.2006.164.993 · euclid:annm/1172614618 · arxiv:math/0406087
[21] S. Kusuoka, ”The nonlinear transformation of Gaussian measure on Banach space and absolute continuity. I,” J. Fac. Sci. Univ. Tokyo Sect. IA Math., vol. 29, iss. 3, pp. 567-597, 1982. · Zbl 0525.60050
[22] K. Kusuoka, ”On the regularity of solutions to SDE,” in Asymptotic Problems in Probability Theory: Wiener Functionals and Asymptotics (Sanda/Kyoto, 1990), Harlow: Longman Sci. Tech., 1993, pp. 90-103. · Zbl 0790.60050
[23] S. Kusuoka and D. Stroock, ”Applications of the Malliavin calculus. III,” J. Fac. Sci. Univ. Tokyo Sect. IA Math., vol. 34, iss. 2, pp. 391-442, 1987. · Zbl 0633.60078
[24] M. Hairer, ”Ergodicity of stochastic differential equations driven by fractional Brownian motion,” Ann. Probab., vol. 33, iss. 2, pp. 703-758, 2005. · Zbl 1071.60045 · doi:10.1214/009117904000000892 · arxiv:math/0304134
[25] T. J. Lyons, ”Differential equations driven by rough signals,” Rev. Mat. Iberoamericana, vol. 14, iss. 2, pp. 215-310, 1998. · Zbl 0923.34056 · doi:10.4171/RMI/240 · eudml:39555
[26] T. J. Lyons and Z. Qian, ”Calculus of variation for multiplicative functionals,” in New Trends in Stochastic Analysis (Charingworth, 1994), World Sci. Publ., River Edge, NJ, 1997, pp. 348-374.
[27] T. J. Lyons and Z. Qian, ”Flow of diffeomorphisms induced by a geometric multiplicative functional,” Probab. Theory Related Fields, vol. 112, iss. 1, pp. 91-119, 1998. · Zbl 0918.60009 · doi:10.1007/s004400050184
[28] M. Ledoux, Z. Qian, and T. Zhang, ”Large deviations and support theorem for diffusion processes via rough paths,” Stochastic Process. Appl., vol. 102, iss. 2, pp. 265-283, 2002. · Zbl 1075.60510 · doi:10.1016/S0304-4149(02)00176-X
[29] T. J. Lyons and Z. Qian, System Control and Rough Paths, Oxford: Oxford University Press, 2002. · Zbl 1044.93009
[30] P. Malliavin, Stochastic Analysis, New York: Springer-Verlag, 1997. · Zbl 0878.60001
[31] B. B. Mandelbrot and J. W. Van Ness, ”Fractional Brownian motions, fractional noises and applications,” SIAM Rev., vol. 10, pp. 422-437, 1968. · Zbl 0179.47801 · doi:10.1137/1010093
[32] D. Nualart, The Malliavin Calculus and Related Topics, Second ed., New York: Springer-Verlag, 2006. · Zbl 1099.60003 · doi:10.1007/3-540-28329-3
[33] D. Nualart and B. Saussereau, ”Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion,” Stochastic Process. Appl., vol. 119, iss. 2, pp. 391-409, 2009. · Zbl 1169.60013 · doi:10.1016/j.spa.2008.02.016
[34] Y. Hu and D. Nualart, ”Differential equations driven by Hölder continuous functions of order greater than 1/2,” in Stochastic Analysis and Applications, New York: Springer-Verlag, 2007, pp. 399-413. · Zbl 1144.34038 · arxiv:math/0601628
[35] J. Picard, ”A tree approach to \(p\)-variation and to integration,” Ann. Probab., vol. 36, iss. 6, pp. 2235-2279, 2008. · Zbl 1157.60055 · doi:10.1214/07-AOP388 · arxiv:0705.2128
[36] I. Shigekawa, Stochastic Analysis, Providence, RI: Amer. Math. Soc., 2004. · Zbl 1064.60003
[37] D. W. Stroock, ”The Malliavin calculus and its application to second order parabolic differential equations. I,” Math. Systems Theory, vol. 14, iss. 1, pp. 25-65, 1981. · Zbl 0474.60061 · doi:10.1007/BF01752389
[38] S. Taniguchi, ”Applications of Malliavin’s calculus to time-dependent systems of heat equations,” Osaka J. Math., vol. 22, iss. 2, pp. 307-320, 1985. · Zbl 0583.35055 · projecteuclid.org
[39] S. A. Üstünel and M. Zakai, Transformation of Measure on Wiener Space, New York: Springer-Verlag, 2000. · Zbl 0974.46044
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