A version of Hörmander’s theorem for the fractional Brownian motion. (English) Zbl 1123.60038

Summary: It is shown that the law of an SDE driven by fractional Brownian motion with Hurst parameter greater than 1/2 has a smooth density with respect to Lebesgue measure, provided that the driving vector fields satisfy Hörmander’s condition. The main new ingredient of the proof is an extension of Norris’ lemma to this situation.


60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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[1] Arous, G. B., Développement asymptotique du noyau de la chaleur hypoelliptique sur la diagonale, Ann l’institut Fourier, 39, 73-99 (1989) · Zbl 0659.35024
[2] Baudoin, F., An Introduction to the Geometry of Stochastic Flows (2005), London: Imperial College Press, London
[3] Baudoin, F., Coutin, L.: Self-similarity and fractional brownian motions on lie groups (2006) (preprint) · Zbl 1189.60083
[4] Bismut, J.-M., Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions, Z. Wahrsch. Verw. Gebiete, 56, 4, 469-505 (1981) · Zbl 0445.60049
[5] Bogachev, V. I., Gaussian Measures. Mathematical Surveys and Monographs, vol. 62 (1998), Providence, RI: American Mathematical Society, Providence, RI
[6] Cirel′son, B.S., Ibragimov, I.A., Sudakov, V.N.: Norms of Gaussian sample functions. In: Proceedings of the Third Japan-USSR Symposium on Probability Theory (Tashkent, 1975), pp. 20-41. Lecture Notes in Mathematics., vol. 550. Springer, Berlin (1976)
[7] Coutin, L.; Qian, Z., Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Relat Fields., 122, 1, 108-140 (2002) · Zbl 1047.60029
[8] Friz, P., Victoir, N.: Euler estimates for rough differential equations (2006) (preprint) · Zbl 1140.60037
[9] Hu, Y., Nualart, D.: Differential equations driven by hölder continuous functions of order greater than 1/2 (2006) (preprint) · Zbl 1144.34038
[10] Hörmander, L., Hypoelliptic second order differential equations, Acta Math., 119, 147-171 (1967) · Zbl 0156.10701
[11] Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. I. In: Stochastic analysis (Katata/Kyoto, 1982 pp. 271-306). North-Holland Mathematics. Library, vol. 32. North-Holland, Amsterdam (1984)
[12] Kusuoka, S.; Stroock, D., Applications of the Malliavin calculus. II, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 32, 1, 1-76 (1985) · Zbl 0568.60059
[13] Kusuoka, S.; Stroock, D., Applications of the Malliavin calculus. III, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 34, 2, 391-442 (1987) · Zbl 0633.60078
[14] Lyons, T. J., Differential equations driven by rough signals. I, An extension of an inequality of L. C. Young. Math. Res. Lett., 1, 4, 451-464 (1994) · Zbl 0835.34004
[15] Malliavin, P.: Stochastic calculus of variations and hypoelliptic operators. Symp. Stoch. Diff. Equations, Kyoto 1976 147-171
[16] Norris, J.: Simplified Malliavin calculus. In: Séminaire de Probabilités, XX, 1984/85, pp. 101-130. Lecture Notes in Mathematics,vol. 1204. Springer, Berlin Heidelberg Newyork (1986)
[17] Nualart, D.; Răşcanu, A., Differential equations driven by fractional Brownian motion, Collect. Math., 53, 1, 55-81 (2002) · Zbl 1018.60057
[18] Nualart, D., Saussereau, B.: Malliavin calculus for stochastic differential equations driven by a fractional brownian motion (2005) (preprint) · Zbl 1169.60013
[19] Nourdin, I.; Simon, T., On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion, Stat. Prob Lett., 76, 907-912 (2006) · Zbl 1091.60008
[20] Nualart, D.: The Malliavin Calculus and Related Topics. In:Probability and its Applications (New York). Springer, Berlin Heidelberg New York (1995) · Zbl 0837.60050
[21] Pipiras, V.; Taqqu, M. S., Integration questions related to fractional Brownian motion, Prob. Theory Relat. Fields, 118, 2, 251-291 (2000) · Zbl 0970.60058
[22] Rothschild, L. P.; Stein, E. M., Hypoelliptic differential operators and nilpotent groups, Acta Math., 137, 3-4, 247-320 (1976) · Zbl 0346.35030
[23] Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon (1993). Theory and applications, Edited and with a foreword by S. M. Nikol′skiĭ, Translated from the 1987 Russian original, Revised by the authors · Zbl 0617.26004
[24] Talagrand, M., Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math., 81, 73-205 (1995) · Zbl 0864.60013
[25] Young, L. C., An inequality of Hölder type connected with Stieltjes integration, Acta Math., 67, 251-282 (1936) · Zbl 0016.10404
[26] Zähle, M., Integration with respect to fractal functions and stochastic calculus. II, Mathe. Nachrich., 225, 145-183 (2001) · Zbl 0983.60054
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