Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion

M. Hairer and N. S. Pillai

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Abstract

We demonstrate that stochastic differential equations (SDEs) driven by fractional Brownian motion with Hurst parameter H>½ have similar ergodic properties as SDEs driven by standard Brownian motion. The focus in this article is on hypoelliptic systems satisfying Hörmander’s condition. We show that such systems enjoy a suitable version of the strong Feller property and we conclude that under a standard controllability condition they admit a unique stationary solution that is physical in the sense that it does not “look into the future.”

The main technical result required for the analysis is a bound on the moments of the inverse of the Malliavin covariance matrix, conditional on the past of the driving noise.

Résumé

Nous démontrons que les équations différentielles stochastiques (EDS) conduites par des mouvements Browniens fractionnaires à paramètre de Hurst H>½ ont des propriétés ergodiques similaires aux EDS usuelles conduites par des mouvements Browniens. L’intérêt principal du présent article est de pouvoir traîter également des systèmes hypoelliptiques satisfaisant la condition de Hörmander. Nous montrons qu’une version adéquate de la propriété de Feller forte est vérifiée par de tels systèmes et nous en déduisons que, sous une propriété de controllabilité usuelle, ils admettent une unique solution stationnaire qui ait un sens physique.

L’ingrédient principal de notre analyse est une borne supérieure sur les moments inverses de la matrice de Malliavin associée, conditionnée au passé du bruit.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist. Volume 47, Number 2 (2011), 601-628.

Dates
First available in Project Euclid: 23 March 2011

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1300887284

Digital Object Identifier
doi:10.1214/10-AIHP377

Mathematical Reviews number (MathSciNet)
MR2814425

Zentralblatt MATH identifier
1221.60083

Subjects
Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G10: Stationary processes 60H07: Stochastic calculus of variations and the Malliavin calculus 26A33: Fractional derivatives and integrals

Keywords
Ergodicity Fractional Brownian motion Hörmander’s theorem

Citation

Hairer, M.; Pillai, N. S. Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 2, 601--628. doi:10.1214/10-AIHP377. https://projecteuclid.org/euclid.aihp/1300887284


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References

  • [1] L. Arnold. Random Dynamical Systems. Springer, Berlin, 1998.
  • [2] F. Baudoin and L. Coutin. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 (2007) 550–574.
  • [3] F. Baudoin and M. Hairer. A version of Hörmander’s theorem for the fractional Brownian motion. Probab. Theory Related Fields 139 (2007) 373–395.
  • [4] J.-M. Bismut. Martingales, the Malliavin calculus and hypoellipticity under general Hörmander’s conditions. Z. Wahrsch. Verw. Gebiete 56 (1981) 469–505.
  • [5] V. I. Bogachev. Gaussian Measures. Mathematical Surveys and Monographs 62. Amer. Math. Soc., Providence, RI, 1998.
  • [6] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002) 108–140.
  • [7] G. Da Prato, K. D. Elworthy and J. Zabczyk. Strong Feller property for stochastic semilinear equations. Stochastic Anal. Appl. 13 (1995) 35–45.
  • [8] G. Da Prato and J. Zabczyk. Ergodicity for Infinite-Dimensional Systems. London Mathematical Society Lecture Note Series 229. Cambridge Univ. Press, Cambridge, 1996.
  • [9] K. D. Elworthy and X.-M. Li. Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125 (1994) 252–286.
  • [10] P. K. Friz. Multidimensional Stochastic Processes as Rough Paths. Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge, 2010.
  • [11] M. Hairer. Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33 (2005) 703–758.
  • [12] M. Hairer. Ergodic properties of a class of non-Markovian processes. In Trends in Stochastic Analysis. LMS Lecture Note Series 353. Cambridge Univ. Press, Cambridge, 2009.
  • [13] M. Hairer and J. C. Mattingly. Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann. of Math. (2) 164 (2006) 993–1032.
  • [14] M. Hairer and J. C. Mattingly. A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Preprint, 2009.
  • [15] Y. Hu and D. Nualart. Differential equations driven by Hölder continuous functions of order greater than 1/2. In Stochastic Analysis and Applications 399–413. Abel Symp. 2. Springer, Berlin, 2007.
  • [16] M. Hairer and A. Ohashi. Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 (2007) 1950–1977.
  • [17] L. Hörmander. Hypoelliptic second order differential equations. Acta Math. 119 (1967) 147–171.
  • [18] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. I. In Stochastic Analysis (Katata/Kyoto, 1982) 271–306. North-Holland Math. Library 32. North-Holland, Amsterdam, 1984.
  • [19] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. II. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 32 (1985) 1–76.
  • [20] S. Kusuoka and D. Stroock. Applications of the Malliavin calculus. III. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 34 (1987) 391–442.
  • [21] T. Lyons and Z. Qian. System Control and Rough Paths. Oxford Univ. Press, Oxford, 2002.
  • [22] P. Malliavin. Stochastic calculus of variations and hypoelliptic operators. In Symp. on Stoch. Diff. Equations, Kyoto 1976 147–171. Wiley, New York, 1978.
  • [23] P. Malliavin. Stochastic Analysis. Grundlehren der Mathematischen Wissenschaften 313. Springer, Berlin, 1997.
  • [24] Y. S. Mishura. Stochastic Calculus for Fractional Brownian Motion and Related Processes. Lecture Notes in Mathematics 1929. Springer, Berlin, 2008.
  • [25] L. Mazliak and I. Nourdin. Optimal control for rough differential equations. Stoch. Dyn. 8 (2008) 23–33.
  • [26] S. P. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability. Springer, London, 1993.
  • [27] B. B. Mandelbrot and J. W. van Ness. Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 (1968) 422–437.
  • [28] J. Norris. Simplified Malliavin calculus. In Séminaire de Probabilités, XX, 1984/85 101–130. Lecture Notes in Math. 1204. Springer, Berlin, 1986.
  • [29] I. Nourdin and T. Simon. On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion. Statist. Probab. Lett. 76 (2006) 907–912.
  • [30] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Stochastic Process. Appl. 119 (2009) 391–409.
  • [31] D. Nualart. The Malliavin Calculus and Related Topics, 2nd edition. Springer, Berlin, 2006.
  • [32] S. G. Samko, A. A. Kilbas and O. I. Marichev. Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon, 1993.
  • [33] G. Samorodnitsky and M. S. Taqqu. Stable Non-Gaussian Random Processes. Chapman & Hall, New York, 1994.
  • [34] L. C. Young. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 (1936) 251–282.