The Annals of Probability

Smoothness of the density for solutions to Gaussian rough differential equations

Thomas Cass, Martin Hairer, Christian Litterer, and Samy Tindel

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We consider stochastic differential equations of the form $dY_{t}=V(Y_{t})\,dX_{t}+V_{0}(Y_{t})\,dt$ driven by a multi-dimensional Gaussian process. Under the assumption that the vector fields $V_{0}$ and $V=(V_{1},\ldots,V_{d})$ satisfy Hörmander’s bracket condition, we demonstrate that $Y_{t}$ admits a smooth density for any $t\in(0,T]$, provided the driving noise satisfies certain nondegeneracy assumptions. Our analysis relies on relies on an interplay of rough path theory, Malliavin calculus and the theory of Gaussian processes. Our result applies to a broad range of examples including fractional Brownian motion with Hurst parameter $H>1/4$, the Ornstein–Uhlenbeck process and the Brownian bridge returning after time $T$.

Article information

Ann. Probab. Volume 43, Number 1 (2015), 188-239.

First available in Project Euclid: 12 November 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 60G15: Gaussian processes 60H10: Stochastic ordinary differential equations [See also 34F05]

Rough path analysis Gaussian processes Malliavin calculus


Cass, Thomas; Hairer, Martin; Litterer, Christian; Tindel, Samy. Smoothness of the density for solutions to Gaussian rough differential equations. Ann. Probab. 43 (2015), no. 1, 188--239. doi:10.1214/13-AOP896.

Export citation


  • [1] Baudoin, F. and Hairer, M. (2007). A version of Hörmander’s theorem for the fractional Brownian motion. Probab. Theory Related Fields 139 373–395.
  • [2] Baudoin, F. and Ouyang, C. (2011). Small-time kernel expansion for solutions of stochastic differential equations driven by fractional Brownian motions. Stochastic Process. Appl. 121 759–792.
  • [3] Baudoin, F., Ouyang, C. and Tindel, S. (2014). Upper bounds for the density of solutions to stochastic differential equations driven by fractional Brownian motions. Ann. Inst. Henri Poincaré Probab. Stat. 50 111–135.
  • [4] Berman, S. M. (1973/74). Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23 69–94.
  • [5] Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge Univ. Press, Cambridge.
  • [6] Cass, T. and Friz, P. (2010). Densities for rough differential equations under Hörmander’s condition. Ann. of Math. (2) 171 2115–2141.
  • [7] Cass, T., Friz, P. and Victoir, N. (2009). Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 3359–3371.
  • [8] Cass, T., Litterer, C. and Lyons, T. (2013). Integrability and tail estimates for Gaussian rough differential equations. Ann. Probab. 41 3026–3050.
  • [9] Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108–140.
  • [10] Cuzick, J. (1982). Continuity of Gaussian local times. Ann. Probab. 10 818–823.
  • [11] Cuzick, J. and DuPreez, J. P. (1982). Joint continuity of Gaussian local times. Ann. Probab. 10 810–817.
  • [12] Friz, P. and Oberhauser, H. (2010). A generalized Fernique theorem and applications. Proc. Amer. Math. Soc. 138 3679–3688.
  • [13] Friz, P. and Shekhar, A. (2013). Doob–Meyer for rough paths. Bull. Inst. Math. Acad. Sin. (N.S.) 8 73–84.
  • [14] Friz, P. and Victoir, N. (2006). A variation embedding theorem and applications. J. Funct. Anal. 239 631–637.
  • [15] Friz, P. and Victoir, N. (2010). Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré Probab. Stat. 46 369–413.
  • [16] Friz, P. and Victoir, N. (2011). A note on higher dimensional $p$-variation. Electron. J. Probab. 16 1880–1899.
  • [17] Friz, P. K. and Victoir, N. B. (2010). Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Cambridge Studies in Advanced Mathematics 120. Cambridge Univ. Press, Cambridge.
  • [18] Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216 86–140.
  • [19] Gubinelli, M. (2010). Ramification of rough paths. J. Differential Equations 248 693–721.
  • [20] Hairer, M. (2011). On Malliavin’s proof of Hörmander’s theorem. Bull. Sci. Math. 135 650–666.
  • [21] Hairer, M. and Mattingly, J. C. (2011). A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. Electron. J. Probab. 16 658–738.
  • [22] Hairer, M. and Pillai, N. S. (2011). Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion. Ann. Inst. Henri Poincaré Probab. Stat. 47 601–628.
  • [23] Hairer, M. and Pillai, N. S. (2013). Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths. Ann. Probab. 41 2544–2598.
  • [24] Hörmander, L. (1967). Hypoelliptic second order differential equations. Acta Math. 119 147–171.
  • [25] Hu, Y. and Nualart, D. (2007). Differential equations driven by Hölder continuous functions of order greater than $1/2$. In Stochastic Analysis and Applications. Abel Symp. 2 399–413. Springer, Berlin.
  • [26] Hu, Y. and Tindel, S. (2013). Smooth density for some nilpotent rough differential equations. J. Theoret. Probab. 26 722–749.
  • [27] Inahama, Y. (2011). Short time kernel asymptotics for Young SDE by means of Watanabe distribution theory. Available at arXiv:1110.2604.
  • [28] Inahama, Y. (2013). Malliavin differentiability of solutions of rough differential equations. Available at arXiv:1312.7621.
  • [29] Li, W. V. and Linde, W. (1998). Existence of small ball constants for fractional Brownian motions. C. R. Acad. Sci. Paris Sér. I Math. 326 1329–1334.
  • [30] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoam. 14 215–310.
  • [31] Lyons, T. J., Caruana, M. and Lévy, T. (2007). Differential Equations Driven by Rough Paths. Lecture Notes in Math. 1908. Springer, Berlin.
  • [32] Malliavin, P. (1978). Stochastic calculus of variation and hypoelliptic operators. In Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976) 195–263. Wiley, New York.
  • [33] Molchan, G. M. (1999). Maximum of a fractional Brownian motion: Probabilities of small values. Comm. Math. Phys. 205 97–111.
  • [34] Monrad, D. and Rootzén, H. (1995). Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Theory Related Fields 101 173–192.
  • [35] Norris, J. (1986). Simplified Malliavin calculus. In Séminaire de Probabilités, XX, 1984/85. Lecture Notes in Math. 1204 101–130. Springer, Berlin.
  • [36] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Springer, Berlin.
  • [37] Sugita, H. (1985). On a characterization of the Sobolev spaces over an abstract Wiener space. J. Math. Kyoto Univ. 25 717–725.
  • [38] Zhang, F., ed. (2005). The Schur Complement and Its Applications. Numerical Methods and Algorithms 4. Springer, New York.