The Annals of Probability

Ergodic theory for SDEs with extrinsic memory

M. Hairer and A. Ohashi

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We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of non-Markovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the Doob–Khas’minskii theorem.

The second part of this article shows how it is possible to apply these results to the case of stochastic differential equations driven by fractional Brownian motion. It follows that under a nondegeneracy condition on the noise, such equations admit a unique adapted stationary solution.

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Ann. Probab. Volume 35, Number 5 (2007), 1950-1977.

First available in Project Euclid: 5 September 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60G10: Stationary processes 26A33: Fractional derivatives and integrals

Non-Markovian processes ergodicity fractional Brownian motion


Hairer, M.; Ohashi, A. Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 (2007), no. 5, 1950--1977. doi:10.1214/009117906000001141.

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  • Alòs, E. and Nualart, D. (2003). Stochastic integration with respect to the fractional Brownian motion. Stochastics Stochastics Rep. 75 129--152.
  • Arnold, L. (1998). Random Dynamical Systems. Springer, Berlin.
  • Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108--140.
  • Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite Dimensional Systems. Cambridge Univ. Press.
  • Elworthy, K. D. and Li, X.-M. (1994). Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125 252--286.
  • Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216 86--140.
  • Hairer, M. (2005). Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33 703--758.
  • Has'minskiĭ, R. Z. (1980). Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn.
  • Hairer, M. and Mattingly, J. C. (2004). Ergodic properties of highly degenerate 2D stochastic Navier--Stokes equations. C. R. Math. Acad. Sci. Paris 339 879--882.
  • Hu, Y. and Nualart, D. (2006). Differential equations driven by hölder continuous functions of order greater than $1/2$. Preprint.
  • Ledoux, M., Qian, Z. and Zhang, T. (2002). Large deviations and support theorem for diffusion processes via rough paths. Stochastic Process. Appl. 102 265--283.
  • Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215--310.
  • Meyn, S. P. and Tweedie, R. L. (1994). Markov Chains and Stochastic Stability. Springer, London.
  • Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422--437.
  • Nualart, D. and Răşcanu, A. (2002). Differential equations driven by fractional Brownian motion. Collect. Math. 53 55--81.
  • Nualart, D. and Saussereau, B. (2005). Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. Preprint.
  • Nualart, D. (1995). The Malliavin Calculus and Related Topics. Springer, New York.
  • Ruzmaikina, A. A. (2000). Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100 1049--1069.
  • Russo, F. and Vallois, P. (2000). Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics Stochastics Rep. 70 1--40.
  • Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach Science Publishers, Yverdon.
  • Samorodnitsky, G. and Taqqu, M. S. (1994). Stable non-Gaussian Random Processes. Chapman and Hall, New York.
  • Stroock, D. W. and Varadhan, S. R. S. (1972). On the support of diffusion processes with applications to the strong maximum principle. Proc. 6th Berkeley Symp. Math. Statist. Probab. III 333--368. Univ. California Press, Berkeley.