# zbMATH — the first resource for mathematics

Ergodic theory for SDEs with extrinsic memory. (English) Zbl 1129.60052
From the authors’ summary: 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 topolgical 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.

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60G10 Stationary stochastic processes 26A33 Fractional derivatives and integrals
Full Text:
##### References:
 [1] Alòs, E. and Nualart, D. (2003). Stochastic integration with respect to the fractional Brownian motion. Stochastics Stochastics Rep. 75 129–152. · Zbl 1028.60048 · doi:10.1080/1045112031000078917 [2] Arnold, L. (1998). Random Dynamical Systems . Springer, Berlin. · Zbl 0906.34001 [3] Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108–140. · Zbl 1047.60029 · doi:10.1007/s004400100158 [4] Da Prato, G. and Zabczyk, J. (1996). Ergodicity for Infinite Dimensional Systems . Cambridge Univ. Press. · Zbl 0849.60052 · doi:10.1017/CBO9780511662829 [5] Elworthy, K. D. and Li, X.-M. (1994). Formulae for the derivatives of heat semigroups. J. Funct. Anal. 125 252–286. · Zbl 0813.60049 · doi:10.1006/jfan.1994.1124 [6] Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216 86–140. · Zbl 1058.60037 · doi:10.1016/j.jfa.2004.01.002 [7] Hairer, M. (2005). Ergodicity of stochastic differential equations driven by fractional Brownian motion. Ann. Probab. 33 703–758. · Zbl 1071.60045 · doi:10.1214/009117904000000892 [8] Has’minskiĭ, R. Z. (1980). Stochastic Stability of Differential Equations . Sijthoff & Noordhoff, Alphen aan den Rijn. [9] 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. · Zbl 1059.60073 · doi:10.1016/j.crma.2004.09.035 [10] Hu, Y. and Nualart, D. (2006). Differential equations driven by hölder continuous functions of order greater than $$1/2$$. · Zbl 1144.34038 · doi:10.1007/978-3-540-70847-6_17 [11] 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. · Zbl 1075.60510 · doi:10.1016/S0304-4149(02)00176-X [12] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215–310. · Zbl 0923.34056 · doi:10.4171/RMI/240 · eudml:39555 [13] Meyn, S. P. and Tweedie, R. L. (1994). Markov Chains and Stochastic Stability . Springer, London. · Zbl 0925.60001 [14] Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422–437. JSTOR: · Zbl 0179.47801 · doi:10.1137/1010093 · links.jstor.org [15] Nualart, D. and Răşcanu, A. (2002). Differential equations driven by fractional Brownian motion. Collect. Math. 53 55–81. · Zbl 1018.60057 · eudml:42846 [16] Nualart, D. and Saussereau, B. (2005). Malliavin calculus for stochastic differential equations driven by a fractional Brownian motion. · Zbl 1169.60013 [17] Nualart, D. (1995). The Malliavin Calculus and Related Topics . Springer, New York. · Zbl 0837.60050 [18] Ruzmaikina, A. A. (2000). Stieltjes integrals of Hölder continuous functions with applications to fractional Brownian motion. J. Statist. Phys. 100 1049–1069. · Zbl 0970.60045 · doi:10.1023/A:1018754806993 [19] Russo, F. and Vallois, P. (2000). Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics Stochastics Rep. 70 1–40. · Zbl 0981.60053 [20] Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives . Gordon and Breach Science Publishers, Yverdon. · Zbl 0818.26003 [21] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable non-Gaussian Random Processes . Chapman and Hall, New York. · Zbl 0925.60027 [22] 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. · Zbl 0255.60056
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.