Ergodicity of stochastic differential equations driven by fractional Brownian motion
Martin Hairer
Source: Ann. Probab. Volume 33, Number 2
(2005), 703-758.
Abstract
We study the ergodic properties of finite-dimensional systems of SDEs driven by nondegenerate additive fractional Brownian motion with arbitrary Hurst parameter H∈(0,1). A general framework is constructed to make precise the notions of “invariant measure” and “stationary state” for such a system. We then prove under rather weak dissipativity conditions that such an SDE possesses a unique stationary solution and that the convergence rate of an arbitrary solution toward the stationary one is (at least) algebraic. A lower bound on the exponent is also given.
First Page:
Show
Hide
Full-text: Open access
Links and Identifiers
Permanent link to this document: http://projecteuclid.org/euclid.aop/1109868598
Digital Object Identifier: doi:10.1214/009117904000000892
Mathematical Reviews number (MathSciNet): MR2123208
Zentralblatt MATH identifier: 02164480
References
Arnold, L. (1998). Random Dynamical Systems. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR1723992
Cerrai, S. (1999). Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces. Probab. Theory Related Fields 113 85--114.
Mathematical Reviews (MathSciNet): MR1670729
Digital Object Identifier: doi:10.1007/s004400050203
Zentralblatt MATH: 0920.60061
Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108--140.
Mathematical Reviews (MathSciNet): MR1883719
Digital Object Identifier: doi:10.1007/s004400100158
Zentralblatt MATH: 1047.60029
Crauel, H. (1991). Markov measures for random dynamical systems. Stochastics Stochastics Rep. 37 153--173.
Mathematical Reviews (MathSciNet): MR1148346
Crauel, H. (2002). Invariant measures for random dynamical systems on the circle. Arch. Math. (Basel) 78 145--154.
Mathematical Reviews (MathSciNet): MR1888416
Digital Object Identifier: doi:10.1007/s00013-002-8228-y
Zentralblatt MATH: 1015.37040
Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge Univ. Press.
Mathematical Reviews (MathSciNet): MR1207136
Duncan, T. E., Hu, Y. and Pasik-Duncan, B. (2000). Stochastic calculus for fractional Brownian motion. I. Theory. SIAM J. Control Optim. 38 582--612 (electronic).
Mathematical Reviews (MathSciNet): MR1741154
Eckmann, J.-P. and Hairer, M. (2001). Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Comm. Math. Phys. 219 523--565.
Mathematical Reviews (MathSciNet): MR1838749
Digital Object Identifier: doi:10.1007/s002200100424
Zentralblatt MATH: 0983.60058
Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes. Wiley, New York.
Mathematical Reviews (MathSciNet): MR838085
Zentralblatt MATH: 0592.60049
Gänssler, P. and Stute, W. (1977). Wahrscheinlichkeitstheorie. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR501219
Hairer, M. (2002). Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124 345--380.
Mathematical Reviews (MathSciNet): MR1939651
Digital Object Identifier: doi:10.1007/s004400200216
Zentralblatt MATH: 1032.60056
Jakšić, V. and Pillet, C.-A. (1997). Ergodic properties of the non-Markovian Langevin equation. Lett. Math. Phys. 41 49--57.
Mathematical Reviews (MathSciNet): MR1455031
Digital Object Identifier: doi:10.1023/A:1007307617547
Zentralblatt MATH: 0894.60051
Jakšić, V. and Pillet, C.-A. (1998). Ergodic properties of classical dissipative systems. I. Acta Math. 181 245--282.
Mathematical Reviews (MathSciNet): MR1668590
Kuksin, S. B. and Shirikyan, A. (2001). A coupling approach to randomly forced nonlinear PDE's. I. Comm. Math. Phys. 221 351--366.
Mathematical Reviews (MathSciNet): MR1845328
Digital Object Identifier: doi:10.1007/s002200100479
Zentralblatt MATH: 0991.60056
Lyons, T. J. (1994). Differential equations driven by rough signals. I. An extension of an inequality of L. C. Young. Math. Res. Lett. 1 451--464.
Mathematical Reviews (MathSciNet): MR1302388
Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215--310.
Mathematical Reviews (MathSciNet): MR1654527
Mandelbrot, B. B. and Van Ness, J. W. (1968). Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10 422--437.
Mathematical Reviews (MathSciNet): MR242239
Digital Object Identifier: doi:10.1137/1010093
Zentralblatt MATH: 0179.47801
Maslowski, B. and Nualart, D. (2002). Evolution equations driven by a fractional Brownian motion. Preprint.
Mathematical Reviews (MathSciNet): MR1994773
Digital Object Identifier: doi:10.1016/S0022-1236(02)00065-4
Zentralblatt MATH: 1027.60060
Maslowski, B. and Schmalfuss, B. (2002). Random dynamical systems and stationary solutions of differential equations driven by the fractional Brownian motion. Preprint.
Mattingly, J. C. (2002). Exponential convergence for the stochastically forced Navier--Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230 421--462.
Mathematical Reviews (MathSciNet): MR1937652
Digital Object Identifier: doi:10.1007/s00220-002-0688-1
Zentralblatt MATH: 1054.76020
Meyn, S. P. and Tweedie, R. L. (1994). Markov Chains and Stochastic Stability. Springer, New York.
Mathematical Reviews (MathSciNet): MR1287609
Zentralblatt MATH: 0925.60001
Øksendal, B. and Hu, Y. (1999). Fractional white noise calculus and applications to finance. Preprint.
Risken, H. (1989). The Fokker--Planck Equation, 2nd ed. Springer, Berlin.
Mathematical Reviews (MathSciNet): MR987631
Zentralblatt MATH: 0665.60084
Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Springer, New York.
Mathematical Reviews (MathSciNet): MR1725357
Zentralblatt MATH: 0917.60006
Samko, S. G., Kilbas, A. A. and Marichev, O. I. (1993). Fractional Integrals and Derivatives. Gordon and Breach Science, Yverdon.
Mathematical Reviews (MathSciNet): MR1347689
Zentralblatt MATH: 0818.26003
The Annals of Probability
