Asymptotic coupling and a general form of Harris' theorem with applications to stochastic delay equations

Probability Theory and Related Fields

Volumn 149, Issue 1, 2011, Pages 223-259

  Hairer, M ^a,b   Mattingly, J C ^c   Scheutzow, M ^d  

^a UNIVERSITY OF WARWICK   (United Kingdom)

^b NEW YORK UNIVERSITY   (United States)

^c DUKE UNIVERSITY   (United States)

^d TECHNISCHE UNIVERSITÄT BERLIN   (Germany)

Author keywords

Asymptotic coupling; Harris' theorem; Invariant measure; Spectral gap; Stochastic delay equation; Weak convergence

Indexed keywords

EID: 79952008924     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00440-009-0250-6     Document Type: Article

References (36)

1. Bakry D., Cattiaux P., Guillin A.: Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré. J. Funct. Anal. 254(3), 727-759 (2008).
2. Bakhtin Y., Mattingly J.: Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations. Commun. Contemp. Math. 7, 553-582 (2005).
3. Bass R.: Diffusions and Elliptic Operators. Springer, New York (1998).
4. Bricmont J., Kupiainen A., Lefevere R.: Ergodicity of the 2D Navier-Stokes equations with random forcing. Comm. Math. Phys. 224(1), 65-81 (2001).
5. Cranston M., Wang F.-Y.: A condition for the equivalence of coupling and shift coupling. Ann. Prob. 28(4), 1666-1679 (2002).
6. Douc R., Fort G., Guillin A.: Subgeometric rates of convergence of f-ergodic strong Markov processes. Stoch. Proc. Appl. 119(3), 897-923 (2009).
7. Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992).
8. Es-Sarhir A., van Gaans O., Scheutzow M.: Invariant measures for stochastic functional differential equations with superlinear drift term. Diff. Int. Equ. 23(1-2), 189-200 (2010).
9. Eckmann J.-P., Hairer M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Comm. Math. Phys. 219(3), 523-565 (2001).
10. Weinan E., Mattingly J. C., Sinai Y.: Gibbsian dynamics and ergodicity for the stochastically forced Navier-Stokes equation. Comm. Math. Phys. 224(1), 83-106 (2001).
11. Hairer M.: Exponential mixing properties of stochastic PDEs through asymptotic coupling, Prob. Theory Relat Fields 124, 345-380 (2002).
12. Hairer M.: How hot can a heat bath get?. Comm. Math. Phys. 292(1), 131-177 (2009).
13. Hairer M., Mattingly J.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 164, 993-1032 (2006).
14. Hairer M., Mattingly J.: Spectral gaps in Wasserstein distances and the 2D stochastic Navier-Stokes equations. Ann. Prob. 36(6), 2050-2091 (2008).
15. Hairer, M., Mattingly, J.: Yet another look at Harris' ergodic theorem for Markov chains. (2008, Submitted).
16. Hairer, M., Mattingly, J.: A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. (2008, Submitted).
17. Itô K., Nisio M.: On stationary solutions of a stochastic differential equation. J. Math. Kyoto Univ. 4, 1-75 (1964).
18. Kuratowski K., Ryll-Nardzewski C.: A general theorem on selectors. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13, 397-403 (1965).
19. Kuksin S., Shirikyan A.: Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. (9) 81(6), 567-602 (2002).
20. Lindvall T.: Lectures on the Coupling Method. J Wiley, New York (1992).
21. Masmoudi N., Young L.-S.: Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs. Comm. Math. Phys. 227(3), 461-481 (2002).
22. Mattingly J. C.: Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230(3), 421-462 (2002).
23. Mattingly, J. C.: Saint Flour Lectures. Summer 2007 version from summer school (in preparation).
24. Mohammed S. E. A.: Non-linear flows of stochastic linear delay equations. Stochastics 17, 207-213 (1986).
25. Mohammed S. E. A., Scheutzow M.: Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales. Part II: Examples and case studies. Ann. Prob. 25, 1210-1240 (1997).
26. Meyn S. P., Tweedie R. L.: Markov Chains and Stochastic Stability. Springer, London (1993).
27. Reiß M., Riedle M., van Gaans O.: Delay differential equations driven by Lévy processes: Stationarity and Feller properties. Stoch. Process. Appl. 116, 1409-1432 (2006).
28. van Renesse, M., Scheutzow, M.: Existence and uniqueness of solutions of stochastic functional differential equations. (2008, Submitted).
29. Revuz D., Yor M.: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol. 293. Springer-Verlag, Berlin (1991).
30. Rey-Bellet L., Thomas L.: Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Comm. Math. Phys. 225(2), 305-329 (2002).
31. Scheutzow M.: Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics 12, 41-80 (1984).
32. Scheutzow M.: Exponential growth rates for stochastic delay differential equations. Stoch. Dyn. 5, 163-174 (2005).
33. Scheutzow M., Steinsaltz D.: Chasing balls through martingale fields. Ann. Prob. 30, 2046-2080 (2002).
34. Veretennikov A. Yu.: On polynomial mixing and the rate of convergence for stochastic differential and difference equations. Theory Probab. Appl. 44(2), 361-374 (2000).
35. Villani C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence (2003).
36. Wagner D. H.: Survey of measurable selection theorems. SIAM J. Control Optim. 15(5), 859-903 (1977).