Advertisement

Probability Theory and Related Fields

, Volume 149, Issue 1–2, pp 223–259 | Cite as

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

  • M. Hairer
  • J. C. Mattingly
  • M. Scheutzow
Article

Abstract

There are many Markov chains on infinite dimensional spaces whose one-step transition kernels are mutually singular when starting from different initial conditions. We give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of Harris’ theorem. The first uses the existence of couplings which draw the solutions together as time goes to infinity. Such “asymptotic couplings” were central to (Mattingly and Sinai in Comm Math Phys 219(3):523–565, 2001; Mattingly in Comm Math Phys 230(3):461–462, 2002; Hairer in Prob Theory Relat Field 124:345–380, 2002; Bakhtin and Mattingly in Commun Contemp Math 7:553–582, 2005) on which this work builds. As in Bakhtin and Mattingly (2005) the emphasis here is on stochastic differential delay equations. Harris’ celebrated theorem states that if a Markov chain admits a Lyapunov function whose level sets are “small” (in the sense that transition probabilities are uniformly bounded from below), then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed. This convergence takes place in a total variation norm, weighted by the Lyapunov function. A second aim of this article is to replace the notion of a “small set” by the much weaker notion of a “d-small set,” which takes the topology of the underlying space into account via a distance-like function d. With this notion at hand, we prove an analogue to Harris’ theorem, where the convergence takes place in a Wasserstein-like distance weighted again by the Lyapunov function. This abstract result is then applied to the framework of stochastic delay equations. In this framework, the usual theory of Harris chains does not apply, since there are natural examples for which there exist no small sets (except for sets consisting of only one point). This gives a solution to the long-standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it.

Keywords

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

Mathematics Subject Classification (2000)

34K50 37A30 60J05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bass R.: Diffusions and Elliptic Operators. Springer, New York (1998)zbMATHGoogle Scholar
  4. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Cranston M., Wang F.-Y.: A condition for the equivalence of coupling and shift coupling. Ann. Prob. 28(4), 1666–1679 (2002)MathSciNetGoogle Scholar
  6. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Da Prato G., Zabczyk J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)zbMATHCrossRefGoogle Scholar
  8. 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)Google Scholar
  9. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hairer M.: Exponential mixing properties of stochastic PDEs through asymptotic coupling, Prob. Theory Relat Fields 124, 345–380 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hairer M.: How hot can a heat bath get?. Comm. Math. Phys. 292(1), 131–177 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Hairer M., Mattingly J.: Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing. Ann. Math. 164, 993–1032 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Hairer M., Mattingly J.: Spectral gaps in Wasserstein distances and the 2D stochastic Navier–Stokes equations. Ann. Prob. 36(6), 2050–2091 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Hairer, M., Mattingly, J.: Yet another look at Harris’ ergodic theorem for Markov chains. (2008, Submitted)Google Scholar
  16. 16.
    Hairer, M., Mattingly, J.: A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs. (2008, Submitted)Google Scholar
  17. 17.
    Itô K., Nisio M.: On stationary solutions of a stochastic differential equation. J. Math. Kyoto Univ. 4, 1–75 (1964)zbMATHMathSciNetGoogle Scholar
  18. 18.
    Kuratowski K., Ryll-Nardzewski C.: A general theorem on selectors. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 13, 397–403 (1965)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Kuksin S., Shirikyan A.: Coupling approach to white-forced nonlinear PDEs. J. Math. Pures Appl. (9) 81(6), 567–602 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lindvall T.: Lectures on the Coupling Method. J Wiley, New York (1992)zbMATHGoogle Scholar
  21. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Mattingly, J.C.: Saint Flour Lectures. Summer 2007 version from summer school (in preparation)Google Scholar
  24. 24.
    Mohammed S.E.A.: Non-linear flows of stochastic linear delay equations. Stochastics 17, 207–213 (1986)zbMATHMathSciNetGoogle Scholar
  25. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Meyn S.P., Tweedie R.L.: Markov Chains and Stochastic Stability. Springer, London (1993)zbMATHGoogle Scholar
  27. 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)zbMATHCrossRefGoogle Scholar
  28. 28.
    van Renesse, M., Scheutzow, M.: Existence and uniqueness of solutions of stochastic functional differential equations. (2008, Submitted)Google Scholar
  29. 29.
    Revuz D., Yor M.: Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften, vol. 293. Springer-Verlag, Berlin (1991)Google Scholar
  30. 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)zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Scheutzow M.: Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics 12, 41–80 (1984)zbMATHMathSciNetGoogle Scholar
  32. 32.
    Scheutzow M.: Exponential growth rates for stochastic delay differential equations. Stoch. Dyn. 5, 163–174 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Scheutzow M., Steinsaltz D.: Chasing balls through martingale fields. Ann. Prob. 30, 2046–2080 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 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)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Villani C.: Topics in Optimal Transportation. Graduate Studies in Mathematics, 58. American Mathematical Society, Providence (2003)Google Scholar
  36. 36.
    Wagner D.H.: Survey of measurable selection theorems. SIAM J. Control Optim. 15(5), 859–903 (1977)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Mathematics InstituteThe University of WarwickCoventryUK
  2. 2.Courant InstituteNew York UniversityNew YorkUSA
  3. 3.Department of MathematicsDuke UniversityDurhamUSA
  4. 4.Institut für MathematikFakultät II, Mathematik und NaturwissenschaftenBerlinGermany

Personalised recommendations