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Law of large numbers and central limit theorem for randomly forced PDE's

  • Armen Shirikyan
Article

Abstract

We consider a class of dissipative PDE's perturbed by an external random force. Under the condition that the distribution of perturbation is sufficiently non-degenerate, a strong law of large numbers (SLLN) and a central limit theorem (CLT) for solutions are established and the corresponding rates of convergence are estimated. It is also shown that the estimates obtained are close to being optimal. The proofs are based on the property of exponential mixing for the problem in question and some abstract SLLN and CLT for mixing-type Markov processes.

Mathematics Subject Classifications (2000)

35Q30 60F05 60H15 60J05 

Key words or phrases

Strong law of large numbers Central limit theorem Rate of convergence Exponential mixing Randomly forced PDE's 

References

  1. 1.
    Bolthausen, E.: The Berry-Esseén theorem for strongly mixing Harris recurrent Markov chains. Z. Wahrsch. Verw. Gebiete 60 (3), 283–289 (1982)MathSciNetGoogle Scholar
  2. 2.
    Bolthausen, E.: Exact convergence rates in some martingale central limit theorems. Ann. Probab. 10 (3), 672–688 (1982)MathSciNetGoogle Scholar
  3. 3.
    Bricmont, J., Kupiainen, A., Lefevere, R.: Probabilistic estimates for the two-dimensional Navier–Stokes equations. J. Statist. Phys. 100 (3–4), 743–756 (2000)Google Scholar
  4. 4.
    Bricmont, J., Kupiainen, A., Lefevere, R.: Exponential mixing for the 2D stochastic Navier–Stokes dynamics. Commun. Math. Phys. 230 (1), 87–132 (2002)MathSciNetGoogle Scholar
  5. 5.
    Bricmont, J.: Ergodicity and mixing for stochastic partial differential equations. Proceedings of the International Congress of Mathematicians. Vol. I (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 567–585Google Scholar
  6. 6.
    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite–Dimensional Systems. Cambridge University Press, Cambridge, 1996Google Scholar
  7. 7.
    E, W., Mattingly, J.C., Sinai, Ya.G.: Gibbsian dynamics and ergodicity for the stochastically forced Navier–Stokes equation. Commun. Math. Phys. 224 (1), 83–106 (2001)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Eckmann, J.-P., Hairer, M.: Uniqueness of the invariant measure for a stochastic PDE driven by degenerate noise. Commun. Math. Phys. 219 (3), 523–565 (2001)MathSciNetGoogle Scholar
  9. 9.
    Ferrario, B.: Ergodic results for stochastic Navier-Stokes equation. Stochastics Stochastics Rep. 60 (3–4), 271–288 (1997)Google Scholar
  10. 10.
    Flandoli, F., Maslowski, B.: Ergodicity of the 2-D Navier-Stokes equation under random perturbations. Commun. Math. Phys. 172 (1), 119–141 (1995)MathSciNetGoogle Scholar
  11. 11.
    Gallavotti, G.: Foundations of Fluid Dynamics. Springer-Verlag, Berlin, 2002Google Scholar
  12. 12.
    Hairer, M.: Exponential mixing properties of stochastic PDE's through asymptotic coupling. Probab. Theory Relat. Fields 124 (3), 345–380 (2002)MathSciNetGoogle Scholar
  13. 13.
    Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Application. Academic Press, New York–London, 1980Google Scholar
  14. 14.
    Hasminskii, R.Z.: Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations. Teor. Verojatnost. i Primenen. 5, 196–214 (1960)MathSciNetGoogle Scholar
  15. 15.
    Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes. Springer-Verlag, Berlin, 1987Google Scholar
  16. 16.
    Kuksin, S.: On exponential convergence to a stationary measure for nonlinear PDE's, perturbed by random kick-forces, and the turbulence-limit. The M.I. Vishik Moscow PDE seminar, AMS Translations, 2002Google Scholar
  17. 17.
    Kuksin, S.: Ergodic theorems for 2D statistical hydrodynamics. Rev. Math. Physics 14 (6), 585–600 (2002)MathSciNetGoogle Scholar
  18. 18.
    Kuksin, S., Shirikyan, A.: Stochastic dissipative PDE's and Gibbs measures. Commun. Math. Phys. 213 (2), 291–330 (2000)MathSciNetGoogle Scholar
  19. 19.
    Kuksin, S., Shirikyan, A.: A coupling approach to randomly forced nonlinear PDE's. I. Commun. Math. Phys. 221 (2), 351–366 (2001)MathSciNetGoogle Scholar
  20. 20.
    Kuksin, S., Piatnitski, A., Shirikyan, A.: A coupling approach to randomly forced nonlinear PDE's. II. Commun. Math. Phys. 230 (1), 81–85 (2002)MathSciNetGoogle Scholar
  21. 21.
    Kuksin, S., Shirikyan, A.: Coupling approach to white-forced nonlinear PDE's. J. Math. Pures Appl. 81 (6), 567–602 (2002)MathSciNetGoogle Scholar
  22. 22.
    Kuksin, S., Shirikyan, A.: Some limiting properties of randomly forced 2D Navier–Stokes equations. Proc. Roy. Soc. Edinburgh Sect. A 133 (4), 875–891 (2003)MathSciNetGoogle Scholar
  23. 23.
    Landers, D., Rogge, L.: On the rate of convergence in the central limit theorem for Markov-chains. Z. Wahrsch. Verw. Gebiete 35 (1), 57–63 (1976)MathSciNetGoogle Scholar
  24. 24.
    Liptser, R.S., Shiryayev, A.N.: Theory of Martingales. Kluwer, Dordrecht, 1989Google Scholar
  25. 25.
    Maruyama, G., Tanaka, H.: Some properties of one-dimensional diffusion processes. Mem. Fac. Sci. Kyusyu Univ. Ser. A. Math. 11, 117–141 (1957)zbMATHMathSciNetGoogle Scholar
  26. 26.
    Maruyama, G., Tanaka, H.: Ergodic property of N-dimensional recurrent Markov processes. Mem. Fac. Sci. Kyushu Univ. Ser. A 13, 157–172 (1959)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Masmoudi, N., Young, L.-S.: Ergodic theory of infinite dimensional systems with applications to dissipative parabolic PDEs. Commun. Math. Phys. 227 (3), 461–481 (2002)MathSciNetGoogle Scholar
  28. 28.
    Mattingly, J.C.: Ergodicity of 2D Navier-Stokes equations with random forcing and large viscosity. Commun. Math. Phys. 206 (2), 273–288 (1999)MathSciNetGoogle Scholar
  29. 29.
    Mattingly, J.C.: Exponential convergence for the stochastically forced Navier–Stokes equations and other partially dissipative dynamics. Commun. Math. Phys. 230 (3), 421–462 (2002)MathSciNetGoogle Scholar
  30. 30.
    Mattingly, J.C.: On recent progress for the stochastic Navier Stokes equations. Journées “Équations aux Dérivées Partielles”, Exp. No. XI, 52 pp., Univ. Nantes, Nantes, 2003Google Scholar
  31. 31.
    Meyn, S.P., Tweedie, R.L.: Markov Chains and Stochastic Stability. Springer-Verlag London, London, 1993Google Scholar
  32. 32.
    Rio, E.: Théorie Asymptotique des Processus Aléatoires Faiblement Dépendants. Springer, Berlin–Heidelberg, 2000Google Scholar
  33. 33.
    Sawyer, S.: Rates of convergence for some functionals in probability. Ann. Math. Statist. 43, 273–284 (1972)zbMATHMathSciNetGoogle Scholar
  34. 34.
    Shirikyan, A.: Analyticity of solutions for randomly forced two-dimensional Navier-Stokes equations. Russian Math. Surveys 57 (4), 785–799 (2002)MathSciNetGoogle Scholar
  35. 35.
    Shirikyan, A.: A version of the law of large numbers and applications. In: Probabilistic Methods in Fluids, Proceedings of the Swansea Workshop held on 14 – 19 April 2002, I.M. Davies et al (eds.), World Scientific, New Jersey, 2003, pp. 263–272Google Scholar
  36. 36.
    Shirikyan, A.: Exponential mixing for 2D Navier–Stokes equations perturbed by an unbounded noise. J. Math. Fluid Mech. 6 (2), 169–193 (2004)MathSciNetGoogle Scholar
  37. 37.
    Shirikyan, A.: Some mathematical problems of statistical hydrodynamics. In: Proceedings of the International Congress of Mathematical Physics, Lisbon, 2003. To appearGoogle Scholar
  38. 38.
    Temam, R.: Navier–Stokes Equations. Theory and Numerical Analysis. North-Holland, Amsterdam–New York–Oxford, 1977Google Scholar
  39. 39.
    Veretennikov, A.Yu.: Estimates of the mixing rate for stochastic equations. Theory Probab. Appl. 32 (2), 273–281 (1987)Google Scholar
  40. 40.
    Vishik, M.I., Fursikov, A.V.: Mathematical Problems in Statistical Hydromechanics. Kluwer, Dordrecht, 1988Google Scholar
  41. 41.
    Watanabe, H., Hisao, M.: Ergodic property of recurrent diffusion processes. J. Math. Soc. Japan 10, 272–286 (1958)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.Laboratoire de MathématiquesUniversité de Paris-Sud XIOrsay CedexFrance

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