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Wong–Zakai Approximations and Long Term Behavior of Stochastic Partial Differential Equations

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Abstract

In this paper we study the Wong–Zakai approximations given by a stationary process via the Wiener shift and their associated long term pathwise behavior for the stochastic partial differential equations driven by a white noise. We prove that the approximate equation has a pullback random attractor under much weaker conditions than the original stochastic equation. When the stochastic partial differential equation is driven by a linear multiplicative noise or additive white noise, we prove the convergence of solutions of Wong–Zakai approximations and the upper semicontinuity of random attractors of the approximate random system as the size of approximation approaches zero.

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References

  1. Arnold, L.: Random Dynamical Systems. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  2. Bally, V., Millet, A., Sanz-Solé, M.: Approximation and support theorem in Holder norm for parabolic stochastic partial differential equations. Ann. Probab. 23, 178–222 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brzezniak, Z., Capinski, M., Flandoli, F.: A convergence result for stochastic partial differential equations. Stochastics 24, 423–445 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brzezniak, Z., Flandoli, F.: Almost sure approximation of Wong–Zakai type for stochastic partial differential equations. Stoch. Process. Appl. 55, 329–358 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Caraballo, T., Langa, J.A., Melnik, V.S., Valero, J.: Pullback attractors for nonautonomous and stochastic multivalued dynamical systems. Set-Valued Anal. 11, 153–201 (2003)

  6. Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100, 365–393 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Deya, A., Jolis, M., Quer-Sardanyons, L.: The Stratonovich heat equation: a continuity result and weak approximations. Electron. J. Probab. 18, 1–34 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Flandoli, F., Schmalfuss, B.: Random attractors for the 3D stochastic Navier–Stokes equation with multiplicative noise. Stoch. Stoch. Rep. 59, 21–45 (1996)

    Article  MATH  Google Scholar 

  9. Flandoli, F.: Regularity Theory and Stochastic Flow for Parabolic SPDEs, Stochastics Monographs, vol. 9. Gordon and Breach Science Publishers SA, Singapore (1995)

    MATH  Google Scholar 

  10. Gao, H., Garrido-Atienza, M., Schmalfuss, B.: Random attractors for stochastic evolution equations driven by fractional Brownian motion. SIAM J. Math. Anal. 46(4), 22812309 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Garrido-Atienza, M., Lu, K., Schmalfuss, B.: Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters \(H\in (1/3,1/2]\). SIAM J. Appl. Dyn. Syst. 15(1), 625–654 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Grecksch, W., Schmalfuss, B.: Approximation of the stochastic Navier–Stokes equation. Mat. Apl. Comput. 15(3), 227–239 (1996)

    MathSciNet  MATH  Google Scholar 

  13. Ganguly, A.: Wong–Zakai type convergence in infinite dimensions. Electron. J. Probab. 18(31), 34 (2013)

    MathSciNet  MATH  Google Scholar 

  14. Gyongy, I., Shmatkov, A.: Rate of convergence of Wong–Zakai approximations for stochastic partial differential equations. Appl. Math. Optim. 54, 315–341 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gyongy, I.: On the approximation of stochastic partial differential equations, I. Stochastics 25, 59–85 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gyongy, I.: On the approximation of stochastic partial differential equations, II. Stochastics 26, 129–164 (1989)

    MathSciNet  MATH  Google Scholar 

  17. Hairer, M., Pardoux, E.: A Wong–Zakai theorem for stochastic PDEs. J. Math. Soc. Jpn. 67(4), 1551–1604 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes, 2nd edn. Noth-Holland, Amsterdam (1989)

    MATH  Google Scholar 

  19. Ikeda, N., Nakao, S., Yamato, Y.: A class of approximations of Brownian motion. Publ. RIMS Kyoto Univ. 13, 285–300 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kelley, D., Melbourne, I.: Smooth approximation of stochastic differential equations. Ann. Probab. 44, 479–520 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Konecny, F.: On Wong–Zakai approximation of stochastic differential equations. J. Multivar. Anal. 13, 605–611 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kurtz, T., Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19, 1035–1070 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kurtz, T., Protter, P.: Wong–Zakai Corrections, Random Evolutions, and Simulation Schemes for SDE. Stochastic Analysis: Liber Amicorum for Moshe Zakai, pp., 331–346. Academic Press, San Diego (1991)

    MATH  Google Scholar 

  24. Lions, J.L.: Quelques Methodes de Resolution des Problemes aux Limites Non Lineaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  25. Lu, K., Wang, Q.: Chaotic behavior in differential equations driven by a Brownian motion. J. Differ. Equ. 251, 2853–2895 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. McShane, E.J.: Stochastic Differential Equations and Models of Random Processes. Dissertation, Springer, Berlin (1972)

  27. Nakao, S.: On weak convergence of sequences of continuous local martingale. Ann. De L’I. H. P. Sect. B 22, 371–380 (1986)

    MathSciNet  MATH  Google Scholar 

  28. Nakao, S., Yamato, Y.: Approximation theorem on stochastic differential equations. In: Proceedings of the International Symposium on SDE, Kyoto, pp. 283–296 (1976)

  29. Nowak, A.: A Wong–Zakai type theorem for stochastic systems of Burgers equations. Panam. Math. J. 16(2), 1–25 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Protter, P.: Approximations of solutions of stochastic differential equations driven by semimartingales. Ann. Probab. 13, 716–743 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  31. Shen, J., Lu, K., Zhang, W.: Heteroclinic chaotic behavior driven by a Brownian motion. J. Differ. Equ. 255, 4185–4225 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Schmalfuss, B.: Backward cocycles and attractors of stochastic differential equations. In: Reitmann, V., Riedrich, T., Koksch, N. (eds.) International Seminar on Applied Mathematics-Nonlinear Dynamics: Attractor Approximation and Global Behavior, pp. 185–192. Technische Universitat, Dresden (1992)

    Google Scholar 

  33. Stroock, D.W., Varadhan, S.R.S.: On the support of diffusion processes with applications to the strong maximum principle. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 3, pp. 333–359 (1972)

  34. Sussmann, H.J.: An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point. Bull. Am. Math. Soc. 83, 296–298 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sussmann, H.J.: On the gap between deterministic and stochastic and stochastic ordinary differential equations. Ann. Probab. 6, 19–41 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  36. Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, New York (1997)

    Book  MATH  Google Scholar 

  37. Tessitore, G., Zabczyk, J.: Wong–Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6, 621–655 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  38. Twardowska, K.: On the approximation theorem of the Wong–Zakai type for the functional stochastic differential equations. Probab. Math. Stat. 12(2), 319–334 (1991)

    MathSciNet  MATH  Google Scholar 

  39. Twardowska, K.: An extension of the Wong–Zakai theorem for stochastic evolution equations in Hilbert spaces. Stoch. Anal. Appl. 10(4), 471–500 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. Twardowska, K.: An approximation theorem of Wong–Zakai type for nonlinear stochastic partial differential equations. Stoch. Anal. Appl. 13(5), 601–626 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  41. Twardowska, K.: Wong–Zakai approximations for stochastic differential equations. Acta Appl. Math. 43(3), 317–359 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, B.: Sufficient and necessary criteria for existence of pullback attractors for non-compact random dynamical systems. J. Differ. Equ. 253, 1544–1583 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang, B.: Existence and upper semicontinuity of attractors for stochastic equations with deterministic non-autonomous terms. Stoch. Dyn. 14(4), 1450009, 1-31 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wang, B.: Random attractors for non-autonomous stochastic wave equations with multiplicative noise. Discrete Contin. Dyn. Syst. Ser. A 34, 269–300 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  45. Wang, B.: Existence, stability and bifurcation of random complete and periodic solutions of stochastic parabolic equations. Nonlinear Anal. TMA 103, 9–25 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Wong, E., Zakai, M.: On the relation between ordinary and stochastic diferetnial equations. Int. J. Eng. Sci. 3, 213–229 (1965)

    Article  MATH  Google Scholar 

  47. Wong, E., Zakai, M.: On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat. 36, 1560–1564 (1965)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

Funding was provided by NSF (Grant No. 1413603) and NSFC (Grant No. 11331007).

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Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Kening Lu

  2. Department of Mathematics, New Mexico Institute of Mining and Technology, Socorro, NM, 87801, USA

    Bixiang Wang

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  1. Kening Lu
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  2. Bixiang Wang
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Correspondence to Kening Lu.

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Dedicated to the memory of George Sell

This work is partially supported by grants from NSF.

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Lu, K., Wang, B. Wong–Zakai Approximations and Long Term Behavior of Stochastic Partial Differential Equations. J Dyn Diff Equat 31, 1341–1371 (2019). https://doi.org/10.1007/s10884-017-9626-y

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  • DOI: https://doi.org/10.1007/s10884-017-9626-y

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