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Invariant Measure and Random Attractors for Stochastic Differential Equations with Delay

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Abstract

In this paper, we consider the long term behavior of solutions to stochastic delay differential equations with additive noise. We first establish the existence of an invariant measure by regarding the stochastic equation as a perturbation of a deterministic equation. If the deterministic part is stable it is plausible to expect the existence of an invariant measure under some conditions on the perturbation. Different from traditional methods, a retarded Ornstein-Uhlenbeck process is introduced to the study of the existence of a continuous random dynamical system and tempered pullback attractors. In particular, we prove pullback asymptotical compactness of solutions as well as the existence and uniqueness of tempered random attractors. We also establish the upper semicontinuity of the random attractors when noise intensity approaches zero, and investigate the exponential stability of stationary solutions in the mean square sense and almost surely. It is found that our theoretical results and methods can be generalized to other stochastic delay differential equations, even with multiplicative noise. Finally, our theoretical results are illustrated by an applications to a stochastic Nicholson’s blowflies model.

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References

  1. Guo, S.: Bifurcation in a reaction-diffusion model with nonlocal delay effect and nonlinear boundary condition. J. Differ. Equ. 289, 236–278 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  2. Guo, S., Li, S.: On the stability of reaction-diffusion models with nonlocal delay effect and nonlinear boundary condition. Appl. Math. Lett. 103, 106197 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  3. Guo, S., Wu, J.: Bifurcation Theory of Functional Differential Equations. Springer, New York (2013)

    Book  MATH  Google Scholar 

  4. Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer-Verlag, New York (1993)

    Book  MATH  Google Scholar 

  5. Li, S., Guo, S.: Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete Contin. Dyn. Syst. Ser. B 26(5), 2693–2719 (2021)

    MathSciNet  MATH  Google Scholar 

  6. Li, S., Guo, S.: Persistence and extinction of a stochastic SIS epidemic model with regime switching and Lvy jumps. Discrete Contin. Dyn. Syst. Ser. B 26(9), 5101–5134 (2021)

    MathSciNet  MATH  Google Scholar 

  7. Li, S., Guo, S.: Permanence and extinction of a stochastic prey-predator model with a general functional response. Math. Comput. Simul. 187, 308–336 (2021)

    Article  MATH  Google Scholar 

  8. Ruelle, D.: Characteristic exponents for a viscous fluid subjected to time dependent forces. Comm. Math. Phys. 93, 285–300 (1984)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Google Scholar 

  12. Babin, A.V., Vishik, M.I.: Attractors of Evolution Equations. North-Holland, Amsterdam (1992)

    MATH  Google Scholar 

  13. Hale, J.K.: Asymptotic Behavior of Dissipative Systems. American Mathematical Society, Providence, RI (1988)

    MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  15. Caraballo, T., Real, J.: Attractors for 2D-Navier-Stokes models with delays. J. Differ. Equ. 205, 271–297 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. Marin-Rubio, P., Real, J.: Attractors for 2D-Navier-Stokes equations with delays on some unbounded domains. Nonlinear Anal. 67, 2784–2799 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Bessaih, H., Garrido-Atienza, M., Schmalfuss, B.: Pathwise solutions and attractors for retarded SPDEs with time smooth diffusion coefficients. Discrete Contin. Dyn. Syst. 34, 3945–3968 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Caraballo, T., Garrido-Atienza, M.J., Schmalfuss, B., Valero, J.: Non-autonomous and random attractors for delay random semilinear equations without uniqueness. Discrete Contin. Dyn. Syst. 21, 415–433 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Caraballo, T., Garrido-Atienza, M.J., Schmalfuss, B., Valero, J.: Asymptotic behaviour of a stochastic semilinear dissipative functional equation without uniqueness of solutions. Discrete Contin. Dyn. Syst. Ser. B 14, 439–455 (2010)

    MathSciNet  MATH  Google Scholar 

  20. Caraballo, T., Real, J., Chueshov, I.D.: Pullback attractors for stochastic heat equations in materials with memory. Discrete Contin. Dyn. Syst. Ser. B 9, 525–539 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, S., Guo, S.: Random attractors for stochastic semilinear degenerate parabolic equations with delay. Physica A 550, 124164 (2020)

    Article  MathSciNet  Google Scholar 

  22. Wu, F., Kloeden, P.E.: Mean-square random attractors of stochastic delay differential equations with random delay. Discrete Contin. Dyn. Syst. Ser. B 18, 1715–1734 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Yan, W., Li, Y., Ji, S.: Random attractors for first order stochastic retarded lattice dynamical systems. J. Math. Phys. 51, 032702 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  24. Butkovsky, O., Scheutzow, M.: Invariant measures for stochastic functional differential equations. Electron. J. Probab. 22(98), 1–23 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Es-Sarhir, A., Gaans, O.V., Scheutzow, M.: Invariant measures for stochastic functional differential equations with superlinear drift term. Differ. Integr. Equ. 23, 189–200 (2010)

    MathSciNet  MATH  Google Scholar 

  26. Bao, J., Yin, G., Yuan, C.: Ergodicity for functional stochastic differential equations and applications. Nonlinear Anal. 98, 66–82 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. Jiang, J., Lv, X.: A small-gain theorem for nonlinear stochastic systems with inputs and outputs I: additive white. SIAM J. Control Optim. 54(5), 2383–2402 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Küchler, U., Mensch, B.: Langevins stochastic differential equation extended by a time-delayed term. Stochastics 40, 23–42 (1992)

    MathSciNet  MATH  Google Scholar 

  29. Bakhtin, Y., Mattingly, J.C.: Stationary solutions of stochastic differential equations with memory and stochastic partial differential equations. Commun. Contemp. Math 7(5), 553–582 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  30. Chow, P.-L., Khasminskii, R.Z.: Stationary solutions of nonlinear stochastic evolution equations. Stoch. Anal. Appl 15(5), 671–699 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Scheutzow, M.: Qualitative behaviour of stochastic delay equations with a bounded memory. Stochastics 12, 41–80 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  32. van Gaans, O., Verduyn Lunel, S.: Long term behavior of dichotomous stochastic differential equations in Hilbert spaces. Commun. Contemp. Math 6(3), 349–376 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  33. Arnold, L.: Random Dynamical Systems. Springer-Verlag, Berlin (1998)

    Book  MATH  Google Scholar 

  34. Caraballo, T., Kloeden, P., Schmalfu\(\beta \), B.: Exponentially stable stationary solutions for stochastic evolution equations and their perturbation. Appl. Math. Optim. 50, 183-207 (2004)

  35. Bates, P.W., Lisei, H., Lu, K.: Attractors for stochastic lattice dynamical system. Stoch. Dyn. 6, 1–21 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  36. Crauel, H., Debussche, A., Flandoli, F.: Random attractors. J. Dyn. Differ. Equ. 9, 307–341 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  38. Karatzas, I., Shreve, S.: Brownian Motion and Stochastic Calculus. Springer Verlag, New York (1988)

    Book  MATH  Google Scholar 

  39. Scheutzow, M.: Chaining techniques and their application to stochastic flows. In: Blath, J., Mörters, P., Scheutzow, M. (eds.) Trends in Stochastic Analysis, LMS Lecture Notes Series 353. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  40. Wang, X.H., Shen, J., Lu, K., Wang, B.X.: Wong-Zakai approximations and random attractors for non-autonomous stochastic lattice systems. J. Differ. Equ. 280, 477–516 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  41. Nicholson, A.J.: An outline of the dynamics of animal populations. Aust. J. Zool. 2(1), 9–65 (1954)

    Article  Google Scholar 

  42. Sri Namachchivaya, N.: Co-dimension two bifurcations in the presence of noise. ASME J. Appl. Mech. 58, 259–265 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of People’s Republic of China (Grant Nos. 11671123 and 12071446).

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Authors and Affiliations

  1. School of Mathematics and Physics, China University of Geosciences, Wuhan, 430074, Hubei, People’s Republic of China

    Shangjiang Guo

  2. Center for Mathematical Sciences, China University of Geosciences, Wuhan, 430074, Hubei, People’s Republic of China

    Shangzhi Li

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  1. Shangjiang Guo
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All authors contributed to the study conception and design. Material preparation and analysis were performed by SG; Numerical simulations were performed by SL. The first draft of the manuscript was written by SG and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to Shangjiang Guo.

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This work supported in part by the National Natural Science Foundation of China (Grant Nos. 12071446 & 12101578) and the Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) (Grant No. CUGST2).

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Guo, S., Li, S. Invariant Measure and Random Attractors for Stochastic Differential Equations with Delay. Qual. Theory Dyn. Syst. 21, 40 (2022). https://doi.org/10.1007/s12346-022-00569-y

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