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Weak mean attractors and invariant measures for stochastic Schrödinger delay lattice systems

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Abstract

In this paper, we study the long term dynamics of the stochastic Schrödinger delay lattice systems when the nonlinear drift and diffusion terms are both locally Lipschitz continuous. Based on the well-posedness of the system, we first prove the existence and uniqueness of weak pullback mean random attractors in a product Hilbert space. We then show the tightness of distribution laws of solutions and the existence of invariant measures. We further prove the set of all invariant measures of the delay system is tight and every limit point of invariant measures of the delay system must be an invariant measure of the limiting system as time delay approaches zero. The idea of uniform tail-estimates is employed to to establish the tightness of distributions of solutions as well as the set of invariant measures of the delay system.

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References

  1. Bao, J., Yin, G., Yuan, C., Wang, L.: Exponential ergodicity for retarded stochastic differential equations. Appl. Anal. 93(11), 2330–2349 (2014)

    MathSciNet  MATH  Google Scholar 

  2. Bates, P.W., Chmaj, A.: A discrete convolution model for phase transitions. Arch. Rat. Mech. Anal. 150, 281–368 (1999)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  4. Bates, P.W., Lu, K., Wang, B.: Attractors of non-autonomous stochastic lattice systems in weighted spaces. Phys. D Nonlinear Phenom. 289, 32–50 (2014)

    MathSciNet  MATH  Google Scholar 

  5. Bell, J.: Some threshold results for models of myelinated nerves. Math. Biosci. 54, 181–190 (1981)

    MathSciNet  MATH  Google Scholar 

  6. Beyn, W.J., Pilyugin, S.Y.: Attractors of reaction diffusion systems on infinite lattices. J. Dyn. Diff. Equ. 15, 485–515 (2003)

    MathSciNet  MATH  Google Scholar 

  7. Bo, L., Yuan, C.: Stochastic delay differential equations with jump reflection: invariant measure. Stochastics 88(6), 841–863 (2016)

    MathSciNet  MATH  Google Scholar 

  8. Brzezniak, Z., Motyl, E., Ondrejat, M.: Invariant measure for the stochastic Navier–Stokes equations in unbounded 2D domains. Ann. Probab. 45, 3145–3201 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Brzezniak, Z., Ondrejat, M., Seidler, J.: Invariant measures for stochastic nonlinear beam and wave equations. J. Differ. Equ. 260, 4157–4179 (2016)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  11. Caraballo, T., Morillas, F., Valero, J.: Attractors of stochastic lattice dynamical systems with a multiplicative noise and non-Lipschitz nonlinearities. J. Differ. Equ. 253, 667–693 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Caraballo, T., Morillas, F., Valero, J.: On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems. Discrete Contin. Dyn. Syst. 34(1), 51–77 (2013)

    MathSciNet  MATH  Google Scholar 

  13. Caraballo, T., Morillas, F., Valero, J.: Asymptotic behaviour of a logistic lattice system. Discrete Contin. Dyn. Syst. 34, 4019–4037 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Chen, L., Dong, Z., Jiang, J., Zhai, J.: On limiting behavior of stationary measures for stochastic evolution systems with small noise intensity. Sci. China Math. 63, 1463–1504 (2020)

    MathSciNet  MATH  Google Scholar 

  15. Chen, Z., Li, X., Wang, B.: Invariant measures of stochastic delay lattice systems. Discrete Contin. Dyn. Syst. Ser. B 26, 3235–3269 (2021)

    MathSciNet  MATH  Google Scholar 

  16. Chen, T., Zhou, S., Zhao, C.: Attractors for discrete nonlinear Schrödinger equation with delay. Acta Math. Appl. Sin. 26(4), 633–642 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Chow, S.N., Mallet-Paret, J.: Pattern formation and spatial chaos in lattice dynamical systems, I, II. IEEE Trans. Circuits Syst. 42, 746–751 (1995)

    MathSciNet  Google Scholar 

  18. Chow, S.N., Mallet-Paret, J., Shen, W.: Traveling waves in lattice dynamical systems. J. Differ. Equ. 149, 248–291 (1998)

    MathSciNet  MATH  Google Scholar 

  19. Chua, L.O., Roska, T.: The CNN paradigm. IEEE Trans. Circuits Syst. 40, 147–156 (1993)

    MATH  Google Scholar 

  20. Da Prato, G., Zabczyk, J.: Stochastic Equations Infinite Dimensions. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  22. Han, X.: Random attractors for stochastic sine-Gordon lattice systems with multiplicative white noise. J. Math. Anal. Appl. 376, 481–493 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Han, X., Kloeden, P.: Non-autonomous lattice systems with switching effects and delayed recovery. J. Differ. Equ. 261(6), 2986–3009 (2016)

    MathSciNet  MATH  Google Scholar 

  24. Han, X., Kloeden, P.: Asymptotic behavior of a neural field lattice model with a Heaviside operator. Phys. D Nonlinear Phenom. 389, 1–12 (2019)

    MathSciNet  MATH  Google Scholar 

  25. Han, X., Shen, W., Zhou, S.: Random attractors for stochastic lattice dynamical systems in weighted spaces. J. Differ. Equ. 250, 1235–1266 (2011)

    MathSciNet  MATH  Google Scholar 

  26. Huang, J., Han, X., Zhou, S.: Uniform attractors for non-autonomous Klein-Gordon-Schrödinger lattice systems. Appl. Math. Mech. 30, 1597–1607 (2009)

    MathSciNet  MATH  Google Scholar 

  27. Keener, J.P.: Propagation and its failure in coupled systems of discrete excitable cells. SIAM J. Appl. Math. 47, 556–572 (1987)

    MathSciNet  MATH  Google Scholar 

  28. Kim, J.: Periodic and invariant measures for stochastic wave equations. Electron. J. Differ. Equ. 2004(05), 1–30 (2004)

    MathSciNet  Google Scholar 

  29. Kim, J.: On the stochastic Burgers equation with polynomial nonlinearity in the real line. Discrete Contin. Dyn. Syst. Ser. B 6, 835–866 (2006)

    MathSciNet  MATH  Google Scholar 

  30. Kim, J.: On the stochastic Benjamin-Ono equation. J. Differ. Equ. 228, 737–768 (2006)

    MathSciNet  MATH  Google Scholar 

  31. Kim, J.: Invariant measures for a stochastic nonlinear Schrödinger equation. Indiana Univ. Math. J. 55, 687–717 (2006)

    MathSciNet  MATH  Google Scholar 

  32. Kloeden, P., Lorenz, T.: Mean-square random dynamical systems. J. Differ. Equ. 253, 1422–1438 (2012)

    MathSciNet  MATH  Google Scholar 

  33. Li, D., Shi, L., Wang, X.: Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete Contin. Dyn. Syst. Ser. B 24(9), 5121–5148 (2019)

    MathSciNet  MATH  Google Scholar 

  34. Li, D., Wang, B., Wang, X.: Limiting behavior of invariant measures of stochastic delay lattice systems. J. Dyn. Differ. Equ. (2021). https://doi.org/10.1007/s10884-021-10011-7

    Article  MATH  Google Scholar 

  35. Li, D., Wang, B., Wang, X.: Periodic measures of stochastic delay lattice systems. J. Differ. Equ. 272, 74–104 (2021)

    MathSciNet  MATH  Google Scholar 

  36. Misiats, O., Stanzhytskyi, O., Yip, N.: Existence and uniqueness of invariant measures for stochastic reaction-diffusion equations in unbounded domains. J. Theor. Probab. 29, 996–1026 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Van Vleck, E., Wang, B.: Attractors for lattice FitzHugh–Nagumo systems. Phys. D Nonlinear Phenom. 212, 317–336 (2005)

    MathSciNet  MATH  Google Scholar 

  38. Wang, B.: Dynamics of systems on infinite lattices. J. Differ. Equ. 221, 224–245 (2006)

    MathSciNet  MATH  Google Scholar 

  39. Wang, B.: Weak pullback attractors for mean random dynamical systems in Bochner spaces. J. Dyn. Differ. Equ. 31, 2177–2204 (2019)

    MathSciNet  MATH  Google Scholar 

  40. Wang, B.: Weak pullback attractors for stochastic Navier–Stokes equations with nonlinear diffusion terms. Proc. Am. Math. Soc. 147(4), 1627–1638 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Wang, B.: Dynamics of stochastic reaction-diffusion lattice systems driven by nonlinear noise. J. Math. Anal. Appl. 477, 104–132 (2019)

    MathSciNet  MATH  Google Scholar 

  42. Wang, B.: Dynamics of fractional stochastic reaction-diffision equations on unbounded domains driven by nonlinear noise. J. Differ. Equ. 268, 1–59 (2019)

    Google Scholar 

  43. Wang, B., Wang, R.: Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise. Stoch. Anal. Appl. 38, 213–237 (2020)

    MathSciNet  MATH  Google Scholar 

  44. Wang, R., Li, Y.: Regularity and backward compactness of attractors for non-autonomous lattice systems with random coefficients. Appl. Math. Comput. 354, 86–102 (2019)

    MathSciNet  MATH  Google Scholar 

  45. Wang, Z.C., Li, W.T., Wu, J.H.: Entire solutions in delayed lattice differential equations with monostable nonlinearity. SIAM J. Math. Anal. 40(6), 2392–2420 (2009)

    MathSciNet  MATH  Google Scholar 

  46. Wang, X., Lu, K., Wang, B.: Exponential stability of non-autonomous stochastic delay lattice systems with multiplicative noise. J. Dyn. Differ. Equ. 28, 1309–1335 (2016)

    MathSciNet  MATH  Google Scholar 

  47. Wu, F., Yin, G., Mei, H.: Stochastic functional differential equations with infinite delay: Existence and uniqueness of solutions, solution maps, Markov properties, and ergodicity. J. Differ. Equ. 262(3), 1226–1252 (2017)

    MathSciNet  MATH  Google Scholar 

  48. Yan, W.P., Li, Y., Ji, S.G.: Random attractors for first order stochastic retarded lattice dynamical systems. J. Math. Phys. 51(3), 032702 (2010)

    MathSciNet  MATH  Google Scholar 

  49. Zhang, K.F., Zhao, X.Q.: Spreading speed and travelling waves for a spatially discrete SIS epidemic model. Nonlinearity 21(1), 97 (2007)

    MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work is partially supported by the NNSF of China (11471190, 11971260), the NSF of Shandong Province (ZR2014AM002), and the PSF (2012M511488, 2013T60661, 201202023).

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Authors and Affiliations

  1. School of Mathematics, Shandong University, Jinan, 250100, People’s Republic of China

    Zhang Chen

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

    Bixiang Wang

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  1. Zhang Chen
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  2. Bixiang Wang
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Correspondence to Bixiang Wang.

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Chen, Z., Wang, B. Weak mean attractors and invariant measures for stochastic Schrödinger delay lattice systems. J Dyn Diff Equat 35, 3201–3240 (2023). https://doi.org/10.1007/s10884-021-10085-3

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