IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2023i1p112-d1309602.html
   My bibliography  Save this article

Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise

Author

Listed:
  • Qiu Lin

    (School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China)

  • Ruisheng Qi

    (School of Mathematics and Statistics, Yancheng Teachers University, Yancheng 224002, China)

Abstract

In this paper, we consider the ergodic semilinear stochastic partial differential equation driven by additive noise and the long-time behavior of its full discretization realized by a spectral Galerkin method in spatial direction and an Euler scheme in the temporal direction, which admits a unique invariant probability measure. Under the condition that the nonlinearity is once differentiable, the optimal convergence orders of the numerical invariant measures are obtained based on the time-independent weak error, but not relying on the associated Kolmogorov equation. More precisely, the obtained convergence orders are O ( λ N − γ ) in space and O ( τ γ ) in time, where γ ∈ ( 0 , 1 ] from the assumption ∥ A γ − 1 2 Q 1 2 ∥ L 2 is used to characterize the spatial correlation of the noise process. Finally, numerical examples confirm the theoretical findings.

Suggested Citation

  • Qiu Lin & Ruisheng Qi, 2023. "Optimal Weak Order and Approximation of the Invariant Measure with a Fully-Discrete Euler Scheme for Semilinear Stochastic Parabolic Equations with Additive Noise," Mathematics, MDPI, vol. 12(1), pages 1-29, December.
    • Handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:112-:d:1309602
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/1/112/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/1/112/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Mattingly, J. C. & Stuart, A. M. & Higham, D. J., 2002. "Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise," Stochastic Processes and their Applications, Elsevier, vol. 101(2), pages 185-232, October.
    2. Yang, Xuehua & Wu, Lijiao & Zhang, Haixiang, 2023. "A space-time spectral order sinc-collocation method for the fourth-order nonlocal heat model arising in viscoelasticity," Applied Mathematics and Computation, Elsevier, vol. 457(C).
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Abdumauvlen Berdyshev & Dossan Baigereyev & Kulzhamila Boranbek, 2023. "Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model," Mathematics, MDPI, vol. 11(17), pages 1-27, September.
    2. Lemaire, Vincent, 2007. "An adaptive scheme for the approximation of dissipative systems," Stochastic Processes and their Applications, Elsevier, vol. 117(10), pages 1491-1518, October.
    3. Elif Tan & Diana Savin & Semih Yılmaz, 2023. "A New Class of Leonardo Hybrid Numbers and Some Remarks on Leonardo Quaternions over Finite Fields," Mathematics, MDPI, vol. 11(22), pages 1-14, November.
    4. Jun Zhang & Jingjing Zhang & Shangyou Zhang, 2023. "Explicit Symplectic Runge–Kutta–Nyström Methods Based on Roots of Shifted Legendre Polynomial," Mathematics, MDPI, vol. 11(20), pages 1-13, October.
    5. Susanne Ditlevsen & Adeline Samson, 2019. "Hypoelliptic diffusions: filtering and inference from complete and partial observations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 81(2), pages 361-384, April.
    6. Song, Renming & Xie, Longjie, 2020. "Well-posedness and long time behavior of singular Langevin stochastic differential equations," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 1879-1896.
    7. Jiaqi Wang & Jianbing Su, 2023. "Boundedness and Compactness of Weighted Composition Operators from α -Bloch Spaces to Bers-Type Spaces on Generalized Hua Domains of the First Kind," Mathematics, MDPI, vol. 11(20), pages 1-27, October.
    8. Jiří Holman, 2023. "Numerical Solution of Transition to Turbulence over Compressible Ramp at Hypersonic Velocity," Mathematics, MDPI, vol. 11(17), pages 1-10, August.
    9. Quentin Clairon & Adeline Samson, 2020. "Optimal control for estimation in partially observed elliptic and hypoelliptic linear stochastic differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 23(1), pages 105-127, April.
    10. Jianyu Wang & Chunhua Fang & Guifeng Zhang, 2023. "Multi-Effective Collocation Methods for Solving the Volterra Integral Equation with Highly Oscillatory Fourier Kernels," Mathematics, MDPI, vol. 11(20), pages 1-19, October.
    11. Gao, Shuaibin & Li, Xiaotong & Liu, Zhuoqi, 2023. "Stationary distribution of the Milstein scheme for stochastic differential delay equations with first-order convergence," Applied Mathematics and Computation, Elsevier, vol. 458(C).
    12. Jeffrey A. Hogan & Joseph D. Lakey, 2023. "Spatio–Spectral Limiting on Replacements of Tori by Cubes," Mathematics, MDPI, vol. 11(23), pages 1-14, November.
    13. Nazim & Nadeem Ur Rehman & Ahmad Alghamdi, 2023. "On Normalized Laplacian Spectra of the Weakly Zero-Divisor Graph of the Ring ℤ n," Mathematics, MDPI, vol. 11(20), pages 1-14, October.
    14. ur Rahman, Ghaus & Badshah, Qaisar & Agarwal, Ravi P. & Islam, Saeed, 2021. "Ergodicity & dynamical aspects of a stochastic childhood disease model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 738-764.
    15. Li Cheng & Wen-Xiu Ma, 2023. "Similarity Transformations and Nonlocal Reduced Integrable Nonlinear Schrödinger Type Equations," Mathematics, MDPI, vol. 11(19), pages 1-8, September.
    16. Casella, Bruno & Roberts, Gareth O. & Stramer, Osnat, 2011. "Stability of Partially Implicit Langevin Schemes and Their MCMC Variants," MPRA Paper 95220, University Library of Munich, Germany.
    17. Sarfraz Nawaz Malik & Nazar Khan & Ferdous M. O. Tawfiq & Mohammad Faisal Khan & Qazi Zahoor Ahmad & Qin Xin, 2023. "Fuzzy Differential Subordination Associated with a General Linear Transformation," Mathematics, MDPI, vol. 11(22), pages 1-17, November.
    18. Jianhai Bao & Feng‐Yu Wang & Chenggui Yuan, 2020. "Ergodicity for neutral type SDEs with infinite length of memory," Mathematische Nachrichten, Wiley Blackwell, vol. 293(9), pages 1675-1690, September.
    19. Haytham M. Rezk & Juan E. Nápoles Valdés & Maha Ali & Ahmed I. Saied & Mohammed Zakarya, 2023. "Delta Calculus on Time Scale Formulas That Are Similar to Hilbert-Type Inequalities," Mathematics, MDPI, vol. 12(1), pages 1-16, December.
    20. Gadat, Sébastien & Panloup, Fabien & Saadane, Sofiane, 2016. "Stochastic Heavy Ball," TSE Working Papers 16-712, Toulouse School of Economics (TSE).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2023:i:1:p:112-:d:1309602. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.