Skip to main content
Log in

Ergodicity of linear SPDE driven by Lévy noise

  • Published:
Journal of Systems Science and Complexity Aims and scope Submit manuscript

Abstract

This paper discusses the ergodicity of a linear stochastic partial differential equation driven by Lévy noise.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. D. Barbu and B. Gheorghe, Approximations to mild solutions of stochastic semilinear equations with non-Lipschitz coefficients, Czechoslovak Math. J., 2002, 52(127): 87–95.

    Article  MATH  MathSciNet  Google Scholar 

  2. G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 1992.

  3. G. Da Prato and J. Zabczyk, Ergodicity for Infinite Dimensional Systems, Cambridge University Press, 1996.

  4. G. Da Prato, A. Debussche, and G. Beniamin, Some properties of invariant measures of non symmetric disspative stochstic systems, Probab. Theory Related Fields, 2002, 123: 355–380.

    Article  MATH  MathSciNet  Google Scholar 

  5. Y. Liu and H. Z. Zhao, Reprensentation of pathwise staionary solutions of stochastic Burgers’ equations, Stochastics and Dynamics, 2009, 9(4): 613–634.

    Article  MATH  Google Scholar 

  6. R. Mikulevicius, H. Pragarauskas, On Cauchy-Dirichlet problem for parabolic quasilinear SPDEs, Potential Analysis, 2006, 25: 37–75.

    Article  MATH  MathSciNet  Google Scholar 

  7. R. Mikulevicius, H. Pragarauskas, and N. Sonnadara, On the Cauchy-Dirichlet problem in the half space for parabolic SPDEs in weighted Hoelder spaces, Acta Appl. Math., 2007, 97: 129–149.

    Article  MATH  MathSciNet  Google Scholar 

  8. S. E. A. Mohammed, T. S. Zhang, and H. Z. Zhao, The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differenial equations, Memoirs of the American Mathematical Society, 2006.

  9. Y. Xie, Existence and uniqueness of solutions of SPDE with non-Lipschitz and non-time-homogeneous coefficients, J. Xuzhou Norm. Univ. Nat. Sci. Ed., 2007, 25(1): 1–5.

    MATH  MathSciNet  Google Scholar 

  10. S. Albeverio, J. L. Wu, and T. S. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl., 1998, 74(1): 21–36.

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Applebauam and J. L. Wu, Stochastic partial differential equations driven by Lévy space-time white noise, Random Oper. Stochastic Equations, 2000, 8(3): 245–259.

    Article  MathSciNet  Google Scholar 

  12. Z. Dong, The uniqueness of invariant measure of the Burgers equation driven by Lévy processes, Journal of Theoretical Probability, 2008, 21(2): 322–335.

    Article  MATH  MathSciNet  Google Scholar 

  13. Z. Dong and Y. Xie, Global solutions of stochastic 2D Navier-Stokes equations with Lévy Noise, Science in China Series A: Mathematics, 2009, 52(7): 1–29.

    Article  MathSciNet  Google Scholar 

  14. Z. Dong and T.G. Xu, One-dimensional stochastic burgers equation driven by Lévy processes, Journal Functional Analysis, 2007, 243: 631–678.

    Article  MATH  MathSciNet  Google Scholar 

  15. E. Hausenblas, SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results, Probab. Theory Related Fields, 2007, 137: 161–200.

    Article  MATH  MathSciNet  Google Scholar 

  16. Erwan Saint Loubert Bié, Étude d’une EDPS conduite par un bruit poissonnien, Probability Theory and Related Fields, 1998, 111(2): 287–321.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Truman and J. L. Wu, On a stochastic nonlinear eqnation arising from 1D integro-differential scalar conservation laws, Journal of Functional Analysis, 2006, 238: 612–635.

    MATH  MathSciNet  Google Scholar 

  18. A. Truman and J. L. Wu, Stochastic Burgers equation with Lévy space-time white noise, Probabilistic Methods in Fluids, proceedings of the Swansea 2002 workshop.

  19. Sandra Cerrai, Second Order PDE’s in Finite and Infinite Demension, springer-Verlag Berlin Heidelberg, 2001.

  20. J. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge University Press, 2001.

  21. F. Flandoli and B. Maslowski, Ergodicity of the 2-D Navier-Stokes equation under random perturbations, Commun. Math. Phys., 1995, 171: 119–141.

    Article  MathSciNet  Google Scholar 

  22. B. Ferrario, Ergodic results for stochastic Navier-Stokes equation, Stochastics and Stochastics Reports, 1997, 60: 271–288.

    MATH  MathSciNet  Google Scholar 

  23. G. Da Prato, Kolmogorov Equations for Stochastic PDEs, Birkhäuser Verlag, 2004.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zhao Dong.

Additional information

The work was supported by the Key Laboratory of Random Complex Structures and Data Scienc, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, 973 Project (2006CB8059000), Science Fund for Creative Research Groups (10721101), the National Science Foundation of China (10671197), and the Science Foundation of Jiangsu Province (BK2006032, 06-A-038, 07-333).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dong, Z., Xie, Y. Ergodicity of linear SPDE driven by Lévy noise. J Syst Sci Complex 23, 137–152 (2010). https://doi.org/10.1007/s11424-010-9269-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11424-010-9269-0

Key words

Navigation