Journal of Theoretical Probability

, Volume 21, Issue 2, pp 322–335 | Cite as

On the Uniqueness of Invariant Measure of the Burgers Equation Driven by Lévy Processes



In this paper, we prove the uniqueness of the invariant measure for one-dimensional Burgers equations perturbed by Lévy processes with Dirichlet boundary conditions.


Burgers equations Poisson process Q-Wiener process Mild solution Invariant measure 

Mathematics Subject Classification (2000)

34D08 34D25 60H20 


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  1. 1.
    Bertini, L., Cancrini, N., Jona-Lasinio, G.: The stochastic Burgers equation. Commun. Math. Phys. 165, 211–232 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Chambers, D.H., Adrian, R.J., Moin, P., Stewartand, D.S., Sung, H.J.: Karhunen-Loève expansion of Burgers model of turbulence. Phys. Fluids 31(9), 2573–2582 (1988) CrossRefGoogle Scholar
  3. 3.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) zbMATHGoogle Scholar
  4. 4.
    Da Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996) zbMATHGoogle Scholar
  5. 5.
    Da Prato, G., Debussche, A., Teman, R.: Stochastic Burgers equation. Nonlinear Differ. Eqn. Appl. 1, 389–402 (1994) zbMATHCrossRefGoogle Scholar
  6. 6.
    Dong, Z., Xu, T.G.: One-Dimensional Stochastic Burgers equation driven by Lévy processes. J. Funct. Anal. 234, 631–678 (2007) CrossRefMathSciNetGoogle Scholar
  7. 7.
    E, W., Khanin, K., Mazel, A., Sinai, Ya.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. 151, 877–900 (2000) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Flandoli, F.: Dissipativity and invariant measures for stochastic Navier-Stokes equation. Nonlinear Differ. Eqn. Appl. 1, 403–423 (1994) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hopf, E.: The partial differential equation u t+uu x=μ u xx. Commun. Pure Appl. Math. 3, 201–230 (1950) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Liu, Y., Zhao, H.Z.: On the stationary solution of stochastic Burgers equation with L 2-noise. (2006, submitted) Google Scholar
  11. 11.
    Mohammed, S.-E.A., Zhang, T., Zhao, H.: The stable manifold theorem for semilinear stochastic evolution equations and stochastic partial differential equation. Mem. Am. Math. Soc. (to appear) Google Scholar
  12. 12.
    Sinai, Ya.C.: Statistical of shocks in solution of inviscid Burgers equation. Commun. Math. Phys. 145, 601–621 (1992) CrossRefMathSciNetGoogle Scholar
  13. 13.
    Sinai, Ya.C.: Two results concerning asymptotic behavior of solutions of the Burgers equation with force. J. Stat. Phys. 64, 1–12 (1991) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Skorokhod, A.V.: Asymptotic Methods in the Theory of Stochastic Differential Equations. American Mathematical Society, Providence (1989) zbMATHGoogle Scholar
  15. 15.
    Truman, A., Wu, J.-L.: Stochastic Burgers equation with Lévy space-time white noise. In: Probabilistic Methods in Fluids, pp. 298–323. World Sci. Publ., River Edge (2003) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.Institute of Applied MathematicsAcademy of Mathematics and Systems Sciences, Academia SinicaBeijingChina

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