Acta Applicandae Mathematicae

, Volume 125, Issue 1, pp 193–208 | Cite as

Hyperbolic Type Stochastic Evolution Equations with Lévy Noise

  • Hongbo Fu
  • Jicheng Liu
  • Li Wan


The existence and uniqueness of the solutions for a class of hyperbolic type stochastic evolution equations driven by some non-Gaussian Lévy processes are obtained. Moreover, an energy equality for the solutions of the equations is established. As examples, theses results are applied to a couple of stochastic wave type equations with jumps.


Hyperbolic type stochastic evolution equations Compensated Poisson random measure Energy equality 

Mathematics Subject Classification (2010)

60H15 35L90 74G25 



We would like to thank Professor Jingqiao Duan for helpful discussions and comments. The research of the authors is supported by China NSF Grant No. 10901065, 11271295.


  1. 1.
    Albeverio, S., Mandrekar, V., Rüdiger, B.: Existence of mild solutions for stochastic differential equations and semilinear equations with non-Gaussian Lévy noise. Stoch. Process. Appl. 119, 835–863 (2009) MATHCrossRefGoogle Scholar
  2. 2.
    Applebaum, D.: Lévy Processes and Stochastic Calculus, 2nd edn. Cambridge University Press, Cambridge (2009) MATHCrossRefGoogle Scholar
  3. 3.
    Bally, V., Gyöngy, I., Pardoux, E.: White noise driven parabolic SPDEs with measurable drift. J. Funct. Anal. 120, 484–510 (1994) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Barbu, V., Da Prato, G.: The stochastic nonlinear damped wave equation. Appl. Math. Optim. 46, 125–141 (2002) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Barbu, V., Da Prato, G., Tubaro, L.: Stochastic wave equations with dissipative damping. Stoch. Process. Appl. 117, 1001–1013 (2007) MATHCrossRefGoogle Scholar
  6. 6.
    Bo, L., Shi, K., Wang, Y.: On a stochastic wave equation driven by a non-Gaussian Lévy noise. J. Theor. Probab. 23, 328–343 (2010) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Brzeźniak, Z., Zhu, J.: Stochastic beam equations driven by compensated Poisson random measures arXiv:1011.5377
  8. 8.
    Brzeźniak, Z., Maslowski, B., Seidler, J.: Stochastic nonlinear beam equations. Probab. Theory Relat. Fields 132, 119–149 (2005) MATHCrossRefGoogle Scholar
  9. 9.
    Chen, Z., Zhang, T.: Stochastic evolution equations driven by stable processes. Osaka J. Math. 48, 311–327 (2011) MathSciNetMATHGoogle Scholar
  10. 10.
    Chow, P.: Stochastic wave equation with polynomial nonlinearity. Ann. Appl. Probab. 12, 361–381 (2002) MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Chow, P.: Asymptotics of solutions to semilinear stochastic wave equations. Ann. Appl. Probab. 16, 757–789 (2006) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Chow, P.: Stochastic Partial Differential Equations. Chapman & Hall/CRC, New York (2007) MATHGoogle Scholar
  13. 13.
    Gyöngy, I.: On stochastic equations with respect to semimartingale III. Stochastics 7, 231–254 (1982) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ikeda, N., Watanable, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland/Kodansha, Amsterdam (1989) MATHGoogle Scholar
  15. 15.
    Kim, J.: Periodic and invariant measures for stochastic wave equations. Electron. J. Differ. Equ. 2004(05), 1–30 (2004) Google Scholar
  16. 16.
    Krylov, N., Rozowskii, B.: Stochastic evolution equations. J. Sov. Math. 14, 1233–1277 (1981) CrossRefGoogle Scholar
  17. 17.
    Lions, J., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972) CrossRefGoogle Scholar
  18. 18.
    Marinelli, C., Prévôt, C., Röckner, M.: Regular dependence on initial data for stochastic evolution equations with multiplicative Poisson noise. J. Funct. Anal. 258, 616–649 (2009) CrossRefGoogle Scholar
  19. 19.
    Pardoux, E.: Sur des équations aux dérivées partielles stochastiques monotones. C. R. Acad. Sci. Paris Sér. A-B 275, A101–A103 (1972) MathSciNetGoogle Scholar
  20. 20.
    Pardoux, E.: Équations aux dérivées partielles stochastiques de type monotone. In: Séminaire sur les Équations aux Dérivées Partielles (1974–1975), III, Exp. No. 2, p. 10. Collège de France, Paris (1975) Google Scholar
  21. 21.
    Pardoux, E.: Stochastic partial differential equations and filtering of diffusion processes. Stochastic. 3, 127–167 (1979) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise. Cambridge University Press, Cambridge (2007) MATHCrossRefGoogle Scholar
  23. 23.
    Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) MATHCrossRefGoogle Scholar
  24. 24.
    Prato, G., Zabczyk, J.: Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge (1996) MATHCrossRefGoogle Scholar
  25. 25.
    Prato, G., Röckner, M., Rozowskii, B., Wang, F.: Strong solutions of stochastic generalized porous media equations: existence, uniqueness and ergodicity. Commun. Partial Differ. Equ. 31, 277–291 (2006) MATHCrossRefGoogle Scholar
  26. 26.
    Prévôt, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Springer, Berlin (2007) MATHGoogle Scholar
  27. 27.
    Ren, J., Röckner, M., Wang, F.: Stochastic generalized porous media and fast diffusion equations. J. Differ. Equ. 238, 118–152 (2007) MATHCrossRefGoogle Scholar
  28. 28.
    Röckner, M., Zhang, T.: Stochastic evolution equations of jump type: existence, uniqueness and large deviation principles. Potential Anal. 26, 255–279 (2007) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Walsh, J.: An Introduction to Stochastic Partial Differential Equations. Springer, Berlin (1986) Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  1. 1.College of Mathematics and Computer ScienceWuhan Textile UniversityWuhanChina
  2. 2.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanChina

Personalised recommendations