Advertisement

Mediterranean Journal of Mathematics

, Volume 7, Issue 1, pp 101–109 | Cite as

Hypercyclic Semigroups Generated by Ornstein-Uhlenbeck Operators

  • José A. Conejero
  • Elisabetta M. ManginoEmail author
Article

Abstract

The chaotic and hypercyclic behavior of the C 0-semigroups of operators generated by a perturbation of the Ornstein-Uhlenbeck operator with a multiple of the identity in \({L^2(\mathbb {R}^N)}\) is investigated. Negative and positive results are presented, depending on the signs of the real parts of the eigenvalues of the matrix appearing in the drift of the operator.

Mathematics Subject Classification (2010)

Primary 47A16 Secondary 47D06 47D07 

Keywords

Ornstein-Uhlenbeck operator chaotic C0-semigroups hypercyclic C0-semigroup 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Banasiak J.: Birth-and-death type systems with parameter and chaotic dynamics of some linear kinetic models. Z. Anal. Anwendungen 24, 675–690 (2005)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Banasiak J., Lachowicz M.: Chaos for a class of linear kinetic models. C. R. Acad. Sci. Paris Ser. II b 329, 439–444 (2001)Google Scholar
  3. 3.
    Banasiak J., Lachowicz M., Moszyński M.: Topological chaos: when topology meets medicine. Appl. Math. Lett. 16, 303–308 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Banasiak J., Moszyński M.: A generalization of Desch-Schappacher-Webb criteria for chaos. Discrete Contin. Dyn. Syst. 12, 959–972 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Banasiak J., Moszyński M.: Hypercyclicity and chaoticity spaces for C 0-semigroups. Discrete Contin. Dyn. Syst. 20, 577–587 (2008)zbMATHMathSciNetGoogle Scholar
  6. 6.
    F. Bayart and T. Bermúdez, Semigroups of chaotic operators. Preprint, 2007.Google Scholar
  7. 7.
    Bermúdez T., Bonilla A., Emamirad H.: Chaotic tensor product semigroups. Semigroup Forum 71, 252–264 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bermúdez T., Bonilla A., Torrea J.L.: Chaotic behavior of the Riesz transforms for Hermite expansions. J. Math. Anal. Appl. 337, 702–711 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Bès J., Peris A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Conejero J.A., Müller V., Peris A.: Hypercyclic behaviour of operators in a hypercyclic C0-semigroup. J. Funct. Anal. 244, 342–348 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Costakis G., Peris A.: Hypercyclic semigroups and somewhere dense orbits. C. R. Math. Acad. Sci. Paris 335, 895–898 (2002)zbMATHMathSciNetGoogle Scholar
  12. 12.
    deLaubenfels R., Emamirad H.: Chaos for functions of discrete and continuous weighted shift operators. Ergodic Theory Dynam. Systems 21, 1411–1427 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    deLaubenfels R., Emamirad H., Grosse-Erdmann K.G.: Chaos for semigroups of unbounded operators. Math. Nachr. 261(262), 47–59 (2003)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Desch W., Schappacher W., Webb G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergodic Theory Dynam. Systems 17, 793–819 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Dyson J., Villella-Bressan R., Webb G.: Hypercyclicity of solutions of a transport equation with delays. Nonlinear Anal. 29, 1343–1351 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    El Mourchid S.: The imaginary point spectrum and hypercyclicity. Semigroup Forum 73, 313–316 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    El Mourchid S., Metafune G., Rhandi A., Voigt J.: On the chaotic behaviour of size structured cell populations. J. Math. Anal. Appl. 339, 918–924 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Emamirad H.: Hypercyclicity in the scattering theory for linear transport equation. Trans. Amer. Math. Soc. 350, 3707–3716 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Herzog G.: On a universality of the heat equation. Math. Nachr. 188, 169–171 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Howard K.E.: A size structured model of cell dwarfism. Discrete Contin. Dyn. Syst. Ser. B 1, 471–484 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Howard K.E.: A size and maturity structured model of cell dwarfism exhibiting chaotic behavior. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 13, 3001–3013 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Kalmes T.: On chaotic C 0-semigroups and infinitely regular hypercyclic vectors. Proc. Amer. Math. Soc. 134, 2997–3002 (2006) (electronic)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Kalmes T.: Hypercyclic, mixing, and chaotic C0-semigroups induced by semiflows. Ergodic Theory Dynam. Systems 27, 1599–1631 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    L. Lorenzi and M. Bertoldi, Analytical Methods for Markov Semigroups. Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math. 283, Boca Raton, FL, 2007.Google Scholar
  25. 25.
    Lunardi A.: On the Ornstein-Uhlenbeck operator in L 2 spaces with respect to invariant measures. Trans. Amer. Math. Soc. 349, 155–169 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Metafune G.: L p-spectrum of Ornstein-Uhlenbeck operators. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4)(30), 97–124 (2001)zbMATHMathSciNetGoogle Scholar
  27. 27.
    Myjak J., Rudnicki R.: Stability versus chaos for a partial differential equation. Chaos Solitons Fractals 14, 607–612 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Protopopescu V., Azmy Y.Y.: Topological chaos for a class of linear models. Math. Models Methods Appl. Sci. 2, 79–90 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Takeo F.: Chaos and hypercyclicity for solution semigroups to chaos and hypercyclicity for solution semigroups to some partial differential equations. Nonlinear Anal. 63, 1943–1953 (2005) (electronic)CrossRefGoogle Scholar

Copyright information

© Birkhäuser / Springer Basel AG 2010

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Pura y AplicadaUniversidad Politécnica de ValenciaValenciaSpain
  2. 2.Dipartimento di Matematica “E. De Giorgi”Università del SalentoLecceItaly

Personalised recommendations