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Strong mixing measures for \(C_0\)-semigroups

  • M. Murillo-Arcila
  • A. PerisEmail author
Original Paper

Abstract

Our purpose is to obtain a very effective and general method to prove that certain \(C_0\)-semigroups admit invariant strongly mixing measures. More precisely, we show that the frequent hypercyclicity criterion for \(C_0\)-semigroups ensures the existence of invariant strongly mixing measures with full support. We will provide several examples, that range from birth-and-death models to the Black–Scholes equation, which illustrate these results.

Keywords

Semigroup of operators Strongly mixing measure Frequently hypercyclic 

Notes

Acknowledgments

We would like to thank the referees whose comments led to an improvement of the article’s presentation.

References

  1. 1.
    Albanese, A., Barrachina, X., Mangino, E., Peris, A.: Distributional chaos for strongly continuous semigroups of operators. Commun. Pure Appl. Anal. 12, 2069–2082 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Aroza, J., Peris, A.: Chaotic behaviour of birth-and-death models with proliferation. J. Differ. Eq. Appl. 1–9 (2011)Google Scholar
  3. 3.
    Badea, C., Grivaux, S.: Unimodular eigenvalues uniformly distributed sequences and linear dynamics. Adv. Math. 211, 766–793 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Banasiak, J., Lachowicz, M.: Chaos for a class of linear kinetic models. Comptes Rendus de l’Académie des Sciences-Series IIB-Mechanics 329, 439–444 (2001)CrossRefGoogle Scholar
  5. 5.
    Banasiak, J., Lachowicz, M., Moszyński, M.: Semigroups for generalized birth-and-death equations in \(l^p\) spaces. Semigroup Forum 73, 175–193 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Banasiak, J., Lachowicz, M., Moszyński, M.: Chaotic behaviour of semigroups related to the process of gene amplification–deamplification with cell proliferation. Math. Biosci. 206, 200–215 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Banasiak, J., Moszyński, M.: A generalization of Desch–Schappacher–Webb criteria for chaos. Discret. Contin. Dyn. Syst. 12, 959–972 (2005)CrossRefzbMATHGoogle Scholar
  8. 8.
    Banasiak, J., Moszyński, M.: Dynamics of birth-and-death processes with proliferation-stability and chaos. Discret. Contin. Dyn. Syst. 29, 67–79 (2011)zbMATHGoogle Scholar
  9. 9.
    Barrachina, X., Peris, A.: Distributionally chaotic translation semigroups. J. Differ. Equ. Appl. 18, 751–761 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Bayart, F.: Dynamics of holomorphic groups. Semigroup Forum 82, 229–241 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Bayart, F., Bermúdez, T.: Semigroups of chaotic operators. Bull. Lond. Math. Soc. 41, 823–830 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Bayart, F., Grivaux, S.: Frequently hypercyclic operators. Trans. Am. Math. Soc. 358, 5083–5117 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Bayart, F., Matheron, É.: Dynamics of linear operators. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar
  14. 14.
    Bayart, F., Matheron, É: Mixing operators and small subsets of the circle. Ergod. Theory Dynam. Syst. (to appear)Google Scholar
  15. 15.
    Black, F., Scholes, M.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973)CrossRefzbMATHGoogle Scholar
  16. 16.
    Bermúdez, T., Bonilla, A., Conejero, J.A., Peris, A.: Hypercyclic, topologically mixing and chaotic semigroups on Banach spaces. Studia Math. 170, 57–75 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Bonilla, A., Grosse-Erdmann, K.-G.: Frequently hypercyclic operators and vectors. Ergod. Theory Dynam. Systems 27, 383–404 (2007). Erratum: Ergodic Theory Dynam. Systems 29 (2009), 1993–1994.CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Conejero, J.A., Mangino, E.M.: Hypercyclic semigroups generated by Ornstein–Uhlenbeck operators. Mediterr. J. Math. 7, 101–109 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Conejero, J.A., Müller, V., Peris, A.: Hypercyclic behaviour of operators in a hypercyclic \(C_0\)-semigroup. J. Funct. Anal. 244, 342–348 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Conejero, J.A., Peris, A.: Hypercyclic translation \(C_0\)-semigroups on complex sectors. Discret. Contin. Dyn. Syst. 25, 1195–1208 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Conejero, J.A., Peris, A., Trujillo, M.: Chaotic asymptotic behavior of the hyperbolic heat transfer equation solutions. Int. J. Bifurc. Chaos 20, 2943–2947 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Desch, W., Schappacher, W., Webb, G.F.: Hypercyclic and chaotic semigroups of linear operators. Ergod. Theory Dynam. Systems 17, 793–819 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Diestel, J., Uhl, J.J.: Vector measures, Mathematical Surveys 15. American Mathematical Society, Providence (1977)CrossRefGoogle Scholar
  24. 24.
    Emamirad, H., Goldstein, G., Goldstein, J.A.: Chaotic solution for the Black–Scholes equation. Proc. Am. Math. Soc. 140, 2043–2052 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Flytzanis, E.: Unimodular eigenvalues and linear chaos in Hilbert spaces. Geom. Funct. Anal. 5, 1–13 (1995)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Goldstein, J.A., Mininni, R.M., Romanelli, S.: A new explicit formula for the solution of the Black–Merton–Scholes equation. Infin. Dimens. Stoch. Analy., World Series Publ., 226–235 (2008)Google Scholar
  27. 27.
    Grivaux, S.: A probabilistic version of the frequent hypercyclicity criterion. Studia Math. 176, 279–290 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear chaos. Universitext. Springer-Verlag London Ltd., London (2011)CrossRefGoogle Scholar
  29. 29.
    Halmos, P.R.: Measure Theory. D. Van Nostrand Company Inc, New York (1950)CrossRefzbMATHGoogle Scholar
  30. 30.
    Ji, L., Weber, A.: Dynamics of the heat semigroup on symmetric spaces. Ergod. Theory Dynam. Systems 30, 457–468 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Kalmes, T.: Hypercyclicity and mixing for cosine operator functions generated by second order partial differential operators. J. Math. Anal. Appl. 365, 363–375 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Mangino, E., Peris, A.: Frequently hypercyclic semigroups. Studia Math. 202, 227–242 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Murillo-Arcila, M., Peris, A.: Strong mixing measures for linear operators and frequent hypercyclicity. J. Math. Anal. Appl. 398, 462–465 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    Parthasarathy, K.R.: Probability measures on metric spaces. Academic Press Inc, New York, London (1967)zbMATHGoogle Scholar
  35. 35.
    Rudnicki, R.: Invariant measures for the flow of a first order partial differential equation. Ergod. Theory Dynam. Systems 5, 437–443 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Rudnicki, R.: Strong ergodic properties of a first-order partial differential equation. J. Math. Anal. Appl. 133, 14–26 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Rudnicki, R.: Gaussian measure-preserving linear transformations. Univ. Iagel. Acta Math. 30, 105–112 (1993)MathSciNetGoogle Scholar
  38. 38.
    Rudnicki, R.: Chaoticity and invariant measures for a cell population model. J. Math. Anal. Appl. 339, 151–165 (2012)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Italia 2014

Authors and Affiliations

  1. 1.IUMPA, Universitat Politècnica de ValènciaValenciaSpain
  2. 2.IUMPA, Universitat Politècnica de ValènciaDepartament de Matemàtica AplicadaValenciaSpain

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