Advertisement

Positivity

, Volume 20, Issue 2, pp 355–367 | Cite as

Constrictive Markov operators induced by Markov processes

  • Yukiko IwataEmail author
Article

Abstract

Consider a constrictive Markov operator \(T:L^1(X, \Sigma , \mu ) \rightarrow L^1(X, \Sigma , \mu )\) defined on a finite measure space \((X, \Sigma , \mu )\). We give a necessary and sufficient condition for a constrictive Markov operator T which is an integral operator with stochastic kernel satisfying \(T\mathbf {1}_X=\mathbf {1}_X\).

Keywords

Constrictive operator Markov operator Harris operator 

Mathematics Subject Classification

Primary 45P05 Secondary 46B09 60J05 

References

  1. 1.
    Emel’yanov, E.Y.: Non-Spectral Asymptotic Analysis of One-Parameter Operator Semigroups, Operator Theory: Advances and Applications, vol. 173. Birkhäuser Verlag, Basel (2007)zbMATHGoogle Scholar
  2. 2.
    Emel’yanov, E.Y., Wolff, M.: Quasi-constricted linear operators on Banach spaces. Studia Math. 144(2), 169–179 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Foguel, S.R.: The Ergodic Theory of Markov Processes, Van Nostrand Mathematical Studies, v+102, vol. 21. Van N ostrand Reinhold Co, New York-Toronto (1969)Google Scholar
  4. 4.
    Foguel, S.R.: Harris operators. Israel J. Math. 33(3–4), 281–309 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Halmos, P.R.: On the set of values of finite measure. Bull. Am. Math. Soc. 53, 138–141 (1947)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Huisinga, W.: The Essential Spectral Radius and Asymptotic Properties of Transfer Operators. ZIB-Report (00-26) (2000)Google Scholar
  7. 7.
    Iwata, Y.: Multiplicative stochastic perturbations of one-dimensional maps. Stoch. Dyn. 13, 2 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Iwata, Y., Ogihara, T.: Random perturbations of non-singular transformation on \([0,1]\). Hokkaido Math. J. 42(2), 269–291 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Komorník, J.: Asymptotic Periodicity of Markov and Related Operators, Dynam. Report. Expositions Dynam. Systems, vol. 2, pp. 31–68. Springer, Berlin (1993)Google Scholar
  10. 10.
    Komorník, J., Lasota, A.: Asymptotic decomposition of Markov operators. Bull. Polish Acad. Sci. Math. 35(321327), 5–6 (1987)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Lasota, A., Mackey, M.C.: Chaos, Fractals, and Noise, Stochastic Aspects of Dynamics, Applied Mathematical Science, xiv+472, vol. 97, 2nd edn. Springer-Verlag, New York (1994)zbMATHGoogle Scholar
  12. 12.
    Lin, M.: Strong ratio limit theorems for Markov processes. Ann. Math. Stat. 43, 569579 (1972)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lasota, A., Mackey, M.C.: Noise and statistical periodicity. Phys. D 28(1–2), 143–154 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Meyn, S., Tweedie, R.L.: Markov Chains and Stochastic Stability, 2nd edn, With a Prologue by Peter W. Glynn, xxviii+594. Cambridge University Press, Cambridge (2009)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Meteorological CollegeKashiwaJapan

Personalised recommendations