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Change of velocity and ergodicity in flows and in Markov semi-groups

  • Michael Lin
  • John Montgomery
  • Robert Sine
Article

Abstract

Let {T(t)}t≧0 be a strongly continuous semi-group of Markov operators on C(X) with generator G. If mC(X) is strictly positive, mG generates a semigroup. If {T(t)} is a group given by a flow, m may have isolated zeros and, under some regularity conditions, mG will still generate a flow, constructed explicitly. The connection between some ergodic properties of the new and original flow is studied. For the Markov semi-groups, the new one is strongly ergodic if and only if the original one is strongly ergodic.

Keywords

Stochastic Process Probability Theory Mathematical Biology Regularity Condition Ergodic Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Alaoglu, L., Birkhoff, G.: General ergodic theorems. Ann. of Math. (2) 41, 293–309 (1940)Google Scholar
  2. 2.
    Chaterji, S.D.: Les Martingales et leurs applications analytiques. In Ecole d'été de Probabilités: Processus stochastique. Berlin-Heidelberg-New York: Springer 1973Google Scholar
  3. 3.
    Chernoff, P., Marsden, J.: On continuity and smoothness of group actions. Bull. Amer. Math. Soc. 76, 1044–1049 (1970)Google Scholar
  4. 4.
    Coddington, E.A., Levinson, N.: Theory of ordinary differential equations. New York: McGraw-Hill 1955Google Scholar
  5. 5.
    Dorroh, J.R.: Contraction semi-groups in a function space. Pacific J. Math. 19, 35–38 (1966)Google Scholar
  6. 6.
    Dunford, N., Schwartz, J.: Linear operators, part I. New York: Interscience 1958Google Scholar
  7. 7.
    Gustafson, K.: A note on left multiplication of semi-group generators. Pacific J. Math. 24, 463–465 (1968)Google Scholar
  8. 8.
    Hartman, P.: Ordinary differential equations. Baltimore: John Hopkins University 1973Google Scholar
  9. 9.
    Humphries, P.D.: Change of velocity in dynamical systems. J. London Math. Soc. (2), 7, 747–757 (1974)Google Scholar
  10. 10.
    Kryloff, N., Bogoliouboff, N.: La théorie générale de là mesure dans son application à l'étude des systèmes dynamiques de la méchanique non-linéaire. Ann. of Math. 38, 65–113 (1937)Google Scholar
  11. 11.
    Lin, M.: On the uniform ergodic theorem II. Proc. Amer. Math. Soc. 46, 217–225 (1974)Google Scholar
  12. 12.
    Lloyd, S.P.: On the mean ergodic theorem of Sine. [To appear]Google Scholar
  13. 13.
    Lumer, G.: Semi-inner product spaces. Trans. Amer. Math. Soc. 100, 29–43 (1961)Google Scholar
  14. 14.
    Lumer, G., Phillips, R.S.: Dissipative operations in a Banach space. Pacific J. Math. 11, 679–780 (1961)Google Scholar
  15. 15.
    Marcus, B.H.: Unique ergodicity of some flows related to axiom A differmorphisms. Notices Amer. Math. Soc. 22, A-233 (1975)Google Scholar
  16. 16.
    Masani, P.: Ergodic theorems for locally integrable semi-groups of continuous linear operators on a Banach space. Advances in Math. (To appear. Abstract presented at International congress of Mathematicians, Vancouver, 1974)Google Scholar
  17. 17.
    Nagel, R.: Mittelergodische Halbgruppen Linearer operatoren. Ann. Inst. Fourier 23, 75–87 (1973)Google Scholar
  18. 18.
    Nemytskij, V., Stepanoff, V.: Qualitative theory of differential equations. Princeton: Princeton University Press 1960Google Scholar
  19. 19.
    Osgood, W.: Beweis der Existenz linear Lösung der Differentialgleichung dy/dx= f(x, y) ohne der Cauchy-Lipshitzschen Belingung. Monatsh. Math. Phys. 9, 331–345 (1898)Google Scholar
  20. 20.
    Oxtoby, J.C.: Stepanoff flows on the torus. Proc. Amer. Math. Soc. 4, 982–987 (1953)Google Scholar
  21. 21.
    Oxtoby, J.C.: Ergodic sets. Bull. Amer. Math. Soc. 58, 116–136 (1952)Google Scholar
  22. 22.
    Sato, K., Ueno, T.: Multi-dimensional diffusion and the Markov process on the boundary. J. Math. Kyoto Univ. 4, 529–605 (1965)Google Scholar
  23. 23.
    Sine, R.: A mean ergodic theorem. Proc. Amer. Math. Soc. 24, 438–439 (1970)Google Scholar
  24. 24.
    Sine, R.: Geometric theory of a single Markov operator. Pacific J. Math. 27, 155–166 (1968)Google Scholar
  25. 25.
    Wallach, S.: The differential equation y'=f(y). Amer. J. Math. 70, 345–350 (1940)Google Scholar

Copyright information

© Springer-Verlag 1977

Authors and Affiliations

  • Michael Lin
    • 1
  • John Montgomery
    • 2
  • Robert Sine
    • 2
  1. 1.The Ohio State UniversityColumbusUSA
  2. 2.University of Rhode IslandKingstonUSA

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