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

  • Michael Lin
  • John Montgomery
  • Robert Sine


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.


Stochastic Process Probability Theory Mathematical Biology Regularity Condition Ergodic Property 
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  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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