Quasi-Stationary Distributions, Eigenmeasures, and Eigenfunctions of Markov Processes

  • Joseph Glover
Part of the Progress in Probability and Statistics book series (PRPR, volume 9)


Let Pt be a submarkov semigroup on a Lusin topological state space E with Borel field E. We call a positive sigma-finite measure m on E an eigenmeasure if mPt = e-ct m for some real number c. We call a positive E-measurable function f an eigenfunction if ptf = e-ctf for some real number c. In each case, we call c the eigenvalue of either the eigenmeasure or the eigenfunction. Eigenmeasures are also known by the name quasi-stationary distributions in the Markov chain literature: see [5], [17], [18].


Potential Density Semi Group Resolvent Equation Discrete State Space Excessive Measure 
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  1. 1.
    R.M. BLUMENTHAL and R.K. GETOOR. Dual process and potential theory. Proc. 12th Biennial Seminar of the Canadian Math Soc. (1970), 137–156.Google Scholar
  2. 2.
    R.M. BLUMENTHAL and R.K. GETOOR. Markov Processes and Potential Theory. Academic Press, New York, 1968.MATHGoogle Scholar
  3. 3.
    K.L. CHUNG and K.M. RAO. Equilibrium and energy. Prob. & Math statistics 1 (1980), 99–108.MathSciNetMATHGoogle Scholar
  4. 4.
    K.L. CHUNG. Probabilistic approach in potential theory to the equilibrium problem. Ann. Inst. Fourier 23 (1973), 313–322.MATHCrossRefGoogle Scholar
  5. 5.
    J.N. DARROCH and E. SENETA. On quasi-stationary distributions in absorbing discrete-time finite Markov chains. J. Appl. Prob. 2 (1965), 88–100.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    R.K. GETOOR and M.J. SHARPE. Last exit times and additive functionals. Ann. Prob. 1 (1973) 550–569.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    R.K. GETOOR. Markov Processes: Ray Processes and Right Processes Lect. Notes in Math. 446. Springer-Verlag, Berlin-Heidelberg- New york, 1975.MATHGoogle Scholar
  8. 8.
    R.K. GETOOR and J. GLOVER. Markov processes with identical excessive measures. Math Zeitschrift Google Scholar
  9. 9.
    R.K. GETOOR and J. GLOVER. Riesz decompositions in Markov process theory. Trans Amer. Math Soc. To appear.Google Scholar
  10. 10.
    J. GLOVER. Topics in probabilistic potential theory. Seminar on stochastic Process 1982, pp. 195–202. Birkhäuser, Boston, 1983.CrossRefGoogle Scholar
  11. 11.
    A.C. LAZER. An extremal characterization of the principal eigenvalue of a non-self-adjoint elliptic operator. (1982) preprint.Google Scholar
  12. 12.
    P.A. MEYER. Note sur l’interprétation des mesures d’équilibre. Seminaire de probabilités VII, pp. 210–216. Lect. Notes in Math 321. Springer-Verlag, Berlin-Heidelberg-New York, 1971.Google Scholar
  13. 13.
    .S. PORT and C. STONE. Brownian Motion and Classical Potential Theory. Academic Press, New York, 1978.MATHGoogle Scholar
  14. 14.
    .M. REED and B. SIMON. Methods of Modern Mathematical Physics Vol 1. Academic Press, New York, 1972.MATHGoogle Scholar
  15. 15.
    D. REVUZ. Mesures associées aux fonctionnelles additives de Markov I. Trans. Amer. Math. Soc. 148 (1970), 501–531.MathSciNetMATHGoogle Scholar
  16. 16.
    F. RIESZ and B. .sz-NAGY. Functional Analysis. F. Ungar Pub. Co. New York, 1955.Google Scholar
  17. 17.
    E. SENETA. Non-negative Matrices and Markav Chains. Springer-Verlag Berlin-Heidelberg-New York, 1981.Google Scholar
  18. 18.
    E. SENETA. Quasi-stationary distributions and time reversion in genetics. J. Royal Stat. Soc. Ser. B. 28 (1966), 253–277.MathSciNetGoogle Scholar
  19. 19.
    D. STROOCK. On the spectrum of Markov semigroups and the existence of invariant measures. Lect. Notes in Math 923. Springer-Verlag, Berlin-Heidelberg-New york, 1982.Google Scholar
  20. 20.
    D. SULLIVAN. λ-potential theory. Preprint.Google Scholar
  21. 21.
    K. YOSIDA. Functional Analysis. Springer-Verlag, Berlin-Heidelberg-New York 1980.MATHGoogle Scholar

Copyright information

© Birkhäuser Boston, Inc. 1986

Authors and Affiliations

  • Joseph Glover
    • 1
  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

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