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On the Identification of Markov Processes by the Distribution of Hitting Times

  • P. J. Fitzsimmons
Part of the Progress in Probability and Statistics book series (PRPR, volume 13)

Abstract

J. Glover [6,7] has recently provided a remarkable generalization of the celebrated Blumenthal, Getoor, McKean theorem [2] concerning the identification of Markov processes up to a time change. To state Glover’s theorem let X = (Xt,Px) and Y = (Yt,Qx) be right Markov processes on a common state space (E,E). Let Δ ε E be a cemetery point used to render the resolvents of X and Y Markovian. Recall that Δ is a trap for X and for Y; the lifetime of X (resp. Y) is then ζ = inf{t: Xt = Δ}(resp. n = inf{t: Yt = Δ}). For B ε E,let T(B) = inf{t>0: XtεB}, S(B) = inf{t>0: YtεB}. Recall that X, for example, is transient provided its potential kernel U is proper.

Keywords

Markov Process Time Change Borel Function Identical Cone Finite Dimensional Distribution 
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References

  1. 1.
    R.M. BLUMENTHAL and R.K. GETOOR. Markov Processes and Potential Theory. Academic Press, New York, 1968.MATHGoogle Scholar
  2. 2.
    R.M. BLUMENTHAL, R.K. GETOOR, H.P. McKEAN, JR. Markov processes with identical hitting distributions. IZZ. J. Math. 6 (1962), 402–420.MathSciNetMATHGoogle Scholar
  3. 3. P.J. FITZSIMMONS. Markov processes with identical hitting probabilities. To appear in Math. Zeit. Google Scholar
  4. 4.
    R.K. GETOOR. Markov Processes: Ray Processes and Right Processes. Lecture Notes in Math. 440. Springer-Verlag, Berlin-Heidelberg-New York, 1975.Google Scholar
  5. 5.
    R.K. GETOOR and J. GLOVER. Markov processes with identical excessive measures. Math. Zeit. 184 (1983), 287–300.MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    J. GLOVER. Markov processes with identical hitting probabilities. Trans. Am. Math. Soc. 275 (1983), 131–141.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    J. GLOVER. Identifying Markov processes up to time change. Seminar on Stochastic Processes 1982, 171–194. Birkhäuser, Boston, 1983.Google Scholar

Copyright information

© Birkhäuser Boston 1987

Authors and Affiliations

  • P. J. Fitzsimmons
    • 1
  1. 1.Department of Mathematical SciencesThe University of AkronAkronUSA

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