Constructing Markov Processes with Random Times of Birth and Death
Kuznetsov  (see also ) introduced a Kolmogorov-type construction in which he constructs a stationary measure Qm from a transition semigroup Pt(x,dy) and an excessive measure m. In fact, his theorem has other interesting consequences outside of the Markovian framework, but we do not discuss these here. While Kuznetsov’s proof is “elementary”, it is rather involved. The purpose of this paper is to give an alternate construction of Qm in the case of right processes. We consider both the time homogeneous and time inhomogeneous cases. Our construction does not extend to cover the other interesting cases of Kuznetsov’s theorem, but our approach may yield some insight into the measures Qm and may aid the reader interested in recent articles [5,10] in which the measure Qm has played an important role. Mitro  has obtained a result similar to ours under duality hypotheses on the underlying processes, but her construction is quite different from ours.
KeywordsMarkov Process Inverse Limit Finite Measure Transition Semigroup Excessive Measure
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- 2.C. Dellacherie et P. A. Meyer. Probabilites et Potentiel. Chap. V-VIII et Chap. IX-XI. Hermann. Paris. 1980 and 1983.Google Scholar
- 3.E. B. Dynkin. Regular Markov processes. Russian Math. Surveys. 28 (1973) 33–64. Reprinted in London Math. Soc. Lecture Note Series 54. Cambridge Univ. Press. 1982.Google Scholar
- 5.P. J. Fitzsimmons and B. Maisonneuve. Excessive measures and Markov processes with random birth and death. To appear in Z. Wahrscheinlichkeitstheorie verw. Geb.Google Scholar
- 6.R. K. Getoor. Markov Processes: Ray Processes and Right Processes. Lecture Notes in Math. 440. Springer. Berlin-Heidelberg-New York. 1975.Google Scholar
- 7.R. K. Getoor. On the construction of kernels. Sem. de Prob. IX. Lecture Notes in Math. 465, 443–463. Springer. Berlin-Heidelberg-New York. 1975.Google Scholar
- 10.R. K. Getoor and J. Steffens. Capacity theory without duality. Submitted to Z. Wahrscheinlichkeitstheorie verw. Geb.Google Scholar
- 14.M. J. Sharpe. General Theory of Markov Processes. Forthcoming book.Google Scholar