Birth and death of a stationary Markov process. (English) Zbl 0701.60072

Let X be a Borel right process with semigroup \((P_ t)\), \(\eta\) an excessive measure and M a multiplicative functional of X with killed semigroup \((K_ t)\). In a recent paper, R. K. Getoor [Ann. Prob. 16, No.2, 564-585 (1988; Zbl 0651.60078)] studied the relationship between the stationary measures \(Q_{\eta}\) associated with \((P_ t)\) and \(Q^*_{\eta}\) associated with \((K_ t)\). He proved that \(Q^*_{\eta}\) is obtained from \(Q_{\eta}\) by birthing and killing the paths according to a homogeneous random measure \(\lambda\) on \({\bar {\mathbb{R}}}\times {\bar {\mathbb{R}}}\) depending on M.
The author studies stationary processes that arise when \(\lambda\) is restricted to suitable subsets of \({\bar {\mathbb{R}}}\times {\bar {\mathbb{R}}}\). In particular, pure killing, pur birthing, and birthing after some random time are considered. For M the multiplicative functional associated with hitting a set B, the restriction to (\(\alpha\),\(\beta\) ]\(\times (\alpha,\beta]\) (where \(\alpha\), \(\beta\) denote the birth, resp. death time of the process) corresponds to a stationary excursion from B; appropriate conditioning yields furthermore excursions of X straddling a fixed time t.
Reviewer: J.Steffens


60J25 Continuous-time Markov processes on general state spaces
60G10 Stationary stochastic processes
60J57 Multiplicative functionals and Markov processes


Zbl 0651.60078
Full Text: DOI


[1] Blumenthal, R. M., and Getoor, R. K. (1968).Markov Processes and Potential Theory. Academic press, New York. · Zbl 0169.49204
[2] Fitzsimmons, P. J., and Maisonneuve, B. (1986). Excessive measures and Markov processes with random birth and death.Prob. Theory Rel. Fields 72, 319-336. · Zbl 0584.60085
[3] Getoor, R. K. (1975). Comultiplicative functionals and the birthing of a Markov process.Z. Wahrsch. verw. Gebiete 32, 245-259. · Zbl 0346.60044
[4] Getoor, R. K. (1975). Excursions of a Markov process.Ann. Prob. 7, 244-266. · Zbl 0399.60069
[5] Getoor, R. K. (1988). Killing a Markov process with random times of birth and death.Theor. Prob. 16, 564-585. · Zbl 0651.60078
[6] Kuznetsov, S. E. (1974). Construction of Markov processes with random times of birth and death.Theor. Prob. Appl. 18, 571-574. · Zbl 0296.60049
[7] Maisonneuve, B. (1975). Exit systems.Ann. Prob. 3, 309-411. · Zbl 0311.60047
[8] Meyer, P. A. (1974). Ensembles aléatoires Markoviens homogènes. I. Sém. de Probab. VIII.Lecture Notes in Mathematics 381. Springer-Verlag, Berlin.
[9] Meyer, P. A., Smythe, R. T. and Walsh, J. B. (1972). Birth and death of Markov processes.Proc. 6th. Berk. Symp. Math. Statist. Probab., Vol. III, pp. 295-305. University of California, Berkeley. · Zbl 0255.60046
[10] Mitro, J. B. (1979). Dual Markov processes: Construction of a useful auxiliary process.Z. Wahrsch. verw. Gebiete 47, 139-156. · Zbl 0406.60067
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