## 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

### MSC:

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

Zbl 0651.60078
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### References:

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