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


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