## Killing a Markov process under a stationary measure involves creation.(English)Zbl 0651.60078

Given a Borel right Markov process X and an excessive measure m, one can construct a stationary measure $$Q_ m$$ on the space of “two-sided paths” (with random birth and death in (-$$\infty,\infty))$$, governing a process with the same transition mechanism as X. $$Q_ m$$ is called the Kuznetsov measure associated to m and X. If the transitions of X are transformed via a multiplicative functional M (the process is “killed”), the same technique applied to the new (killed) transition semigroup together with m yields a corresponding Kuznetsov measure $$Q^*.$$
The goal of this paper is to construct $$Q^*$$ directly from $$Q_ m$$ using certain functionals arising from M. The title reflects the fact that both the birth and death mechanisms of the Kuznetsov process governed by $$Q_ m$$ are affected by this procedure.
Reviewer: J.Mitro

### MSC:

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