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

Getoor, R. K.

Ann. Probab. 16, No. 2, 564-585 (1988).

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

Keywords:

multiplicative functional; stationary measure; killing; Markov process; Kuznetsov measure