Uniformly ergodic jump Markov processes with bounded intensities. (English. Ukrainian original) Zbl 0943.60064

Kartashov, M. V.

Theory Probab. Math. Stat. 52, 91-103 (1996); translation from Teor. Jmovirn. Mat. Stat. 52, 86-98 (1995).

Let \(X=(X_t\), \(t\geq 0)\) be an ergodic jump Markov process taking values in a measurable space. Let also \(P_t\) be the transition operator, let \(\Pi\) be the stationary projector, and let \(G\) be the generator. \(X\) is called uniformly ergodic with respect to the operator norm \(\|\cdot\|\) if \(\int_0^{\infty}\|P_t -\Pi\|dt <\infty.\) Assume that \(G\) has a bounded variation. It is shown that in this case the uniform ergodicity of \(X\) is equivalent to the boundedness of the generalized potential of the process. The author gives estimates of \(\lim\sup_{t\to\infty} t^{-1} \ln \|P_t-\Pi\|.\)

Reviewer: O.K.Zakusilo (Kyïv)

MSC:

60J25	Continuous-time Markov processes on general state spaces
60K25	Queueing theory (aspects of probability theory)
60J75	Jump processes (MSC2010)

Keywords:

Markov process; ergodicity; transition operator