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**On limit characteristics of certain Markov processes with deterministic drift.**
*(English.
Russian original)*
Zbl 0625.60024

Theory Probab. Math. Stat. 34, 57-65 (1987); translation from Teor. Veroyatn. Mat. Stat. 34, 53-61 (1986).

Let \(\xi(t)\) be a Markov process with deterministic drift \(f(x,t)\), where \(f(x,t)\) is a dynamical system additively perturbed by a generalized Poisson process \(\alpha(t)\). Consider the processes \(\eta(t)=\xi(\theta_{\nu(t)}-0)\) and \(\eta_ k=\xi(\theta_ k-0)\) where \(\theta_ k\) is the k-th jump time for \(\alpha(t)\) and \(\nu(t)\) is the number of jumps of \(\alpha\) (t) in [0,t], \(t\geq 0\). Let denote by \(\pi\), \(\mu\) and \(\nu\) the limit distributions of the processes \(\xi(t)\), \(\eta(t)\) and \(\eta_ k\), respectively.

A first result shows that for a process \(\xi(t)\) with deterministic drift \(f(x,t)=x.e^{-at}\), \(a>0\), the existence of one of the distributions \(\pi\), \(\mu\) or \(\nu\) implies the existence of the others. A necessary and sufficient condition for their existence is \(E\;\ln(1+| \xi_ 1|)<\infty\), whre \(\xi_ 1\) is the size of the first jump of \(\alpha(t)\). In this case all the three distributions coincide and do not depend on the initial state \(x_ 0.\)

A second result concerns processes \(\xi(t)\in R_+\) with negative drift. In this case the distributions \(\pi\), \(\mu\) and \(\nu\) exist iff \(\xi(0,t)\) is stochastically bounded. All these distributions coincide and do not depend on initial states.

In the last part of the paper, Markov processes with deterministic drift and multidimensional state space are considered by analogy with the one- dimensional case. It is proved that for a process with deterministic drift \(f(\bar x,t)=\exp \{At\}\bar x\) the existence of one of the distributions \(\pi\),\(\mu\) or \(\nu\) independent of the initial state implies the existence of the others. Necessary and sufficient conditions for their existence are given by: a) \(E\;\ln(1+| {\bar \xi}_ 1|)<\infty\), and b) the eigenvalues of A are in the left half-plane.

A first result shows that for a process \(\xi(t)\) with deterministic drift \(f(x,t)=x.e^{-at}\), \(a>0\), the existence of one of the distributions \(\pi\), \(\mu\) or \(\nu\) implies the existence of the others. A necessary and sufficient condition for their existence is \(E\;\ln(1+| \xi_ 1|)<\infty\), whre \(\xi_ 1\) is the size of the first jump of \(\alpha(t)\). In this case all the three distributions coincide and do not depend on the initial state \(x_ 0.\)

A second result concerns processes \(\xi(t)\in R_+\) with negative drift. In this case the distributions \(\pi\), \(\mu\) and \(\nu\) exist iff \(\xi(0,t)\) is stochastically bounded. All these distributions coincide and do not depend on initial states.

In the last part of the paper, Markov processes with deterministic drift and multidimensional state space are considered by analogy with the one- dimensional case. It is proved that for a process with deterministic drift \(f(\bar x,t)=\exp \{At\}\bar x\) the existence of one of the distributions \(\pi\),\(\mu\) or \(\nu\) independent of the initial state implies the existence of the others. Necessary and sufficient conditions for their existence are given by: a) \(E\;\ln(1+| {\bar \xi}_ 1|)<\infty\), and b) the eigenvalues of A are in the left half-plane.

Reviewer: Şt.P.Niculescu

### MSC:

60F05 | Central limit and other weak theorems |

60K25 | Queueing theory (aspects of probability theory) |

60J25 | Continuous-time Markov processes on general state spaces |

37-XX | Dynamical systems and ergodic theory |