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Teor. Veroyatnost. i Primenen., 1983, Volume 28, Issue 4, Pages 653–667 (Mi tvp2214)  

Representation of a semi-Marcov process as a time changed Markov process

B. P. Harlamov

Leningrad

Abstract: Let $(P_x)$ be a $\lambda$-continuous and $\lambda$-regular semi-Markov process on a complete separable locally compact metric space $X$ and let $A_\lambda=A_0+A_\lambda 1\cdot I$ be the $\lambda$-characteristical operator of the process. If $\lambda R_\lambda\varphi\to\varphi$ ($\lambda\to\infty$) uniformly on $X$ where $\varphi\in C_0$ and $R_\lambda$ is the resolvent operator of the process and if $A_\lambda 1$ is continuous negative function on $X$, $A_{0+}1=0$, $A_\lambda 1\to -\infty$ ($\lambda\to\infty$) then for all $\lambda_0>0$ there exists a Markov process which differs from $(P_x)$ by random change of time only. The operator $\bar A=-(\lambda_0/A_{\lambda_0})$ is an infinitesimal operator of the Markov process and
$$ a_t(\lambda)=\lambda_0\int_0^t\frac{A_{\lambda}1}{A_{\lambda_0}1}\circ\pi_s ds\qquad(\lambda>0) $$
($\pi_s(\xi)=\xi(s)$, $\xi$ is a trajectory of the process) is a Laplace family of additive functionals which determines the random change of time.

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English version:
Theory of Probability and its Applications, 1984, 28:3, 688–702

Bibliographic databases:

Received: 14.02.1981

Citation: B. P. Harlamov, “Representation of a semi-Marcov process as a time changed Markov process”, Teor. Veroyatnost. i Primenen., 28:4 (1983), 653–667; Theory Probab. Appl., 28:3 (1984), 688–702

Citation in format AMSBIB
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\by B.~P.~Harlamov
\paper Representation of a semi-Marcov process as a~time changed Markov process
\jour Teor. Veroyatnost. i Primenen.
\yr 1983
\vol 28
\issue 4
\pages 653--667
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\mathscinet{http://www.ams.org/mathscinet-getitem?mr=726892}
\zmath{https://zbmath.org/?q=an:0561.60094|0539.60088}
\transl
\jour Theory Probab. Appl.
\yr 1984
\vol 28
\issue 3
\pages 688--702
\crossref{https://doi.org/10.1137/1128068}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1984TV66700004}


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