A criterion of the Markov property for continuous semi-Markov processes

B. P. Harlamov

Teoriâ veroâtnostej i ee primeneniâ, Tome 25 (1980) no. 3, pp. 535-548

Résumé

We consider a continuous semi-Markov process on a metric space $X$ and investigate the operator $$ A_\lambda(\varphi|x)=\lim_{r\to 0}\frac{1}{m_r(x)}\biggl(\int_{R_+\times X}e^{-\lambda t}\varphi(x_1)F_{\tau_r}(dt\times dx_1|x)-\varphi(x)\biggr), $$ where $m_r(x)=\int_0^\infty tF_{\tau_r}(dt\times X)$, $F_{\tau_r}(dt\times dx_1|x)$ is the distribution of the time and point of the first exit from the spherical neighbourhood of the initial point $x$, $r$ is the radius of this neighbourhood, $\lambda\geqslant 0$, and $\varphi$ is a measurable bounded function. Under some regularity conditions the semi-Markov process is a Markov process iff $$ A_\lambda(\varphi|x)=A_0(\varphi|x)-\lambda b(x)\varphi(x),\qquad\text{where}\quad0\leqslant b(x)\leqslant 1. $$