## Stationary measure of random process of drift with reflecting barriers in semi-Markov environment.(Ukrainian, English)Zbl 1150.60042

Teor. Jmovirn. Mat. Stat. 74, 108-115 (2006); translation in Theory Probab. Math. Stat. 74, 125-132 (2007).
The process of drift in semi-Markov environment is described by equation $${dv(t)\over dt}=C(k(t),v(t))$$, where $$k(t)$$ is semi-Markov process with phase space $$G=X\cup Y$$, $$X=\{x_1,\dots,x_{n}\}$$, $$Y=\{y_1,\dots,y_{m}\}$$. Let $$V_0,V_1, a_{i}, b_{j}\in \mathbb{R}$$, $$V_0<V_1$$, $$a_{i}>0, b_{j}>0$$, $$i=1,\dots,n$$, $$j=1,\dots,m$$, and for all $$x_{i}\in X, i=1,\dots,n$$: $$C(x_{i},v)=\begin{cases} -a_{i},& V_0<v\leq V_1 ,\\ 0,& v=V_0,\end{cases}$$ for all $$y_{j}\in X, j=1,\dots,m$$: $$C(y_{j},v)=\begin{cases} b_{j},& V_0\leq v< V_1,\\ 0,& v=V_1.\end{cases}$$ The author studies the stationary measure of the process $$\xi(t)=(\tau(t), k(t), v(t))$$, where $$\tau(t)=t-\sup\{u\leq t:k(t)\neq k(u)\}$$.

### MSC:

 60K15 Markov renewal processes, semi-Markov processes
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