ИСТИНА

We consider an irreducible Markov chain $\xi=\{\xi(t),t\geq0\}$ generated by $Q$-matrix $A=(a(x,y))_{x,y\in S}$ where $S$ is a finite or denumerable set. For $x\in S$, let $\tau_x:=\inf\{t\geq0:\xi(t)\neq x\}$ on the set $\{\xi(0)=x\}$ and $\tau_x=0$ otherwise. The stopping time $\tau_x$ (w.r.t. the natural filtration of the process $\xi$) is \emph{the first exit time from $x$} and $\mathbb{P}(\tau_x\leq t|\xi(0)=x)=1-e^{a(x,x)t},$ $t\geq0$, where $a(x,x)\in(-\infty,0)$. Following Chung's notation in [1], Ch.~2, Sec.~11, for an arbitrary, possibly empty set $H\subseteq S$ called henceforth the \emph{taboo set} and for $t\geq0$, denote by $$_H p_{x,y}(t):=\mathbb{P}(\xi(t)=y,\;\xi(u)\notin H,\;min[\tau_x,t]<u<t|\xi(0)=x),\quad x,y\in S,$$ the \emph{transition probability from $x$ to $y$ in time $t$ under the taboo $H$}. In the case $H=\varnothing$ the function ${p_{x,y}(\cdot)=_\varnothing p_{x,y}(\cdot)}$ is an ordinary transition probability. Note that $_H p_{x,y}(\cdot)\equiv0$ for $x\neq y$ and $y\in H$ whereas $_H p_{x,x}(t)=e^{a(x,x)t}$ for $x\in H$ and $_H p_{x,x}(t)\geq e^{a(x,x)t}$ for $x\notin H$, $t\geq0$. Set $$_H\tau_{x,y}:=\mathbb{I}\{\xi(0)=x\}\inf\{t\geq\tau_x:\xi(t)=y,\;\xi(u)\notin H,\;\tau_x<u<t\},\quad x,y\in S,$$ where, as usual, we assume that $\inf\{t\in\varnothing\}=\infty$. The stopping time $_H\tau_{x,y}$ is \emph{the first entrance time from $x$ to $y$ under the taboo $H$} whenever $x\neq y$ and is \emph{the first return time to $x$ under the taboo $H$} when $x=y$. For $H=\varnothing$ the random variable $\tau_{x,y}=_\varnothing\tau_{x,y}$ is just \emph{first entrance time from $x$ to $y$} (or \emph{first return time to $x$ whenever $x=y$}). The moments $_H\tau_{x,y}$, $x,y\in S$, are also called \emph{hitting times} or \emph{first passage times under the taboo $H$}. Let $_H F_{x,y}(t):=\mathbb{P}(_H\tau_{x,y}\leq t|\xi(0)=x)$, $t\geq0$, be (improper) c.d.f. of $_H\tau_{x,y}$. Introduction of taboo probabilities and hitting times under taboo is a powerful tool for establishing results relating to study of functionals of Markov chains (see, e.g., [1], Ch.~2, Sec.~14), potential theory of Markov chains (see, e.g., [2], Ch.~4, Sec.~6), matrix analytic methods in stochastic modeling (see, e.g., [3], Ch.~3, Sec.~5) and many other domains of Markov chains research. Our interest in hitting times under taboo was motivated by their application to analysis of catalytic branching processes with finitely many catalysts. For a single catalyst, the model was proposed in [4]. We extend the results of [4] to the case of any finite number of catalysts. To this end we introduce an auxiliary multi-type Bellman-Harris branching process with the help of hitting times under taboo. Then we may reduce the study of catalytic branching processes to the investigation of such Bellman-Harris processes. On this way we obtain and afterwards employ an explicit formula for $_H F_{x,y}(\infty)$, $x,y\in S$, $H\subseteq S$, via taboo probabilities $\int\nolimits_{0}^{\infty}{_H p_{x,y}(t)\,dt}\in(0,\infty)$ for any nonempty set $H$. We also express $_H F_{x,y}(\infty)$ in terms of $_{H'} F_{x',y'}(\infty)$ with appropriate choice of a collection of states $x',y'\in S$ and a certain set $H'$ such that $H'\subset H$ or $H\subset H'$. Thus, for a finite nonempty set $H$, the evaluation of $_H F_{x,y}(\infty)$ can be reduced to the case when $H$ consists of a single point. At last, we derive another useful representation of $_z F_{x,y}(\infty)$, $z\in S$, for transient Markov chains and compare it with that established in [5] for a certain class of recurrent Markov chains. \begin{thebibliography}{5} \bibitem{1} K.L. Chung, \textit{Markov Chains with Stationary Transition Probabilities}, Springer, 2nd ed., (1967). \bibitem{2} J.G. Kemeny, J.L. Snell, A.W. Knapp, D. Griffeath, \textit{Denumerable Markov Chains}, Springer, 2nd ed., (1976). \bibitem{3} G. Latouche, V. Ramaswami, \textit{Introduction to Matrix Analytic Methods in Stochastic Modeling}, SIAM, (1999). \bibitem{4} L. Doering, M. Roberts, \textit{Catalytic Branching Processes via Spine Techniques and Renewal Theory}, arXiv:1106.5428 [math.PR], (2013) (to appear in: C.~Donati-Martin, A.~Lejay, A.~Rouault (Eds.), S\'{e}minaire de Probabilit\'{e}s XLV, Lecture Notes in Mathematics). \bibitem{5} E.Vl. Bulinskaya, ``Hitting Times with Taboo for Random Walk'', \textit{Siberian Advances in Mathematics}, 22, No.~4, 227--242 (2012). \end{thebibliography}