ИСТИНА

We introduce and study the model of generalized catalytic branching process (GCBP) which is a system of particles moving in space and branching only in the presence of catalysts. More exactly, let at the initial time there be a particle which moves on some finite or countable set $S$ according to a continuous-time Markov process with infinitesimal generator $A$. When this particle hits a finite set $W=\{w_1,\ldots,w_N\}\subset S$ of catalysts, say at site $w_k$, it spends there time having the exponential distribution with parameter $1$. Afterwards the particle either branches or leaves site $w_k$ with probability $\alpha_k$ and $1-\alpha_k$ ($0<\alpha_k<1$), respectively. If the particle branches (at site $w_k$), it may produce a random nonnegative integer number $\xi_k$ of offsprings. It is assumed that all newly born particles behave as independent copies of their parent. The particular case of GCBP was considered in Doering, Roberts(2013) when $W$ consists of a single catalyst. The main tool for the moment analysis of the process was the spine technique, that is “many-to-few lemma”, and renewal theory. Another case, for $S=\mathbb{Z}^d$, $d\in\mathbb{N}$, and the Markov chain being a symmetric, homogeneous and irreducible random walk with a finite variance of jump sizes, was investigated by Yarovaya (2012). There the necessary and sufficient conditions for exponential growth of the mean particles numbers were established due to application of the spectral theory to evolution operators. To implement the moment analysis of GCBP we employ other methods. To this end we involve the hitting times with taboo (see, e.g., Bulinskaya (2013)) and introduce an auxiliary Bellman-Harris process with $N(N + 1)$ types of particles extending the approach proposed by Topchii, Vatutin (2013). Then we use the criticality conditions for a multi-type Bellman-Harris process and the theorems on the long-time behavior of its moments which can be found in Mode (1971). So, this technique allows us to generalize the results by Doering, Roberts (2013) as well as by Yarovaya (2012).