ИСТИНА

We propose and study the model of \textit{generalized catalytic branching process} (GCBP). It describes a system of particles moving on a finite or countable set $S$ and splitting only in the presence of catalysts. Namely, let at the moment $t=0$ a particle start the movement viewed as an irreducible continuous-time Markov chain with 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 an exponentially distributed time with parameter $1$. Afterwards the particle either splits or leaves $w_k$ with respective probabilities $\alpha_k$ and $1-\alpha_k$ (${0<\alpha_k<1}$). If the particle splits, it dies producing a random non-negative integer number $\xi_k$ of offsprings. It is assumed that all newly born particles evolve as independent copies of their parent. When $S=\mathbb{Z}^d$, $d\in\mathbb{N}$, and the Markov chain is a symmetric, homogeneous and irreducible random walk, such model was introduced by S.~Albeverio and L.~Bogachev in 2000. Another particular case of GCBP was investigated by L.~D\"{o}ring and M.~Roberts in 2012 for $W$ consisting of a single catalyst. For GCBP, as for other branching processes, the natural interesting problem is the analysis of behavior (as $t\to\infty$) of the total and local numbers of particles existing at time $t$. Here our approach is based on hitting times with taboo and a certain auxiliary indecomposable Bellman-Harris process with $N(N+1)$ types of particles. Note that the number of particles of type $k=1,\ldots,N$ in the auxiliary process at time $t$ coincides with the number of particles located at $w_k$ at time $t$ in GCBP. The total particles number in these processes is the same up to neglecting the particles in GCBP which ``go to infinity'' and never hit $W$ again. Thus, to study the asymptotic properties of GCBP it is enough to analyze the limit behavior of the specified Bellman-Harris process. It is well-known that an indecomposable multi type Bellman-Harris process is classified as supercritical, critical or subcritical if the Perron root of the mean matrix is $>1$, $=1$ or $<1$, respectively. We call GCBP supercritical, critical and subcritical according to the classification of the corresponding Bellman-Harris process. In this way we solve the following problem. Assume that we fix the Markov chain generator $A$ in GCBP and vary ``intensities'' of catalysts, e.g., $m_k:={\sf E}{\xi_k}$, $k=1,\ldots,N$. What is the set $C\subset\mathbb{R}^N_+$ such that GSBP is critical iff $m=(m_1,\ldots,m_N)\in C$? In particular, what is the proportion of ``weak'' and ``powerful'' catalysts (possessing small or large values of $m_i$, respectively). We obtain a complete description of $C$ by means of equation involving the determinant of a specified matrix and indicate the smallest parallelepiped $[0,M_1]\times \ldots \times [0,M_N]$ containing $C$. One can control the choice of $m\in C$ as follows. Take $m_1^0\in [0,M_1)$ and consider $m$ in the section $C_{m_1^0}=C\cap\{m:m_1=m_1^0\}$. Then $m_2\in [0,M_2(m_1^0)]$. If we take $m_2^0 \in [0,M_2(m_1^0))$ then for $m$ belonging to the section $C_{m_1^0,m_2^0}$ we can claim that $m_3\in[0,M_3(m_1^0,m_2^0)]$. After the choice of $m_1^0,\ldots,m_{N-1}^0$ the value $m_N=m_N^0$ such that $(m_1^0,\ldots,m_{N-1}^0,m_N^0)\in C$ is determined uniquely. If for some step $k=1,\ldots,N-1$ we choose $m_k^0= M_k(m_1^0,\ldots,m_{k-1}^0)$ (set $M_1(\varnothing)=M_1$) then $m_i^0=0$ for all $i=k+1,\ldots,N$. Moreover, if for some $k=1,\ldots,N-1$ we take $m_k>M_k(m_1^0,\ldots,m_{k-1}^0)$ then GCBP is supercritical for any $m_{k+1},\ldots,m_N\in\mathbb{R}^{N-k}_+$. Note that if $m_N^0>0$ then the choice $(m_1^0,\ldots,m_{N-1}^0,m_N)$ with $m_N>m_N^0$ or $m_N<m_N^0$ leads to supercritical or subcritical GCBP, respectively. We also provide the explicit formulae for $M_i$ and $M_k(m_1^0,\ldots,m_{k-1}^0)$, $i,k=1,\ldots,N$.