ИСТИНА

The class of symmetric branching random walks on integer lattice with several branching sources in a non-homogeneous environment with a countable number of phase states and continuous time is the object of the study. Special attention was paid to the state aggregation process in the case when the underlying random walk has an infinite variance of jumps. The aggregation resulted in a "new process" with two states: the first corresponded to the set of states where the particle could procreate or die, the second to the set where the particle had no such possibility. Applying Tauberian theorems we found asymptotic behavior of the tail of the distribution of sojourn time of the underlying random walk in each of the "enlarged" states for different dimensions of phase space, which allowed us to conclude the non-Markovian character of the resulting construction. Thus, it was concluded that the original Markovian property of the process disappears as a result of the aggregation. The theorem on exponential growth of the number of particles for the case of enlarged states was also reformulated in the paper. The research was supported by the Russian Foundation for the Basic Research (RFBR), project № 20-01-00487.