On different approaches to the study of branching random walks

The talk is devoted to continuous-time stochastic processes, which can be described in terms of birth, death and transport of particles. Such processes on multidimensional lattices are called branching random walks, and the points of the lattice at which the birth and death of particles can occur are called branching sources. Particular attention is paid to the analysis of the asymptotic behavior of the particle number at each point of the lattice for a branching random walk, which are based on a symmetric, spatially homogeneous, irreducible random walk on the lattice. The behavior of particle number moments is largely determined by the structure of the spectrum of the evolutionary operator of average particle numbers and requires the use of the spectral theory of operators in Banach spaces to study a number of models. Two ways of proving limit theorems will be considered, one of which is based on the conditions guaranteeing the uniqueness of the limit probability distribution of particle numbers by its moments, and the other is based on the approximation of the normalized number of particles at a lattice point by some non-negative martingale (see, N. V. Smorodina and E. B. Yarovaya, 2022), which makes it possible to prove the mean square convergence of these quantities to the limit under fairly general assumptions on the characteristics of the process.
The study is partly supported by RFBR, project 20-01-00487.