Asymptotic analysis of the moment equations for particle numbers of branching random walks with abandonment of a finite variance of jumps

We consider a continuous-time symmetric branching random walk on a multidimensional lattice. Branching random walks are usually described in terms of birth, death and walk of particles, which makes it easier to use them in statistical physics (Ya. Zeldovich et al.), the theory of homopolymers (R. Carmona et al.) and population dynamic studies (S. Molchanov and J. Whitmeyer). A detailed description of such branching random walks with a finite number of branching sources located in lattice points for the case of finite variance of jumps can be found, for example, in Yarovaya's publications. In the present work we study a branching random walk when intensities of the underlying random walk are subjected to a condition leading to infinite variance of jumps. Quite a number of authors investigated the random walks with infinite variance of jumps, see, for example, a book of A. Borovkov and K. Borovkov and the bibliography in it. Proofs of the global limit theorems for the transition probabilities of a spatially homogeneous symmetric irreducible random walk with infinite variance of jumps in the case, when the temporal and spatial variables jointly tend to infinity, was obtained by A. Agbor, S. Molchanov, and B. Vainberg. The corresponding results were proved under an additional regularity condition imposed on the transition intensities of a random walk. We obtained a multidimensional analog of the well-known Watson's lemma which helps to investigate an asymptotic behaviour of the transition probabilities for fixed spatial coordinates without making any additional assumptions on the transition intensities. Abandonment of the finiteness assumption on the variance of jumps leads to changes in random walk properties: as a result the random walk becomes transient even on one- and two-dimensional lattices. We apply the results to describe the asymptotic behavior of the moments of the numbers of particles in branching random walks with infinite variance of jumps. Employing the scheme, suggested for the case with a finite variance of jumps, we find the generating functions, differential and integral equations for the moments of the numbers of particles, as in an arbitrary lattice point as on the entire lattice, for branching random walks with infinite variance of jumps. We obtain the asymptotic behavior of the moments of the particle numbers based on these equations.
The research was supported by the RFFR, project no. 17-01-00468.