Asymptotic Analysis of Heavy-Tailed Branching Walks

The presented work is a study in the field of the theory of stochastic processes.
The aim of the work is an asymptotic analysis of the numbers of particles,
their integer moments and the survival analysis of a particle population in
models of branching random walks on multidimensional lattices. Until recently,
branching random walks usually were considered under the assumption that a variance of the jumps is finite. In the present study, we impose some fairly
general condition on the intensities of the underlying random walk, which
implies that the variance of jumps becomes infinite. This assumption
significantly expands the range of problems considered in the random walk
theory The relevance of the themes is confirmed by numerous studies in the
area of asymptotic analysis of heavy-tailed random walks. It is sufficient
to cite as an example of the monograph by A.A. Borovkov and K.A. Borovkov (2008)
and bibliography therein. In this direction, it was important to generalize
random walk models by introducing into them the death and birth of particles,
that is to turn to branch random walks, which allowed us to consider the
problems of the spatial distribution of the particle field. One of the important and unsolved problems in this area is the classification of the asymptotic behaviour of heavy-tailed branching random walks. To solve the problem, it was necessary to prove a multidimensional analogue of well-known Watson's Lemma, which was used to obtain the local limit theorem for transition probabilities of symmetric random walks on multidimensional lattices with an infinite variance of jumps. An estimation of the growth rate of the Fourier transform of transition intensities is obtained and used to establish the recurrence properties of a heavy-tailed random walk in terms of the Green function. A nonzero critical point was found for the intensity of the branching source in all dimensions above which an exponential increase of the number of particles is observed. The classification of the asymptotic behaviour of the moments of the number of particles at each lattice point and on the entire lattice is completed. Asymptotics of the extinction probability of the particle population on the lattice is studied.
The asymptotic behaviour of the first moment of the number of particles
populations at each lattice point and of the number of subpopulation generated
by each of the initial particle in a branching random walk with an infinite number of initial particles is obtained.