A mini course on new trends in the theory of branching random walks

Por M González Velasco.- La Profesora Elena Yarovaya (Moscow State University) impartirá un mini-curso titulado «New trends in the theory of branching random walks» entre las 16:30 y las 18:00 horas del lunes 27, miércoles 29, jueves 30 y viernes 31 de enero de 2020 en el Departamento de Matemáticas, Facultad de Ciencias, Universidad de Extremadura, Badajoz. El minicurso será impartido en inglés.

Abstract: We consider time-continuous branching random walks on multidimensional lattices. In the first lecture, the asymptotic results depending on the lattice dimension for symmetric branching random walks with a single source of branching and finite variance of jumps are presented. In particular asymptotic behavior of survival probabilities and limit theorems for the numbers of particles, both at an arbitrary point of the lattice and on the entire lattice, are obtained. After that we discuss the effects for branching random walks in another case when the corresponding transition rates of the random walk have heavy tails. As a result, the variance of the jumps is infinite, and a random walk may be transient even on one- or two-dimensional lattices. Conditions of transience for a random walk and limit theorems for the numbers of particles are discussed.

In the second lecture, we present the results about branching random walks with violation of symmetry of the random walk at a source. Moreover, we introduce the general model of a branching random walk with a finitely many branching sources. In a supercritical case for such processes the phase transitions are discovered. This situation differs substantially from the branching random walks with a single source. 

In the third lecture, the behavior of transition probabilities of a branching random walk in the situation when the space and time variables grow jointly are established. One of the main results here is the limit theorem about limiting properties of the Green function for the transition probabilities. These results are important for the investigation of the large deviations for branching random walks, in particular, for studying of the particle population front.

In the fourth lecture, we compare two models of homogeneous and non homogeneous branching random walks in random environments. In these models, the branching environment are formed of the birth-and-death processes at lattice sites with random intensities. Conditions under which the long-time behavior of the moments averaged by the medium coincide for the both models are obtained. It is shown that these assumptions hold for random potentials having Weibull-type and Gumbel-type upper tails.

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