# Survival Analysis of Particle Populations in Branching Random Walks

## TL;DR

This paper investigates the survival probabilities and phase transitions of symmetric heavy-tailed branching random walks on lattices, emphasizing how jump variance influences the process's long-term behavior.

## Contribution

It provides new insights into the limiting behavior and survival analysis of heavy-tailed branching random walks, highlighting the role of jump variance and Green's function methods.

## Key findings

- Survival probability depends on jump variance properties.
- Existence of phase transitions under parameter changes.
- Green's function approach elucidates transition probabilities.

## Abstract

It is a common practice to describe branching random walks in terms of birth, death and walk of particles, which makes it easier to use them in different applications. The main results obtained for the models of symmetric continuous-time heavy-tailed branching random walks on a multidimensional lattice. We will be mainly interested in studying the problems related to the limiting behavior of branching random walks such as the existence of phase transitions under change of various parameters, the properties of the limiting distribution and the survival probability of a particle population. Emphasis is made on the survival analysis. The answers to these and other questions essentially depend on numerous factors which affect the properties of a branching random walk. Therefore, we will try to describe how the properties of a branching random walk depend on such characteristics of the underlying random walk as finiteness or infiniteness of the variance of jumps. The presented results are based on the Green's function representations of the transition probabilities of a branching random walk.