# Operator Equations of Branching Random Walks

## TL;DR

This paper analyzes the asymptotic behavior of the Green function and eigenvalues in continuous-time supercritical branching random walks on multidimensional lattices, especially near critical source intensities.

## Contribution

It provides new asymptotic results for the Green function and eigenvalues of the evolution operator in branching random walks with multiple sources.

## Key findings

- Asymptotic behavior of the Green function established
- Eigenvalues of the evolution operator characterized near criticality
- Results hold with and without jump variance constraints

## Abstract

Consideration is given to the continuous-time supercritical branching random walk over a multidimensional lattice with a finite number of particle generation sources of the same intensity both with and without constraint on the variance of jumps of random walk underlying the process. Asymptotic behavior of the Green function and eigenvalue of the evolution operator of the mean number of particles under source intensity close to the critical one was established.