# Population Processes with Immigration

## TL;DR

This paper provides a comprehensive analysis of Galton-Watson processes with immigration, including models in random environments and branching random walks, establishing key limit results for correlation functions.

## Contribution

It offers the first complete analysis of Galton-Watson models with immigration across various environments and extends to branching random walks with new limit theorems.

## Key findings

- Existence of limits for the first two correlation functions in branching random walks with immigration
- Complete characterization of Galton-Watson models with immigration in different environments
- Analysis includes stationary and non-stationary processes

## Abstract

The paper contains the complete analysis of the Galton-Watson models with immigration, including the processes in the random environment, stationary or non-stationary ones. We also study the branching random walk on $Z^d$ with immigration and prove the existence of the limits for the first two correlation functions.