Exponential ergodicity of branching processes with immigration and competition

Pei-Sen Li, Zenghu Li, Jian Wang, Xiaowen Zhou

Published 2022-05-31Version 1

We study the ergodic property of a continuous-state branching process with immigration and competition, which is an extension of the models studied by Lambert (2005, Ann. Appl. Probab.), Pardoux (2016, Springer) and Berestycki et al. (2018, Probab. Theory Related Fields) with an additional immigration structure. The exponential ergodicity in a weighted total variation distance is proved under natural assumptions. The main theorem applies to branching mechanisms in the full range of criticality, including all stable types. The proof is based on a Markov coupling process and a nonsymmetric control function for the distance, which are designed to identify and to take the advantage of the dominating factor from the branching, immigration and competition mechanisms in different parts of the state space. The approach provides a way of finding explicitly the ergodicity rate.