Boundary behaviors for a class of continuous-state nonlinear branching processes in critical cases

Shaojuan Ma,Xu Yang,Xiaowen Zhou

doi:10.1214/21-ecp374

Abstract

Using Foster-Lyapunov techniques we establish new conditions on non-extinction, non-explosion, coming down from infinity and staying infinite, respectively, for the general continuous-state nonlinear branching processes introduced in Li et al. (2019). These results can be applied to identify boundary behaviors for the critical cases of the above nonlinear branching processes with power rate functions driven by Brownian motion and (or) stable Poisson random measure, which was left open in Li et al. (2019). In particular, we show that even in the critical cases, a phase transition happens between coming down from infinity and staying infinite.

Highlights

Continuous-state branching processes (CSBPs for short) are nonnegative-valued Markov processes satisfying the additive branching processes
Using Foster-Lyapunov techniques we establish new conditions on non-extinction, non-explosion, coming down from infinity and staying infinite, respectively, for the general continuous-state nonlinear branching processes introduced in Li et al (2019)
These results can be applied to identify boundary behaviors for the critical cases of the above nonlinear branching processes with power rate functions driven by Brownian motion and stable Poisson random measure, which was left open in Li et al (2019)

Summary

Introduction

Continuous-state branching processes (CSBPs for short) are nonnegative-valued Markov processes satisfying the additive branching processes. The extinction, explosion and coming down from infinity behaviors are further discussed in Li et al [11] and some rather sharp criteria in terms of functions a0, a1, a2 and the Poisson random measure are obtained on characterization of different kinds of boundary behaviors for the nonlinear CSBPs as a Markov process. The main goal of this paper is to identify the exact boundary behaviors in the above mentioned critical cases for the solution (Xt)t≥0 to (1.1) For this purpose, we adapt the Foster-Lyapunov approach and select logarithm type test functions to obtain two new conditions under which the nonlinear CSBP never becomes extinct and never explodes, respectively. The proofs of preliminary results on Foster-Lyapunov criteria and the main theorem are deferred to Section 3

Main results

Proofs