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.
The authors thank an anonymous referee for helpful comments. This work was supported by Key research project for North Minzu University (No. 2019KJ28), NSFC (Nos. 11772002, 11771018, 11731012 and 12061004), NSF of Ningxia (No. 2020AAC03230), Major research project for North Minzu University (No. ZDZX201902), the Construction Project of First-Class Disciplines in Ningxia Higher Education (No. NXYLXK2017B09) and NSERC (RGPIN-2016-06704).
"Boundary behaviors for a class of continuous-state nonlinear branching processes in critical cases." Electron. Commun. Probab. 26 1 - 10, 2021. https://doi.org/10.1214/21-ECP374