We consider a class of nonlinear continuous-state branching processes which can be obtained from spectrally positive Lévy processes via Lamperti type time transform. Intuitively, they are the branching processes whose branching rates depend on the current population sizes. The extinction, explosion and coming down from infinity behaviors for such processes have been studied in Li (2016) and Li et al. (2017).
In this talk we further discuss the small time asymptotic behaviors of the processes. By analyzing Laplace transforms of weighted occupation times and fluctuation behaviors for spectrally positive Lévy processes, we solve a one-sided exit problem for the nonlinear branching processes and identify the speeds of coming down from infinity in different scenarios.
This talk is based on joint work with Donald Dawson, Clément Foucart and Pei-Sen Li.
 Li, P. S. (2016): A continuous-state polynomial branching process. preprint. ArXiv: 1609.09593.
 Li, P. S., Yang, X. and Zhou, X. (2017): A general continuous-state nonlinear branching process. ArXiv: 1708.01560.