Behaviors near explosion of nonlinear CSBPs with regularly varying mechanisms

We study the explosion phenomenon of nonlinear continuous-state branching processes (nonlinear CSBPs). First an explicit integral test for explosion is designed when the rate function does not increase too fast. We then exhibit three different regimes of explosion when the branching mechanism and the rate function are regularly varying respectively at $0$ and $\infty$ with indices $\alpha$ and $\beta$ such that $0\leq\alpha\leq \beta$ and $\beta>0$. If $\alpha>0$ then the renormalisation of the process before its explosion is linear. When moreover $\alpha\neq \beta$, the limiting distribution is that of a ratio of two independent random variables whose laws are identified. When $\alpha=\beta$, the limiting random variable shrinks to a constant. Last, when $\alpha=0$, i.e. the branching mechanism is slowly varying at $0$, the process is studied with the help of a nonlinear renormalisation. The limiting distribution becomes the inverse uniform distribution. This complements results known in the case of finite mean and provides new insight for the classical explosive continuous-state branching processes (for which $\beta=1$).