We are interested in the property of coming down from infinity of continuous-state branching processes with competition in a Lévy environment. We first study the event of extinction for such a family of processes under Grey’s condition. Moreover, if we add an integrability condition on the competition mechanism then the process comes down from infinity regardless of the long-time behaviour of the environment.

中文翻译：

---

在 Lévy 环境中具有竞争的无限连续状态分支过程的灭绝和下降

我们对以下财产感兴趣从无限中下来在 Lévy 环境中具有竞争的连续状态分支过程。我们首先研究格雷条件下这样一个过程家族的灭绝事件。此外，如果我们在竞争机制上添加一个可积性条件，那么无论环境的长期行为如何，过程都会从无穷大开始下降。