Extinction and coming down from infinity of CB-processes with competition in a L\'evy environment

TL;DR

This paper investigates the extinction and coming down from infinity of CB-processes with competition in a Lévy environment, establishing conditions under which these processes become extinct or come down from infinity.

Contribution

It provides new results on extinction times and coming down from infinity for CB-processes with competition in Lévy environments, under Grey's condition and additional integrability assumptions.

Findings

Processes become extinct in finite time almost surely.

Under certain conditions, processes come down from infinity regardless of environment.

The results depend on the Lévy environment not drifting towards infinity.

Abstract

In this note, we are interested on the event of extinction and the property of coming down from infinity of continuous state branching (or CB for short) processes with competition in a L\'evy environment whose branching mechanism satisfies the so-called Grey's condition. In particular, we deduce, under the assumption that the L\'evy environment does not drift towards infinity, that for any starting point the process becomes extinct in finite time a.s. Moreover if we impose an integrability condition on the competition mechanism, then the process comes down from infinity regardless the long term behaviour of the environment.