Extinction and coming down from infinity of CB-processes with competition in a Lévy environment
arXiv:1801.04501
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 Lévy environment whose branching mechanism satisfies the so-called Grey's condition. In particular, we deduce, under the assumption that the Lévy 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.
Changes with respect to v2: Due to its length, the paper was cut in half, this version gathers the results concerning CSBP with general competition in a Lévy random environment. Firstly, the results were improved and secondly, results concerning coming down from infinity of the process were added. The second part of the paper is given in a separated preprint