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Local explosions and extinction in continuous-state branching processes with logistic competition
Clément Foucart · 2021-11-11 · via math.PR updates on arXiv.org

We study by duality methods the extinction and explosion times of continuous-state branching processes with logistic competition (LCSBPs) and identify the local time at $\infty$ of the process when it is instantaneously reflected at $\infty$. The main idea is to introduce a certain "bidual" process $V$ of the LCSBP $Z$. The latter is the Siegmund dual process of the process $U$, that was introduced in Foucart (2019), as the Laplace dual of $Z$. By using both dualities, we shall relate local explosions and the extinction of $Z$ to local extinctions and the explosion of the process $V$. The process $V$ being a one-dimensional diffusion on $[0,\infty]$, many results on diffusions can be used and transfered to $Z$. A concise study of Siegmund duality for one-dimensional diffusions and their boundaries is also provided.

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