Local explosions and extinction of continuous state branching processes with logistic competition

Abstract : We study by duality methods the extinction and explosion times of continuousstate branching processes with logistic competition (LCSBPs) and identify the local time at ∞ of the process when it is instantaneously reflected at ∞. 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 [Fou19] 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, ∞], many results on diffusions can be used and transfered to Z. A concise study of Siegmund duality for regular one-dimensional diffusions is also provided.