Local explosions and extinction in continuous-state branching processes with logistic competition
Clément Foucart
TL;DR
The article develops a duality-driven framework for logistic continuous-state branching processes (LCSBPs) by introducing a bidual diffusion V, the Siegmund dual of the Laplace dual U of a logistic CSBP Z. By coupling the two dualities, extinction and explosion times for Z are linked to hitting times and local times of V, enabling diffusion-based analysis of boundary behaviors that are difficult to access directly in Z. The work proves precise Laplace-transform identities for extinction and explosion, equates the local time at ∞ for Z with the local time at 0 for V, derives the Hausdorff dimension of explosion sets, and connects excursion measures between Z and V. It also extends Siegmund duality to this diffusion setting, provides a detailed treatment of boundary classifications, and includes a dedicated discussion of the case without competition.
Abstract
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.