Local explosions and extinction in continuous-state branching processes with logistic competition
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866911198758305792 |
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| author | Foucart, Clément |
| author_facet | Foucart, Clément |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_06147 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Local explosions and extinction in continuous-state branching processes with logistic competition Foucart, Clément Probability 60J50, 60J80, 60J55, 60J70, 92D25 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. |
| title | Local explosions and extinction in continuous-state branching processes with logistic competition |
| topic | Probability 60J50, 60J80, 60J55, 60J70, 92D25 |
| url | https://arxiv.org/abs/2111.06147 |