On the coming down from infinity of continuous-state branching processes with drift-interaction
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916994648899584 |
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| author | Rebotier, Félix |
| author_facet | Rebotier, Félix |
| contents | We study the phenomenon of coming down from infinity - that is, when the process starts from infinity and never returns to it - for continuous-state branching processes with generalized drift. We provide sufficient conditions on the drift term and the branching mechanism to ensure both non-explosion and coming down from infinity, without requiring the associated jump measure to have a finite first moment. Assuming the process comes down from infinity and the drift satisfies a one-sided Lipschitz condition, we show that, as the initial values tend to infinity, the process converges locally uniformly almost surely to the strong solution of a stochastic differential equation. The main techniques employed are comparison principles for solutions of stochastic equations and the method of Lyapunov functions, the latter being briefly reviewed in a broader setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_05958 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the coming down from infinity of continuous-state branching processes with drift-interaction Rebotier, Félix Probability We study the phenomenon of coming down from infinity - that is, when the process starts from infinity and never returns to it - for continuous-state branching processes with generalized drift. We provide sufficient conditions on the drift term and the branching mechanism to ensure both non-explosion and coming down from infinity, without requiring the associated jump measure to have a finite first moment. Assuming the process comes down from infinity and the drift satisfies a one-sided Lipschitz condition, we show that, as the initial values tend to infinity, the process converges locally uniformly almost surely to the strong solution of a stochastic differential equation. The main techniques employed are comparison principles for solutions of stochastic equations and the method of Lyapunov functions, the latter being briefly reviewed in a broader setting. |
| title | On the coming down from infinity of continuous-state branching processes with drift-interaction |
| topic | Probability |
| url | https://arxiv.org/abs/2510.05958 |