AI 中文总结
该研究针对带依赖律迁入的分支SDE及其平均场粒子近似，建立混沌传播界、两阶段尺度极限，给出逻辑斯蒂平均场扩散灭绝/非灭绝判据，明确弱相互作用是时间一致近似的必要条件。
AI 中文摘要
我们研究具有依赖律迁入的分支随机微分方程（SDE）及其平均场粒子近似。在耗散性条件和足够弱的相互作用下，建立了时间阶为$N^{-1/2}$的一致混沌传播界。在每个固定有限时间区间上，对于任意有限相互作用强度，混沌传播均保持相同阶。两阶段尺度极限将连续时间离散状态平均场生死过程连接到相互作用分支扩散，再连接到非线性方程。对于逻辑斯蒂平均场扩散，我们证明了灭绝/非灭绝的精确判据，并进一步表明足够弱的相互作用强度是时间一致近似的必要条件。
英文摘要
We study branching SDEs with law-dependent immigration and their mean-field particle approximations. Under a dissipativity condition and sufficiently weak interaction, a uniform propagation-of-chaos bound in time of order $N^{-1/2}$ is established. On every fixed finite time horizon, the same order of propagation of chaos holds for arbitrary finite interaction strength. A two-stage scaling limit connects continuous-time discrete-state mean-field birth--death processes to interacting branching diffusions and then to the nonlinear equation. For a logistic mean-field diffusion we prove a sharp criterion for extinction/non-extinction, and further show that weak enough interaction strength is necessary for a uniform-in-time approximation.
