Non-Malthusian supercritical Crump-Mode-Jagers processes

这个建议的主题是非马尔萨斯一般（Crump-Mode-Jagers）分支过程。具有马尔萨斯参数的超临界一般分支过程的渐进行为现在已经得到了很好的理解;例如，根据尼尔曼著名的大数定律，它们的增长与马尔萨斯速率呈一阶指数增长。最近，围绕一阶增长的波动也得到了理解。另一方面，对没有马尔萨斯参数的过程知之甚少。该项目的目标是，首先，证明一个大数定律的非爆炸一般分支过程没有马尔萨斯参数。第二个目标是找到一般分支过程在有限时间内爆炸的充分必要条件。这两个问题都涉及到尚未研究过的制度中的平滑变换的函数方程。方法来自分支过程、概率论中的函数方程、鞅论、拉普拉斯变换、（Poisson）点过程和亏损更新方程等领域。

The subject of this proposal is non-Malthusian general (Crump-Mode-Jagers) branching processes. The asymptotic behavior of supercritical general branching processes with a Malthusian parameter is by now well understood; for instance, by Nerman's celebrated law of large numbers, their growth is in first order exponential with Malthusian rate. Recently, the fluctuations around the first-order growth have also been understood. On the other hand, comparatively little is known about those processes without a Malthusian parameter. The goals of the project are, first, to prove a law of large numbers for non-explosive general branching processes without a Malthusian parameter. The second goal is to find necessary and sufficient conditions for the explosion of general branching processes in finite time. Both problems are related to the functional equation of the smoothing transformation in regimes that have not been studied yet. Methods come from the fields of branching processes, functional equations in probability theory, martingale theory, Laplace transformation, (Poisson) point processes and defective renewal equations.