Non-Malthusian supercritical Crump-Mode-Jagers processes

这项建议的主题是非马尔萨斯一般(Crump-Model-Jagers)分支过程。具有马尔萨斯参数的超临界一般分枝过程的渐近行为现在已经被很好地理解了；例如，根据著名的尼尔曼大数定律，它们的增长是以马尔萨斯速率为一阶指数的。最近，围绕一级增长的波动也得到了理解。另一方面，对于那些没有马尔萨斯参数的过程，人们知之甚少。该项目的目标是，首先，证明无马尔萨斯参数的非爆炸性一般分枝过程的一个大数定律。第二个目标是找到一般分枝过程在有限时间内爆炸的充要条件。这两个问题都与光滑化变换在尚未被研究的区域中的函数方程有关。方法来源于分支过程、概率论中的函数方程、鞅理论、拉普拉斯变换、(泊松)点过程和缺陷更新方程等领域。

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.