Non-Malthusian supercritical Crump-Mode-Jagers processes

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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.