Genealogical aspects of symbiotic and mutually catalytic branching processes

在优先计划1590的第一阶段，我们的博士生Fridolin Kielisch建立了一个谱系结构，用于相互催化的分支过程（在粒子水平和连续质量极限下）以及共生分支过程（在粒子水平上）。这种结构是在唐纳利和库尔茨以及最近库尔茨和罗德里格斯的向下结构的精神。下一个目标是获得一个类似的结构，也为连续质量共生分支过程。这将完成博士论文。此外，对于这些模型，当分支率趋于无穷大时，将研究下视过程的极限。虽然我们的主要目标是更深入地了解共生分支过程的谱系，但我们也希望在无限速率过程的重整化分析中获得一些新的见解。这将强调共生分支过程的特殊作用。

In the first period of the priority programme 1590, our PhD student Fridolin Kielisch has established a genealogical construction for mutually catalytic branching processes (both on the particle level and in the continuous masses limit) as well as for the symbiotic branching process (on the particle level). This construction is in the spirit of the lookdown constructions of Donnelly and Kurtz and more recently by Kurtz and Rodrigues. The next goal is to obtain a similar construction also for the continuous masses symbiotic branching process. This would finish the PhD thesis.Furthermore, for these models, the limit of the lookdown processes as the branching rate tends to infinity shall be studied. While primarily the goal is to reach a deeper understanding of the genealogies of symbiotic branching processes, we also hope to get some new insight in the renormalization analysis of the infinite rate processes. This would stress the special role of symbiotic branching processes.