Spatial continuum limit of tree-valued state-dependent spatial branching processes

研究了具有状态依赖分支率的单型和多型分支过程的空间连续极限，并分析了极限过程的特征。特别强调的是在相应的种群谱系，即树值过程进行了研究。典型的例子有分支随机游走、催化分支随机游走、互分支随机游走、自催化随机游走和逻辑分支随机游走以及相关的连续质量极限（相互作用的分支扩散）。目的是展示极限的存在性、它们的长期行为特征以及在小空间和时间尺度上的结构。澄清它们的随机性问题，在确定性极限的情况下，通过波动性或大小偏置以及随后的重新标度来研究“热点”的渐近性。在此基础上，将发展无限可分谱系过程的演算和母系和父系祖先谱系的描述。

The project studies the spatial continuum limit of one- and multitype branching processes with state-dependent branching rates and analyses the characteristics of limit processes. Particular emphasis is placed on genealogies in the corresponding populations, i.e. tree-valued processes are studied. Typical examples are branching random walks, catalytic branching random walks, mutually branching random walks, self-catalytic random walks and logistic branching random walks and the relevant continuous mass limits (interacting branching diffusions). The goal is to show the existence of the limits, the characteristics of their longtime behaviour and the structure on small space and time scales. The question of their stochasticity shall be clarified and in the case of a deterministic limit the asymptotic of ''hotspots'' shall be studied by means of volatility or size biasing and subsequent rescaling. As basis the calculus of infinitely divisible genealogical processes and the description of genealogies of matri- and patrilinear ancestor lines are to be developed.