Multi-scale Clustering for a Non-Markovian Spatial Branching Process

Consider a system of particles which move in $R^d$ according to a symmetric $\alpha$-stable motion, have a lifetime distribution of finite mean, and branch with an offspring law of index $1+\beta$. In case of the critical dimension $d=\alpha / \beta$ the phenomenon of multi-scale clustering occurs. This is expressed in an fdd scaling limit theorem, where initially we start with an increasing localized population or with an increasing homogeneous Poissonian population. The limit state is uniform, but its intensity varies in line with the scaling index according to a continuous-state branching process of index $1+\beta$. Our result generalizes the case $\alpha=2$ of Brownian particles of Klenke (1998), where p.d.e. methods had been used which are not available in the present setting