We develop a general probabilistic approach that enables one to get sharp estimates for the almost-sure short-term behavior of hierarchically structured branching-diffusion processes. This approach involves the thorough investigation of the cluster structure and the derivation of some...

We obtain exact almost-sure estimates for the short-time propagation of the closed support of (2, d, [beta])-superprocesses. Upper estimates are derived by solving a certain singular non-linear evolution equation, whereas lower estimates are obtained by the use of the branching-particle-system...